Wave Propagation in Linear Media | re-examined

DISSERTATION 

Wave Propagation in Linear Media | re-examined 

ausgefuhrt zum Zwecke der Erlangung des akademischen Grades eines 

Doktors der technischen Wissenschaften unter Leitung von 

O.Univ.Prof. Dr. Fritz Paschke 

Institut fur Allgemeine Elektrotechnik und Elektronik 

und 

O.Univ.Prof. Dr. Gottfried Magerl 

Institut fur Elektrische Me technik 

eingereicht an der Technischen Universitat Wien 

Fakultat fur Elektrotechnik 

von 

Thilo Sauter 

Matr.-Nr. 8526692 

Tanbruckgasse 5/29 

A-1120 Wien 

Wien, im April 1998 ................................

Thilo Sauter 

Technische Universitat Wien 

Institut fur Computertechnik 

Gu haustra e 27/E384 

A-1040 Wien 

Tel: +43-1-58801-3985 

email: sauter@ict.tuwien.ac.at

Kurzfassung 

Seit der Entdeckung des quantenmechanischen Tunnele ektes haben Physiker daruber diskutiert, 

wie lange ein Partikel braucht, um eine Barriere zu uberwinden, und ob diese Geschwindigkeit 

jene des Lichts ubersteigen kann. Neue Nahrung erhielt die Debatte in den letzten 

Jahren durch die Einbeziehung von evaneszenten elektromagnetischen Wellen, die mit tunnelnden 

Teilchen formal verwandt und dank der gro eren Zeitkonstanten Messungen leichter 

zuganglich sind. Die durchgefuhrten Experimente schienen tatsachlich " superluminale\ Wellenausbreitung 

zu bestatigen und losten eine heftige Kontroverse aus. Dabei zeigte sich, da 

der Begri der Wellenausbreitung keineswegs klar und einheitlich ist, was eine eingehendere 

Untersuchung sinnvoll erscheinen la t. 

Die vorliegende Arbeit besteht aus zwei Teilen, deren erster in einem kurzen historischen 

Abri die gangigsten Geschwindigkeitsbegri e prasentiert, namlich die Phasen-, Gruppen-, 

Signal- und Energiegeschwindigkeit. Im Anschlu daran wird in einer Reihe von eindimensionalen 

Fallstudien das Verhalten von Wellen in linearen und dispersiven Medien untersucht. 

Darunter fallen elektromagnetische Ubertragungsleitungen, Hohlleiter und ein verlustfreies 

Plasma sowie die klassischen Stufen- und Rechteckbarrieren der Quantenmechanik. Dazu werden 

jeweils zunachst die entsprechenden Di erentialgleichungen fur monochromatische Wellen 

gelost. Allgemeinere Losungen konnen dann mittels Fourierintegralen konstruiert werden. Im 

elektromagnetischen Fall zeigt sich, da ein Signalsprung sich stets exakt mit Lichtgeschwindigkeit 

fortbewegt. Allerdings ist eine derartige Wellenfront immer breitbandig, wohingegen 

die Spektren von Tunnelmoden auf den Bereich unterhalb der Grenzfrequenz beschrankt sein 

sollten. Diese Forderung resultiert in glatten, gaussformigen Signalen, deren Maximum oft 

als Ma fur Ausbreitungsgeschwindigkeit genommen wird. Obwohl diese De nition jener der 

Gruppengeschwindigkeit entspricht, ist sie im Hinblick auf die Bestimmung von Tunnelzeiten 

problematisch, da die Barriere als Hochpa lter wirkt und die Maxima von einfallender und 

transmittierter Welle in keiner Beziehung zueinander stehen. 

Der zweite Teil der Arbeit beschaftigt sich mit der numerischen Auswertung von Fourierintegralen 

im Zusammenhang mit dispersiver Wellenausbreitung. Die Quadratur solcher Integrale 

wird durch den unbeschrankten Integrationsbereich und den stark oszillierenden Integranden 

sehr muhsam. Obwohl es Losungsansatze fur dieses Problem gab, ist keine fertige Software 

verfugbar, weshalb eine spezialisierte Quadraturroutine auf Basis des modernen Softwarepakets 

Mathematica entwickelt wurde. Dabei wird das Integrationsintervall an den Nullstellen 

des Integranden zerlegt und eine Partialsummenfolge aufgestellt, deren Grenzwert durch 

Extrapolation bestimmt wird. Die Funktion der Quadraturroutine wird anhand bekannter 

Testprobleme veri ziert, und abschlie end wird beispielhaft ihre Anwendung auf das Problem 

der Partikelstreuung an einem Stufenpotential gezeigt. 

III

Abstract 

Ever since the quantum mechanical tunnel e ect was discovered, physicists have been discussing 

the question of how fast a tunnelling particle actually traverses a barrier and whether 

this velocity can exceed that of light. In the recent past, the debate was extended to evanescent 

electromagnetic waves which are formally equivalent to tunnelling particles and are more 

amenable to measurements. Experiments were carried out and seemed to prove `superluminal' 

wave propagation. The following discussion brought to light that the velocity ofwave propagation 

is by no means a clearly de ned term | enough to merit some further investigation. 

This work is divided into two parts. In a brief historical survey, the rst part introduces 

the most common velocity concepts associated with wave propagation. These are the phase, 

group, signal and energy velocities. Subsequently, anumber of one-dimensional case studies 

are presented that investigate the behaviour of both electromagnetic and quantum waves and 

in linear, dispersive media. They comprise electromagnetic transmission lines, wave guides, 

and the model of a lossless plasma. The quantum mechanical e ects are examined by means 

of the classical step and square potential barriers. The approach adopted is to solve the 

respective di erential equations for monochromatic waves and to construct Fourier integrals 

characterising waves with a broader spectrum. In the electromagnetic case, it is seen that a 

sudden signal change propagates exactly at the velocity of light. While such awavefront is 

always a broad-band signal, the spectra of tunnelling waves should be con ned to the region 

below the cuto frequency. This gives smooth, Gaussian-like functions, the peak of which 

is often used to measure the propagation velocity. Although this de nition is similar to the 

group velocity, which is applicable for small-band signals, it is problematic when a tunelling 

time is to be determined. The case studies show that the barrier acts as a high-pass lter 

and that the peaks of the incident and transmitted waves are not related to each other. 

The second part of this work is devoted to the numerical evaluation of the Fourier integrals associated 

with dispersive wave propagation. What makes quadrature particularly cumbersome 

in this case is the fact that the integration interval is in nite and the integrands oscillate irregularly. 

There have been approaches to this problem, but no ready-to-use computer routine 

is available. Consequently, a custom quadrature routine is devised using the modern software 

tool Mathematica. It is based on the partition of the integrand at its zeros. Quadrature is 

carried out over the subintervals and the resulting sequence of partial sums is extrapolated 

to nd its limit. The routine is tested with benchmark problems and found to give correct 

answers. Finally, as an exemplary application, the scattering of quantum waves at a step 

potential barrier is presented and tested. 

V

Nullum est iam dictum, quod non sit dictum prius. 

Publius Terentius Afer, Eunuchus 

VII

Alles Gescheite ist schon gedacht worden, man mu nur versuchen, es noch 

einmal zu denken. 

Johann Wolfgang von Goethe, Maximen und Re exionen 

VIII

Preface 

Our popular writers and reporters, when they have to deal with physics, 

indulge in similies of all sorts; the trouble is that they leave the reader 

helpless in nding out how far these pungent analogies cover the real issue, 

and therefore more often lead him astray than enlighten him. 

Hermann Weyl, The Mathematical Way of Thinking, reprinted in [1] 

Rarely does theoretical physics attract the attention of a broad public. Even if news of this 

branch ofscience works its way into the papers, it turns into margin notes rather than eyecatching 

headlines. Of course, there are the annual review articles and features celebrating the 

Nobel laureates, and occasionally there are obituaries of prominent physicists. These reports, 

however, always hinge on the personalities themselves, and physics plays only a subordinate 

role there | as opposed to the striking importance physics, as well as engineering, undeniably 

have in modern life. One could blame the underrating of physics on a general scepticism about 

science and technology. While this is certainly true, it can hardly be the only explanation. 

If it was, whence would all the popular articles on biology, cosmology or geophysics come? 

The revival of the dinosaurs, the pros and cons of genetic manipulation, the intricacies of the 

weather, the possibilities of extraterrestrial life on mars or wherever, and naturally all shades 

of medicine | the list of interesting topics wandering through the media is long and may be 

extended arbitrarily. Theoretical physics, though, hardly ever turns up. 

Another point: stories on physics seem to be relevant only if the reported achievements have 

an impact on technology or | better still | on daily life. A better understanding of quantum 

physics assists in the development of smaller microelectronic devices that permit the invention 

of circuits with ever-increasing complexity, but decreasing power consumption. On the other 

end of the scale, nuclear fusion is still believed to be a possible solution for the growing demand 

for electric power. Now, one might argue that these topics do exert an in uence on our lives, 

and therefore people have the right to be informed about them. This eventually comes down 

to dividing science in two parts; the knowledge-oriented and the application-oriented. The 

former is often regarded as science per se, whereas the latter is commonly labelled as (often 

with a slightly derogatory undertone) engineering. I am not to discuss the gap that separates 

the two sides, which, by the way, was chie y dug by the scientists themselves. I just want 

to state that what is addressed to the broad public is mostly the application of physics and 

rarely mere knowledge-oriented work. This is by no means so in other elds, or is there any 

IX

practical use in the popular reports on how certain insects handle their reproduction under 

evolutional pressure? But, on the other hand, such articles are easy to understand for a 

mildly interested reader. Bare physics is so perplexing that few journalists dare to bother 

their readers and listeners with it. Maybe this is the trivial reason: physics just doesn't sell. 

In view of these arguments, it is even more astonishing that within the last two or three 

years, the predominantly theoretical debate whether or not it is possible to transmit signals 

faster than light has received quite some publicity. Known originally only to a handful of 

researchers, the topic somehow was noticed by journalists and began to spread in newspapers 

and magazines all over the world. 1 Yet the discussion is nothing particularly new | it only 

grew in vigour during the last few years, when experiments were conducted to support earlier 

theoretical work. In fact its roots date back to the beginning of the century, being closely 

linked to the discovery and understanding of wave propagation, which is a fascinating subject 

indeed. 

Water waves have been known and studied for a long time, but it is the electromagnetic waves 

that literally penetrate all aspects of modern life. They had been predicted by James Clerk 

Maxwell in his beautiful, lucid electrodynamic theory. In the 1880's, Heinrich Hertz set out to 

prove their existence, and in 1886 he achieved his goal. 2 Hertz was the rst to generate such 

waves and transmit them over a distance of several metres. He did, however, never consider 

any practical application of his discovery. The recognition of its enormous value was left to 

others, like Popow or Ernest Rutherford. 3 But the merit for the real break-through goes to 

Giuglielmo Marconi, the rst modern radio engineer. In 1901, he succeeded in transmitting 

radio signals across the Atlantic Ocean from Cornwall to Newfoundland and, in a way, laid the 

foundations of modern telecommunications. It was an ambitious, expensive endeavour with 

a fortuitous outcome. Hadn't it been for the dispersive properties of the ionosphere, which 

were still unknown at that time, the experiment would have failed, because electromagnetic 

waves in free space follow linear paths, and America cannot be seen from Europe. Still, this 

unexpected aid does not belittle Marconi's achievement and its importance for science as well 

as engineering. 4 

Now what was the reason that the faster-than-light-debate worked its way into the massmedia? 

It can hardly be the general signi cance of waves in modern physics, for this aspect 

was scarcely addressed in the reports. There were other details that made the story worth 

publishing. Today, it is common knowledge that even a beam of light takes some time to 

travel a certain distance. Likewise, a lightyear is well understood as a synonym for an almost 

incredible length. Only partly do we owe this awareness to physics lessons taught in school 

| science ction series in TV and cinema have been much more e ective in reaching (and 

teaching) a broad public. Actually, the famous `Star Trek' series was often quoted to illustrate 

1 New Scientist, 1. April 1995; Frankfurter Allgemeine Zeitung, 7. July 1995; Die ZEIT, 21. July 1995; Bild 

der Wissenschaft, August 1997; pro l, 15. September 1997 

2 Armin Hermann, Weltreich der Physik, Bechtle Verlag, 1980. 

3 Peter Volkmann, Technikpioniere, vde-verlag, 1990. 

4 The receiver Marconi used (in principle an early semiconductor diode) soon gave rise to a scandal. As 

the origin of the device remained unclear for a long time, several contemporaries accused Marconi and his 

collaborators of intellectual theft. For an account of the a air, see a series of articles in the January issue of 

the Proceedings of the IEEE, 1998. 

X

the quest for superluminality and to speculate what life beyond the speed of light could be 

like. By contrast, it was Einstein who postulated that the velocity of light is the utmost speed 

for the transmission of energy or information, and the bare mention of perhaps the best-known 

physicist ever is su cient toevoke interest. Even more thrilling was the fact that his legacy 

seemed to be at stake. His relativistic theory, perhaps the best-known physical theory ever, 

seemed to be endangered and eventually falsi ed by experimental evidence. What followed 

looked pretty much like a strife between disciples and opponents of Einstein. In fact, the 

entire discussion was and still is rather circling around the question how a signal can be 

de ned. This was, however, much too involved for many commentators, compared with the 

exciting imagination of signal transport faster than light. So they often forgot to point out an 

important limitation: superluminality, as it is currently discussed, is con ned to extremely 

short distances that span but a few centimetres, depending on the frequency range of the 

waves under consideration. This creates an odd contrast to the notion of comprehensive 

unboundedness we are so inclined to associate with the propagation of light. 

The preceding remarks account for the somewhat `scienti cal' ingredients of the story. But 

there is also a rather cultural aspect. At present, the quarrel about superluminal wave 

propagation is maintained by two parties that happen to work in Germany and the United 

States, respectively. 5 Consequently, some reports were interspersed with slightly patriotic 

(but nonetheless super uous and annoying) undertones. One could, at times, not escape the 

feeling that the writer painted the picture of a sports competition rather than an academic 

discussion. So to answer our initial question, I believe itwas the mixture of popular names, 

scienti c rivalry, and the dream of bringing science ction one step closer to reality that made 

the subject so fascinating for journalists and readers alike. Who cares that accuracy was left 

behind in the attempt of making things comprehensible? After all, physics got a mention 

in the media, and I was greatly surprised how many people outside the so-called `scienti c 

community' took notice of it. 

It would be dishonest to deny that this thesis was stimulated by the discussion raging about 

the interpretation of the recent experiments. But by the time the work began, the controversy 

still belonged to the main contenders, and it could not be foreseen that the subject would 

sometime merit public attention. The present text has several objectives. First, it is meant as 

asurvey of the eventful history of wave propagation, its discovery and comprehension, with 

a particular emphasis on the superluminality issue. Furthermore, it provides case studies on 

wave propagation phenomena. They are intended as supplements to earlier theoretical work 

along these lines | in the spirit of what Sir Karl Popper declared to be the core of a scienti cal 

theory: that it must, in principle, be subjectable to falsi cation. 6 Now the idea of carrying 

out case studies is neither ingenious nor spectacular, it is as old as science itself. What 

has changed in the recent past are the ways and means that are at hand to reach the goal. 

While in former times researchers had to use approximation methods to evaluate complex 

expressions by hand, the ever-growing power of computers and their software today enables 

us to numerically solve problems in greater detail and accuracy. The formulation of the 

5 Prof. Nimtz at the University of Cologne (http://www.uni-koeln.de/math-nat-fak/ph2/) and Prof. 

Chiao at the University of Berkeley, CA(http://physics1.berkeley.edu/research/chiao/) 

6 Karl R. Popper, The Logic of Scienti c Discovery, 1934. 

XI

investigated examples was straightforward in terms of classical mathematics. The evaluation 

of the results, conversely, called for numerical computation. This is why a substantial part of 

this work is concerned with the application of a modern software tool to tackle the questions 

arising from wave propagation, speci cally the computation of Fourier integrals. 

Interdisciplinary as this work is, many people have contributed to its successful termination, 

whose support I would like to gratefully acknowledge. I am indebted foremost and above 

all to my adviser, Prof. Fritz Paschke, for the stimulus and superb guidance he provided 

during the years of close, highly enjoyable co-operation. In fact, he initiated the work in 

the most unspectacular way when he came up to me one ne day and asked me for the 

numerical evaluation of a wave integral. Appreciating the e orts needed to cope with this 

task, he proposed to pursue it further and make a doctorate thesis out of it. Although I was 

working on a completely di erent subject at that time and wave propagation always remained 

a part-time occupation, I never regretted to have accepted his o er. 

There are still others I owe thanks to. My second adviser, Prof. Gottfried Magerl, read 

the text very carefully and made many suggestions that helped to make the presentation 

clearer. The contribution of Prof. Helmut Rauch was an indirect yet essential one, for it was 

him who drew Prof. Paschke's attention to the newly arisen discussion on superluminality. 

Also, he provided us with a very useful collection of literature. I appreciated the help of 

Doz. Christoph Uberhuber very much, who patiently and willingly answered my insistent 

questions on numerics in general and quadrature in particular. Special thanks go to my 

supervisor, Prof. Dietmar Dietrich, who granted me su cient freedom to nish my work even 

though the subject was beyond his own eld of interest. I had useful discussions with many of 

my colleagues, but I wish to give those with Niki Kero a special mention because they left a 

distinct mark on the appearance and style of the text. I am glad that Steven Gallop underwent 

the tedious task of proof-reading the manuscript, pitilessly cutting back sentences that had 

grown too long and demanding rewriting of passages when they were unclear. Finally, I wish 

to thank my wife, who bore with me during the last, stressful phase of writing down this 

work, for her patience and encouraging support. 

XII

Contents 

Part I Wave propagation phenomena 1 

1 The many velocities of wave propagation 2 

1.1 Phase and group velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 

1.2 A few notes on dispersion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 

1.3 Signal velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 

1.4 Energy velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 

1.5 Other velocity de nitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 

2 Signals faster than light? 20 

2.1 Superluminal wave propagation . . . . . . . . . . . . . . . . . . . . . . . . . . 21 

2.2 Quantum mechanical tunnelling . . . . . . . . . . . . . . . . . . . . . . . . . . 26 

3 Wave propagation in electromagnetic transmission lines 31 

3.1 Model of a transmission line . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 

3.2 Excursion: a delay line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 

3.3 Re ection due to termination mismatch . . . . . . . . . . . . . . . . . . . . . 37 

3.4 A simple thought experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 

3.5 A dispersive system: the lossless plasma . . . . . . . . . . . . . . . . . . . . . 41 

3.6 Inhomogeneous transmission line . . . . . . . . . . . . . . . . . . . . . . . . . 46 

3.7 Turn-on e ects in a lossless plasma . . . . . . . . . . . . . . . . . . . . . . . . 48 

3.8 Turn-on e ects in a wave guide . . . . . . . . . . . . . . . . . . . . . . . . . . 60 

3.9 A Gaussian pulse in plasma . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 

XIII

4 One-dimensional quantum tunnelling 72 

4.1 The potential step . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 

4.2 Initial wave forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 

4.3 Examples of scattering processes . . . . . . . . . . . . . . . . . . . . . . . . . 81 

4.4 The square barrier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 

4.5 Tunnelling time de nitions for a square barrier . . . . . . . . . . . . . . . . . 96 

4.6 Examples of tunnelling events . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 

Interlude Wave functions in graphical representation 119 

Part II Numerical aspects of wave equations 123 

5 Numerical quadrature and extrapolation 124 

5.1 Univariate numerical quadrature . . . . . . . . . . . . . . . . . . . . . . . . . 125 

5.1.1 Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 

5.1.2 Computer routines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 

5.2 Convergence acceleration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 

6 Towards a quadrature routine 132 

6.1 Partitioning the integration interval . . . . . . . . . . . . . . . . . . . . . . . 133 

6.2 Choosing the rst partition point . . . . . . . . . . . . . . . . . . . . . . . . . 138 

6.3 How to compute the rst integral . . . . . . . . . . . . . . . . . . . . . . . . . 141 

6.4 Asymptotic partition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 

6.5 Considerations for a Mathematica implementation . . . . . . . . . . . . . . . 149 

6.6 Controlling the accuracy of the extrapolation result . . . . . . . . . . . . . . . 155 

7 Mathematica implementation of a quadrature function 161 

7.1 User interface of the function OscInt . . . . . . . . . . . . . . . . . . . . . . . 161 

7.2 Structure of the package . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164 

7.3 Implementation of OscInt and related functions . . . . . . . . . . . . . . . . 165 

7.3.1 OscInt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166 

7.3.2 PartitionTable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 

7.3.3 PartitionOffs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168 

XIV

7.3.4 PartitionPoints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170 

7.3.5 PartInt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 

7.3.6 OscIntControlled . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172 

7.4 Auxiliary functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 

7.4.1 Zero computation for quadratic arguments . . . . . . . . . . . . . . . . 174 

7.4.2 Zero computation for hyperbolic arguments . . . . . . . . . . . . . . . 174 

7.4.3 Approximation error control . . . . . . . . . . . . . . . . . . . . . . . . 178 

7.5 Test of the quadrature routine . . . . . . . . . . . . . . . . . . . . . . . . . . 181 

8 Application of the quadrature routine 190 

8.1 Preparation of the wave integrals for quadrature . . . . . . . . . . . . . . . . 191 

8.2 Outline of the program structure . . . . . . . . . . . . . . . . . . . . . . . . . 197 

8.3 Implementation of the quadrature modules . . . . . . . . . . . . . . . . . . . 200 

8.4 Test of the package . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205 

8.4.1 Numerical tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205 

8.4.2 Formal veri cation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211 

A Mathematica packages 214 

A.1 Numerical quadrature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214 

A.2 Solutions for the step potential . . . . . . . . . . . . . . . . . . . . . . . . . . 221 

A.3 Solutions for the square barrier . . . . . . . . . . . . . . . . . . . . . . . . . . 229 

A.4 Electromagnetic waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232 

A.5 Utilities for displaying points in 3D . . . . . . . . . . . . . . . . . . . . . . . . 234 

Bibliography 237 

Index 247 

XV

XVI

Part I 

Wave propagation phenomena 

Soon after the formulation of Einstein's theory of relativity, a discussion arose whether or not 

certain types of waves can propagate faster than light. These are evanescent electromagnetic 

waves and tunnelling quantum particles that are formally equivalent to some electromagnetic 

cases. In recent years, the fairly old debate grew in intensity when experimental results 

seemed to support `superluminal' theories. The rst part of this work is therefore dedicated 

to the many facets of the discussion. 

To provide an insightinto the subject, the rst chapters give a brief overview on the de nitions 

of wave propagation velocities in general and on the faster-than-light issue in particular. The 

variety of opinions and theories is rather unsatisfactory from a practical point of view, and so 

the subsequent chapters present anumber of one-dimensional case studies to explore the problems 

further. The rst among these are concerned with the propagation of electromagnetic 

waves with a special focus on evanescent modes. We shall begin with simple monochromatic 

examples where closed expressions for a propagation velocity can still be found. We then 

extend the monochromatic investigation to dispersive media and nally solve the wave equation 

for broad-band signals in a lossless plasma as well as in a rectangular wave guide. These 

last examples will demonstrate that sudden signal changes travel at a wave front velocity no 

faster than the speed of light. 

The second portion of these case studies is dedicated to the classical quantum mechanical 

tunnel e ect. Like in the sections before, we start with formulating the problem for a plane 

wave, but immediately extend the results to the general case of signals with arbitrary bandwidth. 

We shall then explore the scattering of a particle o a potential step. Subsequently, 

after having looked into quantum mechanical tunnelling time de nitions for plane waves, we 

deal with a particle tunnelling through a rectangular barrier. In all these cases, we must 

employ numerical techniques to solve the Fourier integrals that constitute the solutions of the 

wave equations. The examples will show that for very small-band pulses, the results could 

be interpreted as superluminal propagation of a wave packet. However, as the bandwidth is 

increased, the tunnelling time becomes even negative, which is quite an absurd result. So we 

eventually arrive at the still open question whether a signal pulse is adequately described by 

its peak or centre of mass, or what actually forms the information being transmitted. 

1

Chapter 1 

1 The many velocities of wave propagation 

The many velocities of wave 

propagation 

When a mathematician thinks of wave propagation, he starts by writing 

a well-known second order di erential equation and discussing its peculiar 

properties. The physicist is interested in these results, but he immediately 

asks some indiscreet questions about waves in a dispersive medium, when 

the velocity of propagation is not a constant, but strongly depends upon 

the frequency. Leon Brillouin [2] 

Ever since its discovery, the phenomenon of wave propagation has raised the question of how 

fast a wave actually proceeds. The answer, though fairly simple-looking at rst sight, is far 

from being evident and has kept physicists busy for a long time. This continued occupation 

with the subject is closely related to the question of the de nition of signals. It is commonly 

understood that a signal transfers information from one part of a medium to another, but this 

description is vague and contains no clueas to how the propagation velocity can be de ned. 

Consequently, there were di erent interpretations, and this lack of a precise de nition is 

mirrored in an unexpected variety ofvelocity concepts. They are of course consistent in that 

they all give the same result for the special case of a pulse travelling with unaltered shape 

through space and time. In the more realistic cases, however, when the pulse is deformed and 

spreads out in the course of time, their results may di er grossly. 

Owing to the multitude of approaches, this chapter is concerned with an overview of several 

concepts of propagation velocity. In particular, we shall concentrate on four concepts that 

have become most widely known and are interesting also from a historical point of view: the 

phase and group velocity, the signal velocity, and the energy velocity. There have been other 

suggestions, too, which did not receive somuch attention and are discussed therefore only in 

a brief survey. 

2

1.1 Phase and group velocity 


It may seem strange, but there is no undisputed, clear-cut de nition of what constitutes a 

wave. In many textbooks, speci cally those dedicated to electromagnetism, they are introduced 

as solutions of a partial di erential equation called the wave equation, 

, 1 

v 2 

@ 2 

=0; (1.1) 

@t2 whose solution is easily found to be (r vt). It appears, though, that this de nition is far 

too restrictive, for many other equations can be thought of that yield similar results. We 

shall therefore adopt the pragmatic approach of Whitham [3] and characterise the wave by 

the solution directly, irrespective of whatever equation might have led to it. So we follow 

the intuitive and familiar notion that a wave is any recognisable function that at di erent 

instances of time can be localised at di erent positions in space, such that the spectator has 

the impression of a continuous movement. This is expressed by 

= (r , vt) (1.2) 

with an arbitrary function , a space coordinate r and some velocity vector v. Although this 

is a fairly general formulation of a wave, a special case is of much more practical interest: 

the harmonic wave ( ) = cos( )or () = sin( ), which is | thanks to the Fourier transform 

| the building block for any other waveform in linear systems. For the simplest case of a 

one-dimensional, monochromatic wave of frequency !, may be written as 

(x; t) =Ae j(kx,!t) ; (1.3) 

with a constant amplitude A and the wave number k. The impression of wave motion becomes 

obvious if we spot one point in the function and determine its trajectory. This implies that 

kx , !t = const : (1.4) 

Obviously the point ismoving in positive direction, and we nd its velocity 

vp = dx 

dt 

= ! 

k 

: (1.5) 

Since the propagation of the crests of this wave actually describes loci of constant phase, vp 

is called phase velocity. 

Remark (Notational detail) A signi cant part of this work will be on electomagnetic 

waves. Following the notational conventions in electrodynamics, we shall use the symbol j 

for the imaginary unit p ,1, as opposed to i which is common in physics and mathematics. 

Next we consider the classical example of two superimposed waves that di er slightly in 

frequency and wave number. The two waves are 

1 = A cos ((k , k)x , (! , !)t) 

2 = A cos ((k + k)x,(!+ !)t) : 

3 

(1.6)

The superposition then yields the well-known result 


= 1 + 2 =2Acos( kx , !t) cos(kx , !t) : (1.7) 

The interfering waves form beats or wave packets with the modulation frequency !. The 

trajectories of the envelope are now determined by 

and thus the wave packets propagate at the velocity 

kx , !t = const ; (1.8) 

vg = ! 

k 

! @! 

@k 

; (1.9) 

which is called group velocity. The carrier, however, moves at the phase velocity like before. 

If the two velocities di er, the precise shape of the wave will alter as it travels along, because 

the oscillations of the carrier seem to move within the wave packets. The original shape is 

retained if and only if group and phase velocity are the same. If we assume that vp is a 

function of ! (or, alternatively, k), we can combine the two de nitions to 

vg = vp + k @vp 

@k 

; (1.10) 

which clearly shows that vp must not depend on k in order to avoid distortion. This condition 

is satis ed for transversal electromagnetic waves in free space, where both phase and group 

velocity are equal to the velocity of light c. In any other medium, vp depends on the frequency, 

and the medium is said to be dispersive. The relationship between wave number and frequency 

is often displayed in a dispersion diagram like the one in g. 1.1 . 

The distinction between phase and group velocity has already been pointed out by Rayleigh 

[4], and he himself admitted that Stokes had treated this subject independently before [5]. 

Brillouin also mentions [2] that the very rst idea of group velocity dates even back to Hamilton, 

who published it as early as 1839. 

Both a monochromatic sinusoid and a periodic series of wave packets are inadequate to transport 

any kind of information. An individual pulse or any other non-periodic function, which 

is of far more practical signi cance, can be described by means of a Fourier integral 

or, alternatively, 

(x; t) = 

(x; t) = 

Z 1 

,1 

Z 1 

,1 

A(!) e j(k(!)x,!t) d! (1.11) 

A(k) e j(kx,!(k)t) dk : (1.12) 

An interesting case is that of a modulated signal, which is composed of an envelope 0 and 

a carrier with frequency !c, 

(x; t) = 0(x; t) e j(kcx,!ct) : (1.13) 

4


! 

!c 

vp = ! 

k 

kc 

vg = @! 

@k 

Figure 1.1: Brillouin diagram. 

!(k) 

Using the Fourier representation of the signal, we can write the envelope as 

0(x; t) = 

Z 1 

,1 

A(k) e j((k,kc)x,(!(k),!c)t) dk : (1.14) 

The dispersion relation !(k) may be expanded in a Taylor series about kc, and if terms of 

higher than linear order can be neglected, we nd (!(k) , !c)t = vg(k , kc)t. This nally 

yields 

0(x; t) = 

Z 1 

,1 

A(k) e j(k,kc)(x,vgt) dk ; (1.15) 

which is exactly the initial waveform 0(x; 0), but displaced by an amount vgt. In other 

words: the envelope preserves its shape and moves with the group velocity. This result is 

exact if and only if the dispersion relation has the form of a straight line ! = vgk + !p [2]. In 

most cases, however, the dispersion relation will have a more general form, and (1.15) is then 

only a rst-order approximation that applies to a su ciently small region of the spectrum 

centred at kc and !c, respectively. If the spectrum of the envelope becomes broader, the 

group velocity di erences of the harmonic components will cause the packet to spread. 

Remark (Modulated versus baseband signals) Note that for modulated signals a 

constant group velocity is a prerequisite so that the pulse shape is not deformed. For 

baseband signals, on the other hand, the phase velocity must be invariant within the 

5 

k


bandwidth of the signal. We see this by means of a similar consideration as before. In 

the Fourier representation of a baseband signal, 

(x; t) = 

Z 1 

A(k) e 

,1 

j(kx,!(k)t) dk ; (1.16) 

we replace the dispersion relation again with ! = vgk + !p, which gives 

(x; t) = 

Z 1 

A(!) e 

,1 

j(k(x,vgt),!pt) dk : (1.17) 

This corresponds to the shifted initial signal (x; 0) if and only if !p = 0, which in turn 

means that vg = vp. 

The conclusion to be drawn from the preceding therefore is that the concept of group velocity 

is meaningful only for narrow-band signals. Needless to emphasise that a narrow spectrum 

implies a large pulse duration, which is undesired in many applications. The notion of wave 

packets propagating at group velocity is not necessarily correct in the presence of dispersion 

(see, for example, Stratton [6]). 

There have also been other interpretations of the group velocity in the literature. Lighthill 

[7] mentions a kinematic derivation of vg starting from a continuity equation for the number 

of wave crests, @k=@t + @!=@x =0. This is a suitable approximation only if the frequency 

components have already been separated by dispersion. Two other approaches regard the 

motion of a centre of gravity. Smith [8] reports a new de nition which takes vg to be the 

velocity of the temporal centre of gravity of the wave packet, 

R 1 

1 ,1 t j (r;t)jdt 

= r R 1 

: (1.18) 

vg ,1 j (r;t)jdt 

Considering the case of a modulated signal (1.13), Prestwich [9] stays with the original de - 

nition of the group velocity, but relates it to the centroid (or spatial centre of gravity) of the 

envelope 0(x; t), 

He then shows that 

hx(t)i = 

R 1 

,1 x 0(x; t) dx 

R 1 

: (1.19) 

,1 0(x; t) dx 

hx(t)i = hx(0)i + @! 

t; (1.20) 

@k k=k0 

which means that the centre of gravity moves at the group velocity. It is of particular interest 

that in contrast to (1.15), this derivation involves no power expansion of the dispersion 

relation. 

Vichnevetsky [10] obtained an alternative derivation of the group velocity by considering the 

centre of gravity of the energy contained in a wave moving through a lossless medium, 

hx(t)i = 

R 1 

,1 x j (x; t)j2 dt 

R 1 

,1 j (x; t)j2 : (1.21) 

dt 

6

1.2 A few notes on dispersion 

He could then demonstrate that the motion of the wave energy can be looked at as the 

superposition of the spectral energy components, each moving at its respective group velocity. 

So far, we considered the media to be energy conservative or non-dissipative, which implies 

that the dispersion relation is a real function. When this condition holds, the concept of group 

velocity is clearly de ned. As soon as dissipation or evanescence enter the stage, however, 

the dispersion relation becomes complex, the imaginary part describing the attenuation of 

the respective monochromatic component. In such a case, the validity of the de nition of 

group velocity is at least questionable. For only weak absorption, Brillouin [2] argued that 

the group velocity may safely be determined based on the real part of the wave number, 

vg = @ Re !=@k. As absorption becomes more marked, this approach is no longer justi ed, 

and alternatives must be sought. 


In general, there are two e ects that cause dispersion: 

Parameters of the medium may be frequency-dependent. This accounts in particular 

for resonance e ects in dielectrics. In fact the characteristics of all real media depend 

on the frequency. The only exception to this rule is vacuum. 

Boundary conditions may impose constraints on the relation between wave number 

and frequency even if no medium is present atall. Atypical example are hollow wave 

guides where waves can propagate only in certain frequency ranges. 

Let us rst explore the e ects of frequency-dependent parameters and consider a dielectric 

with losses and a distinct resonance frequency !0 caused by elastically bound polarisable 

electrons. The relative dielectric constant of such a medium can be written as [2, 11] 

!p 2 

"r(!) =1, 

! 2 ,!0 2 +2j! 

; (1.22) 

where the e ects of the losses are contracted into a phenomenological attenuation constant 

. Fig. 1.2 shows the respective curves for the lossy and lossless ( = 0) case. It is important 

to notice that without losses, "r is monotonically increasing over the entire range with a pole 

at resonance, whereas in the presence of losses, the pole degenerates to a zero at !0 and "r 

consequently decreases in the neighbourhood of this point. For the range where d"r=d! < 0, 

the dispersion is said to be anomalous (Jackson [11]). This is, however, not the only de nition 

for this term, as we shall see shortly. 

Remark (A word of caution) In the derivation of (1.2), the time dependence of the 

physical quantities was taken to be e ,j!t , which is a common assumption in physics. 

In electrical engineering, however, the sign of the phase function is usually reversed, so 

that the time factor reads e j!t . With this setting, the denominator of (1.2) would have 

been ! 2 , !0 2 , 2j! . Consequently, if"ras given in (1.2) is used in standard electrical 

7

15 

10 

5 

0 

-5 

-10 

ε 


2 4 6 8 10 12 ω 

Figure 1.2: Dielectric constant "r of a resonant dielectric with (thin lines) and without losses. The 

solid lines denote the real part, the dotted line is the imaginary part in the lossy case. In this example, 

the resonance frequency is at !0 =5. 

engineering formulas, the sign of the attenuation constant must be taken negative if 

is meant tobea loss factor. Otherwise, one might end up for instance with a negative 

conductivity. 

We nowintroduce some quantities frequently used to characterise the propagation of electromagnetic 

waves in dispersive media. The index of refraction relates the phase velocity in the 

medium and the velocity of light in free space and is thus determined by the relative dielectric 

constant and the relative permeability, 

n(!) = c 

vp 

= p "r r; (1.23) 

although the permeability is mostly omitted because the medium is chosen to be non-magnetic. 

From this equation we nd an alternative expression for the wave number, 

k(!) = ! 

c 

which is again the dispersion relation with ! as independent variable. 

With these settings, the group velocity can also be given as 

vg(!) = 1 

dk 

d! 

= 

n(!); (1.24) 

1 

n(!)+! dn 

d! 

: (1.25) 

The velocities are now complex, but only real-valued quantities are physically reasonable. 

Bearing in mind that the imaginary part of the phase of a wave, Im(k(!)x , !t) =Imk(!)x 

8


v/c 

6 

4 

2 

-2 

2 4 6 8 10 12 ω 

Figure 1.3: Index of refraction (solid line), phase velocity vp=c (dashed line), and group velocity vg=c 

(dotted line) of the resonant dielectric in g. 1.2 . 

leads to an attenuation of the wave in the direction of energy propagation, it seems sensible 

to insert only the real part of the wave number into the velocity de nitions. We thus obtain 

and 

vp = ! 

Re k 

vg = 1 

Re dk 

d! 

= c 

Re n 

(1.26) 

: (1.27) 

The results for the dispersive dielectric, scaled to the vacuum velocity of light, are depicted 

in g. 1.3 together with the refractive index. We note that the group velocity exhibits two 

areas where it grows excessively, and becomes negative between the poles. 

This behaviour extends far beyond the region where anomalous dispersion in the above sense 

occurs. Consequently, many authors gave alternative de nitions. According to Sommerfeld 

[12] and Stratton [6], this is the region where vp < vg, which holds only for a narrow 

range about the poles. Jackson [11] and Schulz-DuBois [13] require dn(!)=d! < 0, which 

is tantamount to the most common de nition of anomalous dispersion in modern literature, 

dvp=d! > 0 (Baldock and Bridgeman [14], Ramo et al. [15], Piefke [16]). 

Seen purely mathematically, the imaginary part of the wave number gives rise to an attenuation 

of the wave along the direction of propagation. Physically, this e ect can be traced back 

to two distinct causes: 

Absorption is associated with various losses in the medium like the dielectric resonance 

in the above example. It is characterised by a complex wave number. Hence wave 

propagation is possible, but a part of the wave's energy is dissipated. 

9


Evanescence, in contrast, is non-dissipative and due to total internal re ection. Because 

the wave number is purely imaginary, a propagation of waves is precluded. Evanescence 

typically stems from boundary conditions that limit wave propagation to a certain 

frequency range like inwave guides. 

The term total internal re ection stems from optics where this e ect was rst observed for 

light rays obliquely incident on the interface of two transparent media with di erent indices 

of refraction, provided that the angle between the interface and the rays is smaller than a 

certain limit. This sort of re ection, however, must not be confused with the partial re ections 

arising from discontinuities or parameter changes in the medium. The latter are localised and 

take place only at the respective interface, whereas the former is distributed over the entire 

evanescent region. This also means that the energy is re ected totally only if the evanescent 

region is su ciently | and in a strict sense in nitely | long (with allusion to optics this 

is frequently called the opaque limit). If the evanescent medium, however, has nite length, 

then a portion of the incident energy will appear at the far side of the barrier. In optics, this 

e ect has the colourful name frustrated total internal re ection [17, 18]. 

In physical reality, evanescence and absorption cannot always be disentangled. Fiber optic 

wave guides, for instance, rely on a cladding material that is radially evanescent for the 

frequencies that make up the signal in the core. Still, a certain amount of loss in the cladding 

as well as in the core is almost inevitable. The same is true for mirrors, which also absorb 

a small fraction of the incident intensity rather than re ecting it. But perhaps the simplest 

example is a common LC- lter, whose components are never ideal reactances and always 

entail losses. 

In the course of the review of wave propagation we shall often come across the terms absorption 

and evanescence. To be consistent with the respective literature, they were retained as the 

corresponding authors employed them. Nevertheless, it is to be kept in mind that many of 

the given statements concerning propagation velocity apply to either of the two. 

1.3 Signal velocity 

Throughout the decades following Rayleigh's formulation of group velocity, itwas commonly 

believed that this was the actual velocity at which a nite signal propagates through a dispersive 

medium. Einstein's theory of relativity, however, eventually raised considerable doubts 

concerning this interpretation. The problem was that in absorptive media the group velocity 

may easily reach an arbitrarily large value. Hence such examples were presented as 

contradictions to the postulate that a signal transmission at a velocity greater than that of 

light in vacuum is impossible. This discussion was the motivation for the classical paper of 

Sommerfeld [12]. He investigated the propagation of a step-modulated signal, 

(0;t)= (t) sin !ct ; (1.28) 

(t) being the Heaviside unit step function. The medium he considered is described by the 

Lorentz model, which assumes that the medium consists of a large number of oscillating 

10


dipoles with a single resonance frequency !0 and some damping coe 

relation is thus given by 

cient . The dispersion 

k = 

s ! 2 

c 2 

!p 2 

1 , 

! 2 , !0 2 +2j! 

; (1.29) 

!p being the plasma frequency. Sommerfeld then showed that the wave front of any signal 

always travels at the velocity of light c. 

Proof The proof is based upon function theoretical arguments. Since the Fourier integral for the 

propagated wave, R 1 

d! 

ej(kx,!t) 

,1 , does not converge if the integration is carried out along the real 

!,!c 

axis due to the pole at !c, one can move the contour of integration to the upper half of the complex 

plane, such that ! 7! ! + j . The integral converges if the argument of the exponential function has 

a negative real part at in nity. Since lim!!1 k = !=c, the argument at in nity is,j!(t , x=c), and 

its real part is negative if and only if t , x=c < 0. The contour of integration may be closed with 

a semicircle to the right at in nity, and as this area contains no singularities, the integral is zero. 

Hence the medium is at rest wherever t , x=c < 0, which means that the wave front propagates at 

the velocity of light. This statement applies to all media whose dispersion relation is characterised by 

kc = !j!!1. 

The work of Sommerfeld was complemented by a paper by Brillouin [19], who elaborated the 

behaviour of the signal. He used the relatively new method of saddle point integration and 

steepest descents to obtain an approximate evaluation of the integral. This method relies on 

the fact that a Fourier-type integral of a complex variable, R e '(!) d!, is dominated by those 

areas of the complex plane where the path of integration passes over a saddle point of'(!). 

Thus an approximate evaluation of the integral may be con ned to the environment of such 

points, which are crossed following the path of steepest descent. In fact, this method is a 

generalisation of the method of stationary phase. 

Remark (Method of stationary phase) This approximation method was introduced 

by Lord Kelvin. Consider a Fourier integral R 1 

,1 f(!)ej(k(!)x,!t) d!, where the spectral 

function f(!) is only slowly varying. If x or t are large, then the main contribution to 

the integral comes from areas where the argument of the exponential function does not 

change, i. e. k0 (!)x , t =0. Outside the vicinity of such points, the integrand oscillates 

strongly and the contribution is practically zero. It is therefore su cient toevaluate the 

integral around the points of stationary phase, which maybe accomplished by expansion 

techniques (see, for example, Lighthill [7] or Wait [20]). 

From a purely mathematical point of view, the evolution of the wave (x; t) is largely determined 

by the location of the saddle points, and the contour of integration must follow 

these points as they move through the complex plane. This is what Brillouin did, and he 

nally found that the saddle points | of which there are two pairs due to the structure of the 

dispersion relation (1.29) | cause two distinct so-called precursors to appear after the arrival 

of the wave front. The rst precursor had already been identi ed by Sommerfeld by means of 

an asymptotic expansion of the dispersion relation, thus valid only for small values of t , x=c 

or high frequencies. Therefore, subsequent authors often referred to the rst precursor as 

Sommerfeld and to the second as Brillouin precursor. 

11

Brillouin precursor 

Sommerfeld precursor 

x 

c 


signal arrival 

Figure 1.4: Evolution of the wave as given by Brillouin. 

Following the precursors, an oscillation with the frequency !c of the input signal evolves, 

and Brillouin considered this to be the signal. As g. 1.4 shows, the point where the signal 

actually arrives is anything but clear to see, so he de ned the signal arrival to occur at the 

moment when the path of integration reaches the singularity at!=!c(the residue of which 

yields an explicit contribution to the integral afterwards, namely the steady-state solution). 

The investigation of this signal velocity showed that in the range of normal dispersion, i. e. far 

enough away from the resonance frequency !0, it equals the group velocity. Near the resonance 

frequency, however, the group velocity grows larger than c and even becomes negative (see also 

g. 1.3), whereas the signal velocity reaches, according to Brillouin, a maximum equal to c. 

This was an error arising from the fact that the approximation method was inexact precisely 

at this point. The mistake was spotted and corrected by Baerwald [21], who pointed out that 

the signal velocity actually reaches a minimum near resonance. In a thorough analysis of the 

signal evolution, he found that for su ciently large values of x, the amplitude of the precursor 

may exceed that of the signal for the signal arrival time de ned by Brillouin. Consequently, 

Baerwald proposed that the signal arrival be the moment when the attenuation of precursor 

and stationary signal are equal. 

In a related article [22], Baerwald discussed the general case of continuous dispersive systems 

consisting of an alternating sequence of stop and pass bands. He con rmed that the wave 

front propagates with the velocity 

! 

vf = lim 

!!1 k(!) 

t 

(1.30) 

and remains undistorted except for some attenuation in the presence of losses. In addition, 

he noticed that when the wave has travelled far enough into the medium, it is split up into its 

frequency components that then propagate with distinct velocities. Later, this phenomenon 

was dubbed mature dispersion (see, for instance, Baldock and Bridgeman [14]). 

12


The arbitrariness that underlies this de nition of the signal velocity has caused considerable 

irritation and dispute over the years. The main objection was the di culty to apply the 

de nitions to any practical case. If one waits for the signal to reach a certain fraction of 

its maximum amplitude, one must know the entire signal before being able to decide when 

it has arrived [8], and still one might catch the precursor instead of the steady-state part. 

The other problem is that the amplitude of the steady-state signal may be much less than 

that of the transient part, so if one required the signal to exceed its precursor, one would 

literally wait forever. Apart from these theoretical arguments, Trizna and Weber [23] carried 

out a simulation based on an expansion technique and attempted an experimental veri cation 

of the signal velocity. To this end, they had to investigate the square of the wave (x; t) 2 . 

They observed a ringing e ect in accordance with the theory, but no pulse delay. Hence they 

concluded that a separation of precursor and steady state signal is not as straightforward as 

Brillouin suggested, and that the signal velocity de nition is not meaningful in practice. 

Despite or perhaps just because of these discussions, the wave propagation in the Lorentz 

medium was the subject of continued research. More than seventy years after the papers of 

Sommerfeld and Brillouin, Oughstun and Sherman [24] published an article that reconsidered 

the classical problem and provided an improved solution. They also used the saddle point 

integration method, but did not follow the paths of steepest descents, which allowed for a 

simpler contour of integration. In their interpretation of the signal velocity they followed 

Baerwald in that they de ned the signal arrival to occur when the real part of the phase 

function along the contour of integration equals that of the residue at ! = !c. Physically, this 

means that before this point, the wave is dominated by the contribution of one of the saddle 

points moving through the complex plane. After the signal arrival, the eld is dominated by 

the steady state signal, and the precursors are negligible. The signal arrival is thus exactly 

the moment when the attenuation of the stationary oscillation is the same as that of the 

dominating saddle point at that time | hence the use of the real part of the phase function. 

A di erent interpretation useful for measurements and numerical experiments is that at this 

time, the wave begins to oscillate with the carrier frequency !c [25, 26]. 

The result of this analysis was that the rst or Sommerfeld precursor oscillates at a high 

frequency, whereas the second or Brillouin precursor is low-frequent [27]. Whether they reach 

a signi cant amplitude, and which of the two is more important, depends on the characteristic 

of the input signal. In particular, the distortion due to the precursors becomes more severe as 

the signal frequency is shifted towards the absorption band or the rise and fall times are made 

smaller (i. e. the spectrum is broadened). Additionally, for rectangular pulses with su ciently 

short pulse width the precursors of the leading and trailing edge may interfere [28, 29]. 

Since the signal velocity has a similar de nition to that of Baerwald, its dependence on the 

signal frequency is similar, too. Consequently, Oughstun also found a minimum near resonance. 

At frequencies well above resonance, however, the spacing between the two precursors 

is large enough that the signal seems to break up into two parts separated by the Brillouin 

precursor. Therefore, Oughstun de ned two distinct velocities for these two parts (namely 

the prepulse and the actual main signal), which adds a certain amount of absurdity to the 

entire concept. Balictsis and Oughstun extended the research to Gaussian-shape pulses [30] 

and found that even such a pulse may evolve into a pair of pulses provided that the initial 

13


pulse width is small enough (and the bandwidth thus large). At any rate, the signal velocity 

is di erent from the speed of the maximum of the envelope. Extensions to multiple-resonance 

media also yielded consistent results [31]. 

Remark (Signal velocity for Gaussian pulses) When the carrier frequency is below 

the absorption band, such that 0 !c !0, the signal velocity is (approximately) equal 

to the energy velocity discussed in the next section. A similar criterion holds if !c lies 

above the absorption range [29, 30]. 

There is, at least, one detail where many authors are of the same opinion: the signal velocity 

and the group velocity are practically the same for lossless systems (i. e. = 0) or systems 

with small losses such that the dispersion relation has a dominant real part [22, 23]. Only 

when losses become dominant as in the case of resonant absorption, this concept breaks down. 

1.4 Energy velocity 

While the de nition of the signal velocity sounds very clumsy and far-fetched, there exists 

another velocity de nition that can hardly be surpassed in clarity and brevity. The energy 

velocity is given as the ratio of the mean propagated energy P (which for electromagnetic 

waves is the Poynting vector integrated over the cross-section) and the mean energy W stored 

in the medium per unit length in the direction of propagation, 

ve = P 

W 

; (1.31) 

where the mean values are obtained by averaging over a period. The total energy density W 

includes not only the electric and magnetic elds, but also the potential and kinetic energy 

that resides in the dipoles of the dielectric. 

Remark (Spatial averages) There is also the | at least theoretical | possibility to 

calculate the mean energies as averages over a wavelength (Mainardi [32]). This may be 

applicable as long as the wave is not attenuated. If it is, however, the spatial periodicity 

is lost, and the respective average as such no longer exists. 

The origin of this velocity de nition is somehow unclear. Apparently it entered common 

engineering knowledge so successfully that hardly anyone ever cared about giving references 

to where he got the de nition from. From the few who did, Loudon [33] attributed it to 

Brillouin, whereas Brillouin [2] noted that this de nition was given much earlier by Havelock 

[34]. Anyhow, a similar expression appears already in a paper by Rayleigh [5]. Although 

he considered the ratio of propagated and stored energy, he did not regard it as a separate 

velocity, but rather proved it to be equal to the group velocity for the example of water waves. 

In this connection he mentioned Reynolds having raised this question, and so it seems that 

the concept of energy velocity | like that of group velocity | was conceived more or less 

independently by either of the two [35]. 

14

1.4 Energy velocity 

For electromagnetic waves, Geppert [36] showed how the velocity of energy transport can be 

derived from the continuity equation, which in a general form reads 

r ( v)+ @ 

@t 

+ S =0; (1.32) 

where is some physical quantity moving with mean velocity v and S is the density of sources 

(S < 0) and sinks (S > 0) within the medium. For energy conservative systems, this term 

vanishes. As second ingredient, the Poynting theorem may be written as 

r (E H)+ @ 

@t 

" 

2 E2 + 2 H 2 + S =0: (1.33) 

In this notation, the correspondence between the two expressions is so striking that the 

de nition of the energy velocity is almost self-evident. 

Remark (Energy storage in dispersive media) The second term in (1.33) is the 

energy density in the electromagnetic eld. Schulz-DuBois [13] divided this total energy 

further into a part associated with the eld itself, 

We = "0 

2 E2 + 0 

2 H 2 ; (1.34) 

and a part describing the energy stored in the degrees of freedom of the dispersive medium, 

Ws = 1 @ , 

!(" , "0)E 

2 @! 

2 + !( , 0)H 2 

: (1.35) 

The last formula is valid in a strict sense only for monochromatic waves, as was already 

pointed out by Borgnis [37]. It is, however, a useful approximation if the energy spectrum 

is narrow. 

Although the de nition of the energy velocity looks very plain, the calculation of the energy 

density requires some re ection in order not to forget a contribution. In fact, there are several 

examples of authors who did not take every possibility of energy storage into account and 

obtained surprising but false results. One of them was Brillouin [2] himself, who for the 

Lorentz medium did not properly consider the energy stored in the elastic dipoles, which 

is particularly important at resonance. Loudon [33] corrected this mistake and in addition 

found the energy velocity to be in good agreement with the signal velocity de ned by Brillouin 

[2, 19] and Baerwald [22]. 

A second example is the controversy between Borgnis and Kleen. Borgnis [38] investigated the 

propagation velocity of electromagnetic waves along delay lines. The structure he considered 

consisted of a plane conductor with slots perpendicular to the direction of propagation. These 

slots hold a certain amount of energy that contributes to the total energy density. Borgnis 

did not account for this contribution and found that ve was equal to the phase velocity. 

Kleen and Poschl [39] discovered the aw and published a correct treatment of the problem 

that re-established the well-known result ve = vg. Furthermore, they provided a proof that 

the energy and group velocities are identical for all lossless transmission lines with periodic 

structure. 

15


The equivalence of group and energy velocity for lossless transmission lines was of particular 

importance for the evolution and design of travelling-wave tubes and therefore was widely used 

[40, 41]. In these devices, an electron beam should interact with an electromagnetic wave. 

To this end, the velocity of the electrons and the phase velocity had to be approximately the 

same. Unfortunately, the electromagnetic wave normally travels at the speed of light, whereas 

the electrons are much slower than c. Hence delay lines were needed to slow down the wave 

and to allow for a coupling of the electric eld and the beam. But although the design goal 

was to match the electron speed and vp of the wave, the energy is actually propagated at vg. 

Maybe because of this practical signi cance, the group velocity is often believed to describe 

the motion of a wave's energy. An appealing argument for this identity is that the energy of 

the wave is localised in the wave packet. Since the wave packet moves with the respective 

group velocity, so does the energy. This is, however, not universally true and restricted to 

cases where dissipation is negligible. Borgnis [42] proved comparatively early the equivalence 

ve = vg for monochromatic waves in a dielectric with no dispersion other than that imposed 

by the geometrical constraints of a wave propagating along a conducting plane, which gives 

a dispersion relation of k 2 = ! 2 " , 2 , being an eigenvalue of the corresponding system 

of partial di erential equations. The investigation of Broer [43] provided a general proof. He 

obtained the equation 

@E 

@t 

+ @ 

@x (vgE) =0 (1.36) 

by means of the method of stationary phase, which imposes two important restrictions on the 

wave integrals in order to be applicable. First, the dispersion relation must be a real function 

| this is where the absence of losses comes in. Second, the waves must have separated 

so far that the expansion of the dispersion relation is justi ed and the wave is described 

accurately enough by its local wave number. In other words, the spectrum of the wave at this 

point must be narrow. Biot [35] went one step further and showed that the identity holds 

also for inhomogeneous media without dissipation. The parameters of the medium may be 

frequency-dependent (which he called anomalous dispersion), but they must not depend on 

the coordinate in the direction of propagation. 

The convenient equivalence of group velocity and energy propagation velocity is no longer 

applicable when the medium becomes absorbing. For the meanwhile well-understood Lorentz 

model, Loudon [33] showed that the classical signal velocity and the energy velocity are in 

excellent agreement even in the area of resonance. In the discussion of the results, however, he 

considered only monochromatic waves. An investigation of signals with broader bandwidths 

was carried out by Sherman and Oughstun [44]. They demonstrated that for each point in 

time and space, the wave function (x; t) is dominated by a single frequency component. This 

frequency !e is found to be determined by the equation x=t = ve and | if the solution is 

not unique | the additional constraint that the attenuation of the partial wave be the least 

possible. 

Remark (Interrelation between energy and group velocity) The relation x=t = ve 

is in a certain respect an extension of the method of stationary phase to lossy systems. 

16

1.5 Other velocity de nitions 

For negligible losses, the Fourier integral of (x; t) is determined by frequency components 

where the derivative ofthe phase function k(!)x , !t vanishes, which yields x=t = vg. 

Thus ve takes the role of vg in the presence of losses. 

In a subsequent paper, Oughstun and Shen [45] extended the analysis to multiple-resonance 

media. In their most recent publication, Sherman and Oughstun [46] provided a model that 

allows more physical insight into the pulse propagation than did the purely mathematical 

treatment with the saddle point integration. It applies if the medium is not too highly 

absorbing and if only the region of mature dispersion is regarded. At anygiven time and in 

a small region of space, the pulse is then predominantly made up of two components. One 

of these is always quasi-monochromatic, the other may either be also quasi-monochromatic 

or non-oscillatory. The two components signify the Sommerfeld and Brillouin precursors, 

respectively, and move together with the same energy velocity. As the pulse propagates, 

these monochromatic components decay according to their attenuation coe cients, and in 

addition, the entire pulse spreads because the partial waves travel at di erent velocities. 

Note that since the authors set up the model for a Dirac delta pulse (0;t)= (t), there is no 

third contribution from a steady-state solution like the one that appeared in the investigation 

of the step-modulated input eld. 

There are still other examples where ve 6= vg. Mainardi [32] mentions the case of a uniform 

transmission line characterised by frequency-independent distributed inductance, resistance, 

capacitance, and shunt conductance. These parameters form the classical equivalent circuit 

of a transmission line we shall treat in more detail in chapter 3.1. Using this model, it can be 

shown that the energy velocity equals the phase velocity. If the transmission line is assumed 

to be loss-free, the validity of this statement is trivial since then the transmission line is also 

dispersion-free. If, however, losses are taken into account, the wave number becomes complex, 

and we obtain ve = != Re k = vp in accordance with the de nition of (1.26). 

Like all other velocity de nitions, the energy velocity was criticised, too. Smith [8] complained 

that although the de nition is meaningful, it gives no hint astohow one can actually measure 

the velocity. In particular, unlike the signal velocity, it does not de ne a signal arrival. 

Concerning laser phenomena, Schulz-DuBois [13] noted that for emissive dispersive media 

the energy velocity in the classical de nition can exceed the velocity of light. He therefore 

proposed an alternative approach that distinguishes the propagation of purely electromagnetic 

energy from the energy stored in the resonant medium. 


The aforementioned three di erent velocities of propagating waves are the oldest and thus 

classical ones. As if this had not been enough, a number of additional de nitions emerged in 

the course of time. Many of these approaches considered in one way or another the movement 

of a moment of a distribution or, in allusion to classical dynamics, the motion of some centre 

of gravity. We have already encountered one such de nition. It was discussed by Smith [8] as 

a proposed substitute for the group velocity and concerned the temporal centre of gravity of 

the wave packet's amplitude j (x; t)j. Smith himself suggested still another alternative: the 

17


centrovelocity. Thereby he meant the temporal centre of gravity of the intensity (x; t) 2 of 

the wave, 

1 

vc 

= r 

R 1 

,1 t (r;t)2 dt 

R 1 

,1 (r;t)2 dt 

; (1.37) 

which is identical to the centre of gravity of the energy carried by the eld. As the motion of 

this point at least partly describes what happens to the energy ow, he advocated the use of 

this de nition instead of the classical energy velocity. 

The moments of a wave packet, which are undoubtedly useful for characterising the shape 

of a pulse (see, for example, Baird [47]), were also exploited by Anderson and Askne [48] to 

derive a pulse velocity. Their starting point was the well-known linear approximation of the 

dispersion relation, which they augmented by higher order terms in order to cope with either 

strongly dispersive media or large signal bandwidths. Their de nition of the propagation 

velocity is similar to that of Smith, except that they used the spatial distribution of energy 

in the envelope of the signal to obtain the moment of inertia 

hx(t)i = 

R 1 

,1 0 (x; t) x 0(x; t) dx 

R 1 

,1 0 (x; t) : (1.38) 

0(x; t) dx 

The pulse velocity is simply the temporal derivative of this moment, v = @hx(t)i=@t. For the 

special case of a Gaussian pulse, they demonstrated that the peak of the envelope does not 

travel with the group velocity, but with vmax vg( )+v00 g ( )= 2 , where is the initial spatial 

width of the pulse and = x , vgt is a combined space-time parameter. 

The just mentioned movement ofanenvelope's maximum is a very common measure for the 

pulse propagation if the pulse exhibits a pronounced peak. Obviously, there are two di erent 

types of peaks that can be regarded. One possibility is to determine the temporal maximum at 

a given distance and followitonitswayinto the medium. The alternative is to seek the spatial 

maximum for a xed time and analyse its evolution with time. In general, both trajectories 

will depend on the parameters of the medium and be neither identical nor simple straight 

lines. Anderson et al. [49] discussed both de nitions and concluded that from a practical 

point of view, the temporal velocity is more interesting because it is easier to measure in an 

experiment. 

In a way related to the observation of pulse peaks is the correlation velocity described by Bloch 

[50]. It is based on computing the cross-correlation of the received pulse and its undistorted 

shape 

R( )= 1 

T 

Z T 

0 

(0;t) (x; t + )dt ; (1.39) 

where the wave packet is assumed to be periodic with period T , such that the computation can 

be accomplished by e cient techniques like the Fast Fourier Transform (FFT). The correlation 

function will have a peak at the time delay where the distorted wave packet resembles most 

closely its initial shape, and this delay is then used for the de nition of the velocity. It is 

18


evident that this procedure is in fact nothing but a ltering of the pulse, apt to make it 

smoother and reduce the in uence of noise. Unfortunately, it fails to give a unique result 

whenever the pulse is split into several parts on its passage through the medium. 

Anyhow, it might be indeed more meaningful in such a case to treat the distinct parts separately 

than to construe an arti cial centre of gravity, which Bloch compared drastically to 

the fact that, ìf the earth were blown to pieces by a thermonuclear device, the center of mass 

would still serenely orbit the sun'. On the other hand, is there any good reason for treating 

a pulse di erently only because it is incidentally torn apart by dispersion? After all, it is still 

one single signal. These nal thoughts re ect, in essence, the nature of the entire discussion 

on wave propagation velocity. 

19

Chapter 2 

Signals faster than light? 

2 Signals faster than light? 

Wie wird es sein, wenn wir mit der Schnelligkeit des Blitzes Nachrichten 

uber die ganze Erde werden verbreiten konnen, wenn wir selber mit gro er 

Geschwindigkeit und in kurzer Zeit an die verschiedensten Stellen der Erde 

werden gelangen, und wenn wir mit gleicher Schnelligkeit gro e Lasten werden 

befordern konnen? Werden die Guter der Erde da nicht durch die 

Moglichkeit des leichten Austausches gemeinsam werden, da allen alles 

zuganglich ist? Adalbert Stifter, Der Nachsommer, 1857 [51] 

One of the fundamental tenets of modern physics is the principle of relativistic causality, 

the postulate that no energy may be transmitted at a speed greater than that of light in 

free space. Another puzzling phenomenon, almost as old as Einstein's discovery and closely 

linked to wave propagation, is the tunnel e ect. It permits quantum particles to penetrate 

a barrier too high to overcome within the scope of classical mechanics, and electromagnetic 

waves to cross regions where they normally cannot exist. This e ect has been given much 

attention, and from the very beginning, a central question was how much time a particle 

or pulse spends in the barrier, or how fast it traverses the tunnel. Many answers have 

been attempted. Particular in recent years, theoretical investigations as well as experimental 

evidence seemed to suggest that the tunnelling process takes place in almost no time, and 

that the wave actually propagates faster than light if only the barrier is su ciently thick. 

This chapter shall illustrate that the discussion on superluminal wave propagation resembles 

closely the long-lasting debate on the velocity of wave propagation. In fact, the arguments 

are in many respects identical, and one cannot escape a feeling of deja vu. We shall start with 

a review of superluminal aspects of electromagnetic waves, both with respect to microwave 

propagation in waveguides and the more recent eld of quantum optics, where the behaviour 

of individual photons is of importance. The second part is devoted to the `classical' quantum 

mechanical tunnel e ect and various approaches to describe the corresponding traversal time. 

20

2.1 Superluminal wave propagation 


Causality states that the propagation of signals and energy is upper bounded by c, the velocity 

of light invacuum. This does, of course, not a ect the phase velocity ofawave, and numerous 

thought experiments have been invented to illustrate how vp can exceed c (perhaps the most 

lucid example is that of a rotating light source being observed in a su ciently large distance). 

Some of them have been described in an introductory article by Rothman [52]. He resolved 

the puzzles in the usual way by pointing out the di erence between phase and group velocity. 

Furthermore, he mentioned that in electric wave guides, the group velocity is always smaller 

than c due to the relation vp vg = c 2 . This is, however, not universally true and applies only 

to waves governed by very simple boundary conditions in lossless media. We have already 

mentioned that the group velocity may grow beyond the velocity of light in an absorptive 

medium. Such examples eventually led to Sommerfeld's proof that a wave front always travels 

at c, and to the formulation of the signal velocity. 

Remark (Phase and group velocity) We can most easily derive a criterion necessary 

for the relation vp vg = c 2 to hold. If we replace vp and vg with their de nitions from the 

previous chapter and separate the variables, we obtain 

Integration of both sides gives 

!d!=c 2 kdk : (2.1) 

! 2 = c 2 k 2 + C; (2.2) 

with some integration constant C that must be determined from the boundary conditions. 

The dispersion relation therefore is of the form ! = c p k 2 + C, which is true for hollow 

wave guides and a lossless plasma. The expression vp vg = c 2 is not valid, on the other 

hand, in delay lines, as we shall see in the following chapter. In an article by Borgnis [42], 

this relation appears as vp ve = c 2 , which would be equivalent whenever the velocity of 

energy transport equals the group velocity. In delay lines, the group and energy velocities 

are indeed identical, and hence the above relation is false in these cases. 

For some decades, all remained quiet. With the invention of lasers and thus emissive media, 

experiments were carried out, the results of which allowed interpretations of `superluminal' 

pulse velocities. Anderson et al. [49] refer to such publications. As they were considering the 

motion of the maximum of a pulse, they also noted that this velocity can exceed c for certain 

parameter values in an atomic medium. At the same time, however, they conceded that in 

this case (when the peak moves faster than the pulse front) the underlying assumption of 

their analysis | i. e. that the amplitude of the pulse envelope is only slowly varying | will 

eventually be violated. 

Only recently, Enders and Nimtz [53, 54] carried out microwave experiments to study an 

analogy of the quantum mechanical tunnel e ect | the propagation of evanescent electromagnetic 

waves in a wave guide. It is well known that the basic mode in a rectangular wave 

guide is governed by the dispersion relation 

k = 1 

c 

p ! 2 , !c 2 ; (2.3) 

21

d 

a1 a2 a1 

!c1 = c 

a1 

!c2 = c 

a2 


Figure 2.1: Wave guide used in the Enders-Nimtz experiment. The centre frequency of the pulse is 

chosen to satisfy !c1


a 

!c1= 

"r1; r1 

c 

a p "r1 r1 

d 

"r2; r2 

!c2 = 

c 

a p "r2 r2 

"r1; r1 

Figure 2.2: Wave guide discussed by Martin and Landauer. The dielectrics and the centre frequency 

of the pulse must satisfy !c1 d) is proportional to the input pulse 

(x; t) / (x , d , tc;0) ; (2.6) 

if the pulse was initially centred about x = 0. This means that the pulse seems to arrive 

behind the barrier at a time t (x , d)=c, without spending any time in the barrier itself. 

The derivation was based on a causal theory. However they appeared to be uncertain about 

the interpretation and called the result àn apparent violation of causality'. 

Soon afterwards, Hass and Busch [63] came up with a discussion of the same model. They 

focused on the e ect of an increasing barrier thickness by considering only the di erence 

23


between two tunnelling experiments with di erent barrier lengths. With a rst-order approximation 

of the dispersion relation (2.3) and for narrow-band Gaussian pulses they could show 

that the pulse indeed needs no extra time for the additional barrier length. Apart from this 

veri cation of the measurements, they also found that any wave front does travel, at most, 

with the speed of light. Considering the analogous quantum mechanical tunnelling problem, 

Azbel' [64] obtained the same result. He examined the superposition of smooth wave functions 

and small perturbations and found the latter to be travelling at c. Hence no violation of 

causality occurs. Recently, Emig [65] veri ed the results of Martin and Landauer by means 

of computer simulation based on the solution of the Maxwell equations. 

Remark (Causality and dispersion) Although the concept of group velocity is usually 

regarded as physically meaningless in the presence of absorption, it is still widely used in 

connection with Gaussian pulses. When vg then is not smaller than c inside an absorption 

band (vg >c,vg=1,or vg < 0), the group velocity is simply termed abnormal, but 

nonetheless used further to describe the propagation of the maximum of the pulse. This 

habit clearly calls for an explanation concerning the respectation of causality. Bolda et al. 

[66] proved that any causal dispersive medium must have at least one frequency where the 

group velocity becomes abnormal. In particular, this is a frequency where the absorption 

is an absolute maximum. The proof is based in principle on the dispersion relations 

formulated by Kramers and Kronig (see, for instance, Jackson [11]) that relate the real 

and imaginary parts of the index of refraction n(!) =nr(!)+jni(!), n(!) =ck(!)=!, 

nr(!) =1+ 2 C 

ni(!) =, 2! C 

Z 1 

0 

Z 1 

0 

! 0 ni(! 0 ) 

! 02 ,! 2 d!0 

nr(! 0 ),1 

! 02 ,! 2 d!0 ; 

(2.7) 

where the integrals are taken to be the Cauchy principal value. All media satisfying these 

relations are causal (including the Lorentz medium investigated by Sommerfeld). An even 

earlier treatment was provided by Toll [67]. 

An enlightening contribution to the discussion came from Steinberg [68], who himself had 

investigated the tunnelling of single photons through opaque barriers [58]. In these experiments, 

too, the photon is best described by a Gaussian pulse, and its peak also appeared at 

the tunnel exit sooner than the velocity of light would have permitted. The confusion about 

the Gaussian pulses stems largely from the fact that they are not exactly localised as might 

be expected, but in fact have an in nite duration. Yet their pronounced maximum is often 

mistaken for the signal itself, such that its receipt is associated with a signal arrival. As 

well the emission of the peak is thought to be the signal transmission. In reality, however, 

a Gaussian pulse has neither a beginning nor an end (and is therefore, in a strict sense, in 

itself a violation of causality). Even if the signal has only approximately Gaussian shape, it 

must be switched on long before the peak actually is generated. The turn-on point would be 

a better, if not the only choice as reference for delay calculation. This point, however, travels 

with the velocity c. An illustrative | in the spirit of Lewis Carroll | explanation of both 

this e ect and the quantum optical experiments was provided by Chiao et al. [69]. A critical 

review of these experiments was published by Landauer [70]. Nimtz objected vigorously to 

24


this argument intwo recent publications [71, 72]. Real signals are always band-limited, they 

argued, so there exists no actual wave front to run ahead of the pulse at the speed of light. 

Consequently, superluminal transport of both signals and energy is possible. 

Apart from the misconception regarding the actual pulse width, such e ects are well known as 

`pulse reshaping'. This peculiar behaviour of the evanescent region can be explained in a very 

illustrative manner [61]. Consider a Gaussian wave packet that has already travelled some 

distance in the medium outside the tunnel. Owing to dispersion, the leading tail of the pulse 

will be dominated by higher-frequency components that propagate faster. In addition, these 

components are less attenuated when moving through the evanescent region. Hence the pulse 

appearing on the far side of the barrier will be caused mainly by the front of the wave packet. 

The trailing tail, on the other hand, consists of lower-frequency components that are mostly 

absorbed or re ected. Consequently, the peak of the transmitted pulse seems to be shifted 

forward in time, and in addition, its centre frequency has increased. The overall impression is 

that the pulse has moved through the barrier at an abnormally great velocity. In the extreme 

case of a medium with gain, the velocity of the pulse may even become negative (see Chiao 

et al. [73, 74, 75]). 

Remark (Negative pulse velocities) Formally, a medium that provides gain can be 

described analogous to the Lorentz medium with a negative oscillator strength. This 

medium in uences the wave inversely to the e ect just described, such that the peak of 

the pulse at the end of the evanescent region leaves the medium even before the peak of 

the incident pulse enters it. Owing to the fact that the wave can temporarily borrow 

energy from the inverted atomic states in the medium, the energy velocity in the classical 

sense is negative. This does, however, not interfere with causality, and a wave front due 

to a turn-on of the signal still propagates with the velocity of light [74]. 

Another approach was pursued by Yun-ping and Dian-lin [76], who explained the reshaping 

e ect with interference between di erent possible paths along which aphoton can travel. If 

an incident Gaussian wave 0(t) can take several paths, each characterised by a delay time 

i and a probability j ij2 , then the output wave can be written as 

(t) = X 

i 0(t, i): (2.8) 

i 

If the delay times do not di er too much, the original shape of the pulse is approximately 

preserved despite the interference, and the peak of the pulse is advanced with respect to that 

of a reference pulse propagating through free space. The only problem with this formulation 

is that it obviously neglects the dispersion e ects of the medium. Yet, the superposition of 

undistorted waves yields an overall dispersive transfer function. 

Remark (Multipath propagation) The concept of electromagnetic waves travelling 

along di erent paths from a sender to a receiver plays an important role in the modelling 

of mobile communication systems. Particularly in urban areas, the signal at the receiver 

antenna is a combination of waves scattered from and di racted around the surrounding 

buildings. Depending on the phase delays, the interference can be constructive or 

25


destructive, changing rapidly as the receiver moves on. Thus the signal amplitude undergoes 

signi cant variations over short distances. This phenomenon is known as short-term 

or Rayleigh fading [77], and it is dispersive in nature even though the propagation of the 

individual waves is not. 

Considering a monochromatic signal and keeping the receiver xed (thus sidestepping the 

problem of a Doppler shift introduced by amoving receiver), we can describe the output 

signal of a multipath channel by a sum of delayed, time-shifted copies of the input signal 

like in (2.8). Correspondingly, the transfer function of the channel is given by [78] 

H(j!)= X 

aie ,jkli = X 

ai e ,j! li c = X 

ai e ,j! i ; (2.9) 

i 

i 

where we used the de nition of the frequency-independent phase velocity vp = c = !=k. 

In order to be dispersive, the phase of H(j!) must depend other than linearly on !, so 

that the phase delay arg H(j!)=! is not constant. This is most likely the case in practice, 

although the li can be chosen so as to result in a transfer function with linear phase (which 

provides an interesting connection to the theory of linear FIR lters). 

2.2 Quantum mechanical tunnelling 

Although the knowledge on electromagnetic wave propagation in uenced the development of 

quantum mechanics to a great extent, the analogy between the respective tunnelling phenomena 

was not given too much attention in the past. Only recently, Martin and Landauer 

[61] demonstrated that the continuity conditions for TE and TM modes at the interfaces 

between di erent dielectrics in a uniform waveguide ( g. 2.2) are exactly the same as the ones 

that must be satis ed by the wave function at the edges of a rectangular potential barrier 

for the classical tunnel e ect. In the TE case and if the permeabilities of both regions are 

equal, the continuity of the longitudinal magnetic and the transverse electric eld components 

correspond to the continuity of the quantum mechanical wave function. On the other hand, 

the continuity of the transverse components of the magnetic eld is equivalent to the continuity 

ofthederivative ofthe wave function in a tunnelling problem. Thus the investigation 

of evanescent electromagnetic waves also provides insight into the behaviour of tunnelling 

electrons. 

Apart from the formal aspects there is also a good experimental reason why this correspondence 

is interesting. Quantum mechanical measurements are always invasive and disturb the 

process being observed. This is particularly annoying if the arrival of a wave packet is to be 

measured just ahead of the barrier in order to determine its traversal time. Electromagnetic 

wave packets, on the contrary, consist of many photons, some of which may safely be used 

for a measurement without exerting too great an in uence on the rest of the wave. This 

reasoning was the departure point for the experiments discussed in the previous section. 

One might expect that the equivalence of the two tunnelling phenomena is su cient to discuss 

the problem of tunnelling times, yet the exclusive consideration of evanescent waves 

gives only part of the picture of the controversy. It is nonetheless worthwile to sketch other 

approaches to the traversal time that originated from the discussion of the `classical' tunnel 

26 

i


e ect. In quantum mechanics, a particle is described in terms of its corresponding complex 

wave function (x; t), which satis es a particular wave equation, the Schrodinger equation: 

~ 2 

2m 

+ j~ @ 

@t 

, V =0: (2.10) 

In this equation, ~ = h=(2 ) is Planck's quantum of action, m is the mass of the particle, and 

V is a usually time-invariant potential. In the absence of transient processes the probability 

density j (x; t)j 2 is independent of time, and one can obtain a simpler version by separating 

the variables and setting 

(x; t) = (x)e ,j!t ; (2.11) 

which gives the time-independent Schrodinger equation or energy eigenequation 

~ 2 

2m 

+ E ,V =0; (2.12) 

where E = !~ is the energy of the particle. Obviously, this equation has one-dimensional 

solutions of the form 

(x) =Ae 

j 1 

~ xp 2m(E,V) : (2.13) 

The amplitude A is a normalisation constant and chosen such that the overall probability density 

of the particle equals one, which means that the particle is guaranteed to be somewhere. 

Hence, 

Z 1 

,1 

(x) (x) dx =1: (2.14) 

pThe case we are interested in occurs whenever E < V , so that the wave number k = 

2m(E , V )=~ becomes imaginary. Then the oscillation of the stationary wave turns into an 

exponential decay and we have an e ect similar to evanescence in the electromagnetic case. 

The simplest and classical arrangement leading to the tunnel e ect is a rectangular potential 

barrier with 

V = 

8 

>< 

>: 

0 if ,1


Remark (Scattering and escaping) The potential barrier considered above isonly 

one possibility where tunnelling can occur. It is a scattering con guration in that an 

individual particle impinging on the barrier can be re ected or transmitted. A stream of 

particles will therefore be scattered. The other scenario is the escape from bound states 

(see, for example, Landauer [79]). A particle caught in a potential well surrounded by 

walls with nite height can leave the well by tunnelling through the wall. The classical 

problem, however, is that of scattering, and so we restrict the survey to the controversy 

on tunnelling times in such processes. 

In the course of time, there were quite a number of survey articles on various de nitions of 

quantum mechanical tunnelling times, and we shall browse here only through the best-known 

approaches. Perhaps the most intuitive one is the phase time. It is based on the method 

of stationary phase and thus related to the group velocity concept of classical waves. The 

motion of the wave packet is again de ned via the evolution of a particular point. Usually the 

maximum is chosen as reference. Hence this concept is limited to narrow-band signals with 

a clearly observable peak like Gaussian pulses (which in this context are also referred to as 

minimum uncertainty pulses). For suchawave packet, Hartman [80] found that the wave does 

not, as had previously been assumed, traverse a potential barrier in no appreciable time, but 

rather experiences a distinct delay that is in priciple independent of the barrier thickness. This 

gave rise to the already mentioned superluminality discussions and also to the Enders-Nimtz 

experiment, although Hartman himself emphasised that the e ect was caused by a frequency 

shift in the wave packet emerging from the tunnel and was thus in perfect agreement with 

causality. Fletcher [81] con rmed these observations for the even simpler case of a cosh ,1 -like 

wave packet. Another variant of the phase time uses the centroid of the wave function rather 

than its peak (Martin and Landauer [61]). Derived from the phase time concept is that of 

group delay, where the propagation delay di erence between two identical wave packets is 

considered | one passing through a barrier, its twin travelling the same distance in free 

space. This approach appears chie y in quantum optics and re ects a classical experimental 

setup (Steinberg and Chiao [18]). 

The phase time concept has been, and still is, very controversial. It has been argued that 

its vividness stems from a too classical understanding of particle motion (Thornber et al. 

[82]) and disregards the wave aspect that is essential for the tunnelling phenomenon. In fact, 

the wave is collapsed into a single point whose trajectory is traced. This is a second point of 

attack: it is by no means clear that the peak of the wave function ahead of the barrier and the 

one behind are identical, let alone relatable in a physically sensible way. On the other side, 

reviews like those by Collins et al. [83], Barker [84], or Hauge and St vneng [85] advocated 

the phase time as appropriate description of the tunnel e ect at least for Gaussian pulses. 

The other group of phase time proponents consists of researchers dealing with electromagnetic 

waves like in quantum optics, as we have already seen in the previous section. Ranfagni et al., 

for example, found the phase time most suitable to reproduce their microwave experiments 

[55]. Nonetheless, particularly in recent articles, the phase time as relevant tunnelling time is 

entirely denied (Landauer and Martin [86]) or restricted to very special cases, i. e. where the 

trajectories of the pulse peaks that undeniably exist outside the barrier can be extrapolated 

into the barrier to give a simple description of the tunnelling event (Leavens et al. [87]). 

28


Remark (Interference in front of the barrier) There is still another subtlety inhibiting 

the straightforward assumption that the phase time gives precisely the time spent 

in the barrier by the particle. Since the phase time (like the group velocity) is asymptotic 

in nature, it requires that the wave packet contains only few frequency (or energy) 

components, which in turn makes the packet very broad in space. When this long packet 

approaches the barrier, a portion of its leading tail will inevitably be re ected and interfere 

with the main body of the pulse. This slows down the approach ofthe pulse, and 

therefore the extrapolated phase time consists of two parts that cannot be separated: the 

delay due to self-interference and the interaction time with the barrier itself [85]. 

Another tunnelling time approach is the dwell time, which is de ned as the ratio of the 

probability density in the barrier to the incident ux J, 

D = 1 

J 

Z 

B 

j (x; t)j 2 dx : (2.17) 

The integration extends over the barrier region and gives the time a particle spends in the 

tunnel, averaged over all incoming particles. The ux or probability density current can be 

derived from a continuity equation for the probability density and is in its most general 

form 

J = j~ , 

r , r 

2m 

: (2.18) 

For the special case of plane waves = Ae jkx , this expression is simpli ed to J = ~k=m. In 

a way this de nition resembles the velocity of energy propagation for classical waves, where 

we had the ratio of transported and stored enery. The dwell time, however, su ers from a 

severe drawback in that it cannot distinguish whether a particle is ultimately re ected or 

transmitted. It has therefore been argued that while it is an appropriate description of the 

mean interaction time of a particle with the barrier, it may not be used to characterise the 

actual transmission time (Buttiker [88], Collins et al. [83]). On the other hand, re ection 

and transmission are mutually exclusive events, and hence one can at least formally de ne 

distinct re ection and transmission times R and T that must be related by the respective 

re ection and transmission coe cients, R + T =1, 

D=T T+R R : (2.19) 

This formulation has become widely accepted, and Hauge and St vneng [85] made it a focal 

criterion of their review article. 

Remark (Objections against (2.19)) The decomposition of the dwell time is not 

undisputed. Landauer and Martin [86] argued that (2.17) and (2.19) imply 

Z 

B 

j (x; t)j 2 dx = JT T +JR R: (2.20) 

This comes down to the addition of probability densities. In quantum mechanics, however, 

it is usual to add only complex wave functions, and therefore (2.19) lacks any physical 

meaning. 

29


The conditional average approach of (2.19) was recently extended by Steinberg [89, 90] on the 

basis of conditional probabilities to include the e ects of real-world measurements. To this 

end, the conventional de nition of the dwell time had to be extended by an imaginary part 

describing the backaction of the measurement device on the tunnelling particle. In addition, 

he found that the major contributions to the barrier interaction time come from the edges of 

the barrier, but hardly from its centre. A transmitted particle therefore spends equal amounts 

of time near either of the interfaces, a re ected particle stays of course only close to the tunnel 

entrance. 

Apart from the phase time de nition there are several other approaches that apply the classical 

concept of trajectories to quantum mechanics. They are derived from di erent formal ways 

to describe quantum mechanics, such as Bohm's trajectory formulation (Leavens and Aers 

[91]) or Feynman paths. The former treats a wave packet always like a particle in the classical 

sense, whereas the latter yields complex tunnelling times, which makes many researches feel 

uneasy because results in quantum mechanics are usually real-valued quantities. An overview 

of the trajectory approaches was given by Landauer and Martin [86] as well as Leavens et al. 

[87]. 

The last group of tunnelling time de nitions is that of quantum clocks. This approach associates 

the motion of the particle with some physical degree of freedom that plays the role of 

a clock and can be used to measure the time elapsed during the tunnelling event. There have 

been several proposals like oscillating barriers, oscillating incident amplitudes or the Larmor 

precession. The idea of the last one is to exploit the precession of the spin of a particle in 

a magnetic eld, provided that the eld is in nitesimally small and con ned to the barrier 

region (Buttiker [88]). The time it takes the particle to tunnel through then determines the 

rotation of the spin. The acceptance of quantum clocks is not unanimous: they were criticised 

as unreliable [85], but also strongly favoured [86], at any rate they are well established today. 

30

Chapter 3 

Wave propagation in 

electromagnetic transmission lines 

One cannot escape the feeling that these mathematical formulae have an 

independent existence of their own, that they are wiser than we are, wiser 

even than their discoverers, that we get more out of them than was originally 

put into them. Heinrich Hertz, quoted in [1] 

As the historical review shows, opinions are divided on the question of wave propagation. 

To gain some further insight, we shall explore several examples in the sequel, ranging from 

the simple case of a homogeneous transmission line to transient phenomena in wave guides. 

The main interest of the analysis lies on signals in the evanescent region or stop band of 

transmission lines, and among the many de nitions of the propagation velocity, we shall 

concentrate on that of energy velocity. Not only is its de nition clear-cut and intuitive, we 

shall also nd that it gives reasonable results where the concept of group velocity must fail. 

At the beginning of the chapter, we study the homogeneous and lossless transmission line 

in the pass band and con rm the equality of group and energy velocity. We then digress 

to explore a classical delay line known from the literature, where this identity also holds. 

Gradually complicating the model, we consider evanescence in connection with a mismatched 

termination of a nitely long transmission line. The logical extension of this investigation is to 

consider a frequency-dependent medium. We thus evaluate the behaviour of waves in a lossless 

plasma. A short section on an inhomogeneous transmission line concludes the treatment of 

monochromatic signals. A large part is then devoted to the evaluation of transients in a 

plasma and a rectangular wave guide. In both cases we shall encounter the propagation of a 

wave front, whose velocity will be equal to the velocity of light. The remainder of the chapter 

explores the propagation of a Gaussian pulse in plasma. In connection with this comparatively 

small-band signal, we shall also touch upon the question of causality inevanescent media. 

31

U 

3Wave propagation in electromagnetic transmission lines 

I 

jX 0 dx 

dx 

jB 0 dx 

Figure 3.1: Equivalent circuit for a lossless transmission line 

3.1 Model of a transmission line 

In section 1.4 we already mentioned that the energy velocity equals the group velocity for lossless 

media and for narrow-band signals. Let us now verify this statement for a simple example. 

We consider a general homogeneous transmission line without losses. In addition, we assume 

an ideal termination with the characteristic impedance, so that we need not worry about 

re ections. The transmission line can be described by means of its well-known equivalent 

circuit as depicted in g. 3.1 with arbitrary distributed reactances X 0 (!) and susceptances 

B 0 (!). The impedances may exhibit any dependence on the frequency provided that they 

satisfy Foster's theorem [92] 

dX 0 

d! 

dB 0 

d! 

jX 0 j 

! 

jB 0 j 

! 

(3.1) 

which expresses the physical necessity that the energy stored in such an impedance is always 

non-negative. The di erential equations for voltage and current are obtained as 

and may be solved by setting 

dU 

dx = ,jX0 I 

dI 

dx = ,jB0 U 

U = Ue j(!t,kx) 

I = Ie j(!t,kx) : 

Remark (Time dependence of the solutions) Note that in contrast to the preceeding 

chapters, the waves in (3.3) have been formulated with a time dependence e j!t . Throughout 

the case studies, we shall use this standard electrodynamical notation because it is 

32 

(3.2) 

(3.3)

3.1 Model of a transmission line 

the foundation of the well-known correspondence @ 

@t 7! j! for stationary oscillations and 

consequently the basis for the equivalent circuit in g. 3.1. 

We then obtain easily the propagation constant 

and the characteristic impedance 

k = p X 0 B 0 (3.4) 

Z0 = U 

I = 

r X 0 

: (3.5) 

B0 Wave propagation is possible only if X 0 B 0 > 0, i. e. the signs of the distributed reactances 

and susceptances must be equal. From the dispersion relation (3.4) we nd the reciprocal of 

the group velocity 

1 

vg 

= dk 

d! 

= 1 

2k 

X0 dB0 

d! 

+ B0 dX0 

d! 

: (3.6) 

Remark (Direction of propagation) As can be seen from (3.6), X 0 > 0 and B 0 > 0 

result in sign vp = sign vg, so that the wave crests and wave packets move in the same 

direction. If we choose negative impedances, X 0 < 0 and B 0 < 0, respectively, the phase 

and group velocities have opposite signs, which ischaracteristic for a backward wave. 

In the absence of re ection, which was our initial assumption, the average of the propagated 

energy is simply determined by the amplitude of the current I and the characteristic 

impedance, 

P = I2 

2 

r X 0 

: (3.7) 

B0 On the other hand, the mean stored energy per unit length, with U as voltage amplitude, is 

given by 

W = 1 

4 

2 dX0 2 dB0 

I + U 

d! d! 

I2 dX0 dB0 

= B0 + X0 

4B0 d! d! 

; (3.8) 

where we used the de nition of the characteristic impedance (3.5) to obtain the second expression. 

From (3.7) and (3.8) we see immediately that the energy velocity ve = P=W equals 

the group velocity (3.6). 

Now let us brie y examine the situation when the transmission line is characterised by a 

frequency-independent inductance L 0 and a capacitance C 0 per unit length. We then have 

X 0 = !L 0 ; B 0 = !C 0 : (3.9) 

33


Losses are accounted for by a series resistance R 0 andashunt conductance G 0 . The propagation 

constant aswell as the characteristic impedance then become complex, 

k = p (!L 0 , jR 0 )(!C 0 , jG 0 )= +j (3.10) 

Z 0 = 

s !L 0 , jR 0 

!C 0 , jG 0 = Z0 + jX0 : (3.11) 

With these de nitions, we nd the propagated and stored energies 

P =Re UI 

2 

W = 1 

4 

and subsequently the energy velocity 

= 1 

2 I2 Z0 

, I 2 L 0 + U 2 C 0 = 1 

4 I2 , L 0 + jZ 0j 2 C 0 

ve = P 

W = 

(3.12) 

(3.13) 

2Z0 

L0 +(Z0 2 +X0 2 : (3.14) 

)C0 We can now use the relation k = Z 0(!C 0 , jG 0 ) to express Z0 in terms of and . Furthermore, 

we take the square of (3.10) to obtain an expression for . The absolute value of Z 0 

is readily found from (3.11). Taking all these relations together and inserting them into the 

expression for the energy velocity, we get after some algebra the astonishing result 

ve = ! ; (3.15) 

which is the phase velocity de ned by the real part of the complex dispersion relation. This 

observation has also been reported by Mainardi [32] (see section 1.4). Needless to emphasise 

that in the absence of losses (R 0 = G 0 = 0), the phase, group, and energy velocities are 

identical, and no dispersion occurs. In this connection and for the sake of completeness, we 

also recall the well-known condition R 0 C 0 = L 0 G 0 for a dispersion-free transmission line even 

in the presence of losses. 

Remark (Frequency dependence of losses) At rst sight, the propagation constant 

(3.10) reveals an interesting property for ! !1. Within the scope of this model, the 

real part is roughly proportional to !, whereas the imaginary part remains constant. 

This indicates the existence of a distinct wave front velocity (see section 1.3). Thus a 

step in a signal may be attenuated, but the wave front itself would never be distorted. 

Unfortunately, the assumption of frequency-independent losses are no longer justi ed 

at high frequencies. In fact, the series resistance is increased by the skin e ect (thus 

R 0 / p !), whereas the shunt conductance is deteriorated by dielectric losses (G 0 / !). 

Consequently, the imaginary part will increase with growing frequency, and the wave 

front will be distorted. 

An analysis based on an equivalent circuit is incomplete without discussion of its limits of 

validity. The derivation of the equivalent circuit relies on the fact that adjacent in nitesimal 

34

3.2 Excursion: a delay line 

sections of the transmission line do not in uence each other. As the voltage and current in the 

line stand for the electric and magnetic elds, respectively, this is tantamount to the exclusive 

occurence of TEM waves [93]. A slightly less restrictive de nition based on the alternative 

description of the elds by a scalar and a vector potential was given by Paschke [92]. The use 

of the equivalent circuit is then justi ed when the contribution of the vector potential to the 

electric eld is negligible compared with that of the scalar potential, so that the electric eld 

is almost curl-free (r E ' 0). Apart from TEM waves, where the requirement is identically 

satis ed, it is a good approximation also for slow waves characterised by vp c. 

3.2 Excursion: a delay line 

As a second example, we review now the model of a delay line shown in g. 3.2 , which was 

already analysed by Borgnis [38] as well as Kleen and Poschl [39]. The walls of the wave 

guide consist of conducting combs with narrowly spaced teeth. For the sake of analytical 

simplicity, we assume that the extention in z-direction is in nite and all eld components in 

this direction are constant. In addition, since the boundaries are perfectly conducting, there 

is no component of the electric eld in z-direction. The magnetic eld, on the contrary, does 

exist in this direction only. Hence we have 

E z=0; H x=H y=0; 

@ 

@z 

=0: (3.16) 

With these restrictions and the assumption of a linear and isotropic medium, the Maxwell 

equations reduce to (from r H =@D=@t) 

and (from r E=,@B=@t) 

@Hz @y = " @Ex @t 

; 

@Hz @x = ," @Ey @t 

@Ey @x , @Ex @y = , @Hz @t 

(3.17) 

: (3.18) 

Note that the other two Maxwell equations are also satis ed. It follows directly from our 

assumptions (3.16) that rH = 0, and rE = 0 can be shown by di erentiating this equation 

with respect to t and those of (3.17) with respect to x and y. 

The di erential equations can be combined to yield a wave equation for the transverse magnetic 

component 

@2H z 

@x2 + @2Hz @y2 = " @2H z 

@t2 : (3.19) 

Like before, we set H z = Hze j(!t,kx) , and with c 2 =1=( ") and = p k 2 , (!=c) 2 we nally 

obtain the two possible solutions for the amplitude of the magnetic eld, an antisymmetric 

and a symmetric mode, 

Hz = C sinh y ; Hz = C cosh y (3.20) 

35

z 

y 

x 

l 

2d 


Figure 3.2: Structure of the delay line. 

as well as the corresponding electric eld components 

Ey = kC 

!" sinh y ; Ey = kC 

!" 

Ex = C 

j!" cosh y ; Ex = C 

j!" 

cosh y ; (3.21) 

sinh y : (3.22) 

As for the boundary conditions, the conducting combs act like a shorted strip line of length 

l, and therefore the elds see at y = d the corresponding well-known impedance. This 

structure was quite commonly used and is known as ìnductive wall'. The boundary condition 

then becomes 

, Ex 

r 

= jX(!) =j 

Hz y=d 

" tan !l 

: (3.23) 

c 

With the normalisation K = kd, = !d=c, and L = l=d we nally obtain the implicit 

dispersion relations for the antisymmetric mode, 

p p 

K2 , 2 coth K2 , 2 = tan( L) (3.24) 

and for the symmetric mode 

p p 

K2 , 2 tanh K2 , 2 = tan( L) : (3.25) 

The numerical evaluation of these relations for the base band is given in g. 3.3 . Note that 

the boundary condition (3.23) relies on the fact that the teeth of the combs are in nitely close 

together. If this distance is not small compared to the wavelength, then spatial harmonics 

would have tobetaken into account. 

For the calculation of the stored energy it is important to consider not only the space jyj d 

where the propagation occurs, but also the inductive walls d jyj d + l. It was this 

component that Borgnis [38] disregarded in his rst analysis and that led to the false result 

ve = vp. The average energy stored in the upper wall is given by 

W w = 1 

4 

jHzj 2 

36 

y=d 

dX 

d! 

(3.26)

3.3 Re ection due to termination mismatch 

1.5 

1.25 

1 

0.75 

0.5 

0.25 

Ω 

0 2 4 6 8 10 12 14 

Figure 3.3: Dispersion relations for the delay line of g. 3.2 for L =1. The solid curve corresponds 

to the symmetric low-pass mode, the dashed curve to the antisymmetric band-pass mode. The thin 

lines are the linear approximations for low and high frequencies. Note that the diagonal line marks 

the velocity of light, =K = 1. Thus the band-pass mode exhibits a phase velocity greater than c 

slightly above the lower frequency limit. It is clear to see that the relation vp vg = c 2 mentioned in 

the previous chapter is not valid here. 

with X(!) taken from (3.23). The proof of the identity ve = vg is then straightforward but 

tedious. 

3.3 Re ection due to termination mismatch 

If we want to examine evanescence, it is not very sensible to consider a steady-state excitation 

of the transmission line without re ection. In such a case, the characteristic impedance (3.5) 

as well as the wave number (3.4) are imaginary, resulting in total internal re ection, but no 

energy transport. Therefore we deem it meaningful to study the simple example of re ection 

due to an ohmic load. The transmission line is again described by the equivalent circuit of 

g. 3.1, but has now a nite length and a terminating resistor ( g. 3.4). 

Before we start with evanescence, let us brie y investigate the behaviour of the pass band 

to gain some physical insight. If we solve the di erential equations (3.2) for the boundary 

condition U2 = RI2 at the end of the line, we obtain the well-known spatial voltage and 

current evolution with respect to the voltage across the load, 

U = U2 cos k(l , x)+j Z0 

R 

I = U2 

Z0 

Z0 

R 

sin k(l , x) 

cos k(l , x)+jsin k(l , x) : 

37 

K 

(3.27)

I 0 


Figure 3.4: Transmission line with real-valued termination 

With these results we can calculate the averaged propagated and stored energies 

P =Re UI 

2 

; W = UU 

4 

dB 0 

d! 

l 

+ II 

4 

which after some simple manipulations yields the velocity of energy transport 

ve = 

x 

dX 0 

d! 

R 

; (3.28) 

2k 

, 2 sin k(l , x)+ 2cos2 k(l , x) B0 dX0 d! + , cos2 k(l , x)+ 2sin2 k(l , x) X0 dB0 d! 

(3.29) 

with the termination factor = Z0=R. This complicated expression is reduced dramatically 

if we evaluate it for the special case of a dispersionless line with 

Then (3.29) becomes, with 1=c = p L 0 C 0 , 

B 0 = !C 0 ; X 0 = !L 0 : (3.30) 

ve 

c 

= 2 Z0 

R 

1+ Z0 

R 

If we introduce the re ection factor of the current 

r = 

we can rewrite (3.31) in a more plausible way, 

ve 

c 

Z0 

R 

Z0 

R 

, 1 

: (3.31) 

+1 ; (3.32) 

1 , r2 

= : (3.33) 

1+r2 This invokes the picture of an overall transport velocity weighted by the power the waves 

carry: the incident wave transports a certain amount of energy per unit time, P0, whereas the 

re ected wave carries a part r 2 P0 of it back. Apart from this interpretation, (3.33) mirrors 

exactly the basic idea of the energy velocity de nition: a net propagated energy per unit 

length divided by the total energy content of that unit length. We notice that the energy 

velocity isnot equal to the group velocity vg = c. In the limit of a matched termination r =0, 

however, we haveve=vg. Expression (3.33) also holds for an unterminated or short-circuited 

transmission line R = 1 or R = 0, thus r 2 = 1, where ve = 0 indicates that an energy 

transport is impossible at all. 

38

3.4 A simple thought experiment 

I0 

Z0 

Figure 3.5: Transmission line charged with a short input current pulse. 

3.4 A simple thought experiment 

To judge whether (3.31) is indeed meaningful, we explore yet another example. Consider 

the terminated transmission line like before, but with a pulse source rather than with a 

monochromatic excitation ( g. 3.5). Let the source emit a current pulse with a duration 

T l=c much shorter than the propagation time needed to reach the end of the line. Then 

a total energy 

W1 = I0 2 TZ0 

l 

R 

(3.34) 

will be transmitted to the load after a propagation delay = l=c. It will, however, only partly 

be absorbed due to partial re ection. The re ected part will be entirely re ected in turn by 

the ideal in nite internal impedance of the source. Therefore, the amount of energy that is 

by and by absorbed by the load can be expressed by a series of contributions 

W (t) =I0 2 T(1 + r) 2 R , (t , )+r 2 (t,3 )+r 4 (t,5 )+::: (3.35) 

with the step function (t). The current (1 + r)I0 is the superposition of the incident and 

re ected wave seen by the load resistance. For t !1,the in nite series in (3.35) of course 

gives (3.34). The de nition of an energy velocity in this case is more or less arbitrary, but 

may in a sensible way be given by 

ve = l 

e 

; W ( e) 

W1 

1 , 1 

; (3.36) 

e 

where e is the instance when a given portion of the total energy has been absorbed, the 

remainder being determined by the base e of the natural logarithm. Thus the time-dependent 

function of (3.35), 

f(t) = 

1X 

i=0 

(t , (2i +1) )r 2i ; (3.37) 

is evaluated only to the index i = N where this condition is rst met. Since the summation 

starts with index i = 0, we have at this point N +1 terms included. Knowing that 

limt!1 f(t) =f(1)=1=(1 , r 2 ) and with the summation rule for nite geometric series, we 

obtain 

f( e) 

f(1) = W ( e) 

=1,r 

W1 

2(N+1) ; e =(2N+1) : (3.38) 

39

ve/c 

1 

0.8 

0.6 

0.4 

0.2 

0 

2.5 5 7.5 10 12.5 15 17.5 20 Z0/R 


ve/c 

1 

0.8 

0.6 

0.4 

0.2 

0 

0.2 0.4 0.6 0.8 1 Z0/R 

Figure 3.6: Comparison of the energy velocities obtained for steady-state signals and short pulses. 

The energy velocity thus becomes 

ve 

c = 

1 

: (3.39) 

2N +1 

The results of (3.39) compared with the monochromatic case (3.31) are shown in g. 3.6 . 

The correspondence is satisfactory for a relatively bad termination, i. e. when Z0=R 1 or 

Z0=R 1. If the termination is close to its optimum Z0=R 1, the energy transfer will 

surpass the limit (3.36) with the rst time the pulse reaches the resistor. 

Remark (Relation between steady-state and pulse signal) The good correspondence 

between the curves in g. 3.6 is by no means fortuitous. It is rather due to the 

careful choice of the energy limit in (3.36). In fact, the value 1=e for the still missing 

part of the transferred energy gives exactly the time constant of the corresponding quasistationary 

model. In such an approach the e ect of the re ections is collapsed into one 

single parameter determining the overall shape of the output signal just like the charging 

of a capacitance. Thus the underlying process of wave propagation completely disappears, 

and the curve is smooth, behaving like 1,e ,t= e . The exact model that accounts for the 

wave propagation converges to the quasi-stationary model when there are many re ections 

or only small steps in the output signal, i. e. when the termination is far away from the 

optimum. 

The reasoning just given is of course not the whole truth. After all, the smooth curve in 

the gure was drawn using (3.31), which was derived from a steady-state signal. The pulse 

of the example, however, can be seen as the superposition of monochromatic components. 

From Parseval's theorem we know that 

Z 1 

,1 

ji(t)j 2 dt = 1 

2 

Z 1 

jI(!)j 

,1 

2 d! ; (3.40) 

which means that the energy propagation of the signal is determined by the energy of its 

frequency components. If these components travel with a frequency-independent velocity 

ve as in our example, then so does the entire energy of the pulse. 

40

3.5 A dispersive system: the lossless plasma 


After the above re ections to ensure the plausibility of (3.29), we may apply it to dispersive 

and evanescent systems. Remaining with our transmission line model, we assume it this time 

to be lled with a lossless plasma. This has no e ect on the wave type, but only on its 

propagation velocity. We still consider a TEM wave, and so the equivalent circuit is still 

valid, although c becomes frequency-dependent. If we neglect the motion of ions and focus 

on the contribution of the electrons, the relative dielectric constant is given by 

with the plasma frequency [94] 

"r =1, !p 

! 

!p = 

s q 2 n0 

"0m 

2 

(3.41) 

: (3.42) 

In this formula, q is the elementary charge, n0 the electron density, m the electron mass and 

"0 the dielectric constant ofvacuum. Hence we have 

and the dispersion relation 

X 0 =!L 0 ; B 0 = !C 0 1 , !p 

! 

k(!) = !p 

c 

s ! 

!p 

2 

2 

; c = 1 

p L 0 C 0 

(3.43) 

,1: (3.44) 

We see immediately that the wavenumber becomes imaginary in the evanescent region !

whereas the energy velocity in the evanescent region is 

ve 

c = 

,1 , 2 + , !p 

! 

2 + , !p 

! 

If we chose a matched termination 

then (3.46) would reduce to 


2 

, !p 

! 

2 ,1+ 2 + , !p 

! 

R = Z0 = 

ve 

c = 

r L 0 

s 

C 0 

2 , 1 

2 cosh 

1 

q 1 , , !p 

! 

1 , !p 

! 

2 

2 

2(X , L) 

r 

1 , ! 

!p 

! : (3.47) 

2 

; (3.48) 

; (3.49) 

which is exactly the group velocity vg = d!=dk referred to the speed of light. Numerical 

examples of the energy velocity in dependence on the position and the termination ratio are 

given in g. 3.7 for the pass band as well as in g. 3.8 for the evanescent region. It is worth 

noticing that (3.46) and (3.47) are continuous at the plasma frequency. They do not exhibit 

a zero, but the nite value 

ve(!p) 

c 

= 2 

2+ 

2 : (3.50) 

The gures clearly show that the energy velocity depends on the position along the transmission 

line. As could have been expected, the local velocity in the pass band oscillates 

uniformly. In the stop band, however, the velocity nearly vanishes almost everywhere, but 

increases sharply towards the end of the line within a characteristic distance that is practically 

independent of the signal frequency. This behaviour stems from the fact that the stored 

energy per unit length in the transmission line decays exponentially, whereas the transmitted 

energy remains constant due to the absence of losses. Consequently, the energy velocity must 

rise as the wave approaches the end of the line. 

From a macroscopic point of view, the local energy velocity is of only minor importance | 

an overall energy propagation time through the transmission line is much more interesting. 

This can be de ned via 

e = 

Z l 

and we obtain an è ective' velocity like in (3.36), 

0 

1 

ve 

ve;e = l 

42 

e 

dx ; (3.51) 

: (3.52)


0 

0.66 

ve/c 0.4 .4 

0.2 .2 

0 

100 

2 

4 

X 

6 

10 

8 

10 

1 

ρ 

0.1 

0.01 

Figure 3.7: Evaluation of the energy velocity (3.46) in the pass band of a lossless plasma depending 

on the spatial coordinate X and the termination factor . The length of the line is L = 10, the signal 

frequency is given by !p=! =0:8. Note the partially logarithmic scale. 

0 

100 

2 

10 

4 

X 

6 

1 

ρ 

8 

10 

0.1 

0.01 

0.2 

0 

0.6 

0.4 

ve/c 

Figure 3.8: Evaluation of the energy velocity (3.47) in the evanescent region of a lossless plasma 

depending on the spatial coordinate X and the termination factor . The parameters are L = 10 and 

!p=! =1:25 . Note the partially logarithmic scale. 

43

Straightforward integration yields for the pass band 

ve;e 

c = 

and for the stop band 

ve;e 

c = 

1+ 2 , , !p 

! 

,1 , 2 + , !p 

! 

2 + , !p 

! 


2 + , !p 

! 

2 

2 1 , , !p 

! 

2 

2 1, 2 , , !p 

! 

, !p 

! 

2 , 1 

2 ,1+ 2 + , !p 

! 

2 

2 

r 

sin 2L 

!!p 

r 

2L 

!!p 

r 

sinh 2L 1, ! 

!p r 2 

2L 1, ! 

!p 

2 

,1 

2 

,1 

2 

(3.53) 

: (3.54) 

These velocities are depicted in g. 3.9 and g. 3.10, respectively. We see that if the line is 

terminated with the characteristic impedance, = p 1 , (!p=!) 2 , the energy velocity in the 

pass band is independent ofLand equal to the group velocity. Apart from very short lines, 

the velocity is virtually independent of the length. 

Remark (Velocity maximum) It is interesting to notice that for short lines, the 

maximum velocity of energy propagation does not occur when the termination is optimal, 

but for larger values of . For the example in g. 3.9, the value would be m 1:28 in 

the limit L ! 0. It is not clear where this behaviour comes from. 

A similar e ect appears for evanescence. Although a matched termination is not de ned 

in this case, one might expect the velocity maximum to be related to the propagating 

mode. In fact, for L ! 0 the value where the optimum is reached in g. 3.10 is m 1:6, 

which has nothing p whatsoever to do with any other parameter. For large values of !p=!, 

we nd m !p=!. 

The result for the stop band ( g. 3.10) seems very plausible. Evanescence slows down the 

energy transfer, and the energy velocity gradually diminishes. At any rate, energy transport 

through an evanescent region is possible even in the strictly monochromatic case, when we 

can guarantee that the spectrum of the signal is con ned to the stop band | unlike signals 

with a broader spectral width, where we must always worry about components lying in the 

pass band and thus distorting the result. On the other hand, no energy can be transferred if 

the line is open ( ! 0) or shorted ( !1), which naturally re ects only our precondition 

that the energy is to be dissipated at the termination resistance. 

Remark (Formal equivalence to a wave guide) The results obtained for the transmission 

line with a lossless plasma are formally identical with the H01 mode of a rectangular 

wave guide with width b. The respective boundary conditions are that the excitation 

at x = 0 is monomodal and that at x = l, 

, Ez(l) 

= R; (3.55) 

Hy(l) 

when x is the direction of propagation. The results from (3.46) onwards then apply with 

the replacements !p 7! c=b and p L 0 =C 0 7! p =". 

44


0 

L 

2 

4 

6 

8 

10 

0.01 

0.1 

1 

ρ 

10 

0 

100 

0.6 0 

0.4 0. ve/c 

Figure 3.9: Evaluation of the è ective' energy velocity (3.53) in the pass band of a lossless plasma 

depending on length L and the termination factor . The signal frequency is !p=! =0:8. Note the 

partially logarithmic scale. 

0.01 

0.6 

0.4 

ve/c 

0.2 

0 

0 

0.1 

2 

ρ 

1 

4 

10 

L 

100 

6 

Figure 3.10: Evaluation of the è ective' energy velocity (3.54) in the evanescent region of a lossless 

plasma depending on the length L and the termination factor for !p=! =1:25. Note the partially 

logarithmic scale. 

45 

8 

10 

0.2

3.6 Inhomogeneous transmission line 


So far, the transmission lines we investigated were homogeneous, so that their parameters did 

not depend on the position along the line. We now extend our analysis to inhomogeneous 

transmission lines and brie y sketch the treatment of a so-called exponential line. Such 

transmission lines are not only of academic interest | they have actually been used in the early 

days of integrated circuit design for impedance transformation [95]. The equivalent circuit 

of g. 3.1 and the respective di erential equations (3.2) are still valid, but the distributed 

reactance and susceptance are now position-dependent, 

X 0 = !L0 0 e x ; B 0 = !C0 0 e , x ; c = 

1 

p 

L0 0 : (3.56) 

0 

C0 

Inserting solutions of the form (3.3) into the di erential equations yields a propagation constant 

0 

k = @j 

2 

s 

2! 

c 

1 

2 

, 1A 

(3.57) 

and a complex, position-dependent wave impedance 

Z 0 = 

0 

@,j 0 + 

2!C0 

With the abbreviations and normalisations 

= 2! 

c 

s L0 0 

C0 0 , 2!C0 0 

1 

2 

A e x : (3.58) 

; X = x ; L = l ; (3.59) 

the general solution in the pass band consists of a right- and left-going wave for the voltage 

and a corresponding current 

U = K e X 2 (1,j p 2 ,1)+j!t +(U0 ,K)e X 2(1+j p 2 ,1)+j!t 

I = j2!C0 0 K= 

1+j p e 

2 ,1 ,X 2(1+j p 2 ,1)+j!t j2!C0 

+ 0 (U0 , K)= 

1 , j p 2 , 1 

e , X 2 (1,j p 2 ,1)+j!t 

(3.60) 

(3.61) 

with a complex constant K that must be determined to comply with the boundary conditions. 

With the de nitions of the propagated energy, 

and stored energy, 

W = 1 

4 

P = 1 

Re U I ; (3.62) 

2 

I I dX0 

d! 

46 

+ U U dB0 

d! 

; (3.63)

3.6 Inhomogeneous transmission line 

0 

1 

0.75 75 

ve/c 

0.5 .5 

0.25 25 

0 

100 

2 

X 

4 

10 

6 

8 

1 

ρ 

0.1 

0.01 

Figure 3.11: Evaluation of the energy velocity (3.65) in the pass band of an exponential line depending 

on the spatial coordinate X and the termination factor . The length of the line is L = 8, the signal 

frequency is given by 1= =0:5. Note the partially logarithmic scale. 

we nd for the special case of a matched termination the energy velocity 

ve 

c = 

p 

2 , 1 

; (3.64) 

which is exactly the group velocity we would have obtained from the dispersion relation. 

If we can manage to terminate the transmission line with a real-valued multiple of the 

characteristic impedance Z0, the energy velocity within a line of length l in the pass band, 

2 > 1, becomes after a lengthy calculation 

q 

( 2 3 

, 1) 

ve 

c = 

2 

2 (1 + 2 ) , 2 , ( , 1) 2 cos e X , ( 2 , 1) sin e X ; (3.65) 

with the abbreviations = p 2 , 1 and e X = X , L. In the stop band, where 2 < 1, 

a termination with the wave impedance makes no sense because no energy could then be 

transmitted due to evanescence, so we assume a purely ohmic termination R. With the 

settings = p 1 , 2 and r = R= p L0 0 =C0 0 ,we obtain the corresponding energy velocity 

ve 

c = 

r , 1 , 2 

(cosh e X + sinh e X , 2 )e L + r 2 (cosh e X , sinh e X , 2 )e ,L 

47 

: (3.66)

0 

100000 

2 

10000 

X 

4 

1000 

6 

r 

8 

100 


10 

1 

0.2 

0 

0.4 

ve/c 

Figure 3.12: Evaluation of the energy velocity (3.66) in the stop band of an exponential line depending 

on the spatial coordinate X and the termination factor r. The length of the line is L = 8, the signal 

frequency is given by =0:5. Note the partially logarithmic scale. 

The energy velocity in the pass band is depicted in g. 3.11. Its dependence on the position 

along the line and the termination is very similar to that in a lossless plasma ( g. 3.7). Except 

for matched termination, = 1, spatial harmonics appear in the local velocity. The situation 

in the stop band, however, is a bit di erent, as can be seen by comparing g. 3.12 and g. 3.8. 

While the local velocity still increases towards the end of the line, its maximum | at least in 

a broad range of termination factors | is no longer located directly at x = l but signi cantly 

ahead of this point. Note that in this case, r = 1 has nothing at all to do with matched 

termination. Like in the previous example, we could eliminate the position dependence by 

introducing the e ective velocity (3.51). However, this provides little additional insight and 

has therefore been omitted. 

3.7 Turn-on e ects in a lossless plasma 

In the examples investigated so far we restricted ourselves predominantly to monochromatic 

waves, which is an instructive special case and easy to analyse. On the other hand, we usually 

apply waves to transmit signals, and as a strictly monochromatic wave cannot carry any 

information, this case has no practical relevance. What we need instead, is a modulation of the 

wave to form pulses we can identify with the information. A simple way to do so is to switch 

48


a carrier oscillation on and o , which is in fact a crude amplitude modulation also known as 

on-o -keying. Such apulse can then transport one bit of information. Owing to its sharply 

limited duration, however, its spectrum now spans a broad range of frequencies. Irrespective 

of any measures taken to limit the spectrum, it still widens as the pulse is shortened. Hence 

the narrow-band approximation quickly becomes senseless. 

Remark (Relation between spectral and pulse width) Like in quantum mechanics, 

an inequality can be given that establishes a lower bound for the product of a pulse 

p x 2 , x 2 

duration t and the width of the dominant peak of its spectrum !. If x = 

denotes the variance of the respective intensities (i. e. the squared signal value and the 

power spectrum), then this uncertainty principle reads 

holds for Gaussian distributions [11]. 

t ! 1=2. The equal sign 

We now turn to a fairly classical problem: the propagation of a pulsed wave in a dispersive 

medium. This has been scrutinised over and over again, beginning with Sommerfeld [12] and 

Brillouin [19]. Yet we shall add a new facet to the plethora of results. 

The model we examine is again that of a lossless plasma like in the previous section. But 

this time we leave it unbounded for x>0 so that we need not bother about re ections. We 

also adhere to the transmission line model of g. 3.1 . In principle we could as well calculate 

the electric and magnetic elds in the plasma directly, but as we are concerned only with the 

basic TEM wave, we prefer to use the convenient and proven equivalent circuit. The voltage 

then corresponds for instance to the electric eld vector Ey and the current to the magnetic 

component Hz, ifxis the direction of propagation. All other components do not exist. 

The plasma shall be excited by a monochromatic current source being switched on at t =0 

and turned o an integer multiple of periods later, at t = =2n , 

I(0;t)= 

8 

>< 

>: 

0 if t 

: (3.67) 

The latter requirement entails no loss of generality but only facilitates the analytical treatment. 

It is evident that due to the linearity of the model, the pulse can also be described as 

superposition of two delayed step functions of opposite sign, 

I(0;t)=I0 (t) cos !0t , I0 (t , ) cos !0t ; (3.68) 

such that we may restrict our analysis to the step response of the system. Similar investigations 

were carried out for example by Knop [96] or Haskell and Case [97]. In contrast to 

the approach pursued here, they used modulated sine waves, whereas we consider a cosine 

wave, which has the convenient side bene t that the wave front of the signal can be identi ed 

without any doubt. 

Remark (A glance at the literature) The original articles of Sommerfeld and Brillouin 

were concerned with the propagation of light through a resonant Lorentz-type medium 

49


and had been of purely academic interest at that time, as Brillouin himself admitted [2]. 

The work on wave propagation in plasma, however, was stimulated by the research on the 

properties of the ionosphere, which plays an important role in the transmission of radio 

signals. A survey of this work and the dispersive e ects in the ionosphere was given by 

McIntosh and Malaga [98]. 

In the previous chapter we mentioned microwave experiments that were thought to prove 

superluminal wave propagation in the evanescent region of a wave guide or plasma that 

exhibits the same dispersion properties. As a follow-up to this discussion we choose the 

frequency of our signal source to be below the cuto frequency of the plasma. Of course the 

spectrum of the pulse will extend into the pass band, however the spectral peak about the 

carrier frequency can be con ned to the stop band. 

At rst, we seek the steady-state solution of the wave. With a strictly monochromatic source 

I(0;t)=I0cos !0t, wewould obtain a current 

where 0 is the imaginary part of the wave number at !0, 

Is 

I0 

0 = !p 

c 

= e , 0x cos !0t ; (3.69) 

s 

1 , !0 

!p 

2 

: (3.70) 

Bearing in mind that the characteristic impedance is imaginary, we readily obtain the voltage 

from U = Z0I with the relation Re je j!t = , sin !t, 

Us p 

I0 L0 =C0 = , 

e , 0x 

r 

2 

!p 

!0 

, 1 

sin !0t : (3.71) 

If the signal source is never turned o , then the complete expressions of voltage and current 

must converge to this steady-state solution for t !1. To get the e ect of turn-on, we need 

additional transient solutions such that the wave complies with the initial conditions for t =0 

I(x; 0) = 0 ; U(x; 0) = 0 ; x>0: (3.72) 

The voltage in (3.71) already satis es this requirement, the current does not. Hence we resort 

to a little trick: we add a function that cancels the current everywhere except for the origin 

x = 0, where the wave front I0 emerges at t = 0. We construct this desired function by 

expanding the exponential function ,e , 0x antisymmetrically into the negative half space 

x


Each of the partial waves that make up the function has a distinct frequency, just like the 

steady-state solution (3.69). We can thus immediately write down the transient part of the 

current, 

It 

= , 2 Z 1 

sin x cos !( )t 

d : (3.74) 

I0 

0 

2 + 0 2 

Note that in (3.74) we used a spatial decomposition of the transient solution and not a 

temporal one as might have been expected. This is the second signi cant di erence of the 

present approach as compared with those known from the literature. All available publications 

start from the calculation of the frequency spectrum of the initial conditions. By contrast, 

we have determined the wave number spectrum of the boundary conditions. Whether one 

chooses one possibility or the other is essentially a matter of opinion. In our special case, the 

wave number method was particularly appealing because the transient solution could readily 

be found. 

We still need the transient solution of the voltage. To that end we recall the di erential 

equations that relate current and voltage in the evanescent region, 

@U 

@x 

= ,L0@I 

@t 

@I !p 

= C0 

@x 

If we insert (3.74) in the rst one, we obtain 

Ut 

U0L0 = 2 Z 1 

0 

! 

2 

, 1 @U 

@t : 

(3.75) 

!( ) cos x sin !( )t 

d : (3.76) 

2 + 0 2 

We recognise that the transient voltage also complies with the boundary conditions (3.72) 

and vanishes everywhere for t =0. The dispersion relation !( ) used in (3.74) and (3.76) is 

q 

!( )= !p 2 +( c) 2 ; (3.77) 

which is readily obtained by inverting (3.44). 

We havenow collected all components of the wave and can put together the complete solution. 

For the numerical evaluation of the wave functions, however, it is advisable to dispense with 

the small physical constants and to use scaled variables. To this end we introduce a normalised 

integration variable 

= c 

!p 

(3.78) 

and scale the space and time coordinates to the wavelength and period of the plasma frequency, 

T = !pt (3.79) 

X = !px 

c 

= !0 

!p 

51 

(3.80) 

: (3.81)


After some simple rearrangements, we nally obtain the complete expression for the current 

along the line 

I 

I0 

Likewise, we nd the voltage 

I0 

= e ,Xp1, 2 

cos T , 

, 2 Z 1 

0 

sin X cos T p 1+ 2 

1+ 2 , 2 

U 

p 

L0 =C0 = ,e,Xp 1, 2 1 

p sin T + 

1= 2 , 1 

+ 2 Z 1 

0 

d : 

p 1+ 2 cos X sin T p 1+ 2 

1+ 2 , 2 

d : 

(3.82) 

(3.83) 

Before giving numerical evaluations of the above expressions, we take a look at the their 

range of validity. The lower end of the evanescence region is !0 = 0, i. e. a DC excitation. It 

can be readily seen by setting =0that this raises no particular problem, and the special 

case of a non-oscillating step source is covered by our model as we could have expected. 

Things are di erent for a source oscillating with the plasma frequency !p, which is exactly 

the boundary between evanescent and propagation mode. The steady-state part of the current 

(3.69) reduces to the unattenuated oscillation cos !pt, and for t =0,we nd with the identity 

sign x = 2 R 1 sin x 

d that the current still complies with the boundary conditions. The 

0 

voltage, however, does not, because the steady-state solution (3.71) becomes unbounded. 

The deeper reason for this is our assumption of a lossless plasma, which entails a pole of the 

characteristic impedance at ! = !p. We could have circumvented this particular problem 

by choosing a voltage instead of a current source, thereby forcing the current to vanish at 

resonance. 

Numerical evaluations of (3.82) and (3.83) are shown in g. 3.13 for the DC case = 0, g. 3.14 

for = 0:2 , and g. 3.15 for = 0:8. The most striking property of the result is that the wave 

front indeed goes straight through the medium with a velocity ofX=T = 1 or, expressed with 

the unscaled variables, x=t = c. Thus the plasma is completely at rest before the disturbance 

arrives with the speed of light. This nding is, in principle, nothing spectacular in that it 

only con rms the more general results obtained by Sommerfeld. It emphasises, however, that 

even in the evanescent region not the faintest evidence of superluminal wave propagation can 

be found. 

Apart from the wave front, there are other similarities between the three cases as well. The 

in uence of the forced oscillation applied by the signal source diminishes with the attenuation 

distance 1= 0 = c p 1 

, so that further down the line, the wave is determined exclusively 

!p 1, 2 

by the transient solutions (3.74) and (3.76). Note that the transients consist solely of frequency 

components above the plasma frequency. The evanescent region does not come into play here, 

which is of course merely a mathematical e ect and bears no physical signi cance. The 

free oscillations of the plasma far away from both the entrance (as for the current) and 

52


30 

30 

0 

20 

0 

20 

10 

X 

10 

X 

T 

10 

T 

10 

20 

20 

30 

1 

0 

0 

0.5 

I/I0 

0 

30 

1 

0.5 

U/U0 

Figure 3.13: Evolution of the current (3.82) and voltage (3.83) along a one-dimensional transmission 

line with a lossless plasma for a non-oscillating step as input current ( = 0). 

53 

0


the wave front are dominated by the plasma frequency, which again is not surprising. This 

becomes most clear by inspection of the voltage at the source at low signal frequencies, 

where the in nitely high source impedance permits U to oscillate freely. It shows therefore 

a superposition of the high resonance frequency and a component determined by the signal 

( gs. 3.13 and 3.14). 

Remark (Dispersion e ects) That the plasma frequency ultimately dominates the 

free oscillations is not at all unexpected. The resonance frequency has the smallest group 

velocity vg = d!=d (note that we may safely use this de nition here since the wave 

number k = + j is purely real in the non-evanescent mode). The fastest components, 

on the other hand, are the high-frequent ones that constitute the sharp wave front. They 

rush through, and the slow components around !p remain. That the local frequency 

equals exactly the plasma frequency holds of course only in the limit t !1. 

The local wavelength exhibits a complementary e ect. If we take a snapshot of the wave 

along the line at some given time, we nd that a long way behind the wave front, the 

oscillation gradually fades away. Here, it is the wavenumber that tends to zero as ! ! !p, 

and thus the local wavelength grows in nitely. 

The explanations given so far have been of a rather mathematical nature. There is, however, 

also a physical interpretation why the wave front travels undistorted through the medium. 

The electrons in the plasma have a nite mass and consequently a certain amount of inertia. 

Hence they cannot react to the arriving wave front instantaneously, but only with a short 

delay, and the wave front at the very instant t = x=c remains una ected. This explanation 

was already given by Sommerfeld, who in turn owed it to Voigt [2, 12]. 

Remark (Confession of a graphical retouch) In the gures, the values of the waves 

at X = T have been set to unity. This is, admittedly, not the correct numerical result 

and has been introduced deliberatly to illustrate the actual height of the wave front. The 

true value, as dictated by Fourier, naturally is 0:5 , and 1 is only the left-sided limit, as 

will be demonstrated below. 

The space-time evolution diagrams have a sampling grid that has been chosen su ciently 

coarse to keep the computational e ort at a reasonable value. This impaired particularly the 

area about the wave front, which we examine more closely now. A re ned computation of the 

wave front shows that it remains indeed unaltered even for large values of X and T , as depicted 

in g. 3.16 for the current and g. 3.17 for the voltage. The graphs show the spatial evolution 

of the wave for di erent instances in time and reveal two remarkable features. First, voltage 

and current are in phase for some time after X = T . This is plausible if we consider that 

the high frequency components at the wave front see a real-valued characteristic impedance. 

Thus any phase shift is impossible. 

Second, the trailing edge of the wave front becomes steeper as it travels down the line. The 

left point of the time interval in each of the gures indicates to a good approximation a 

constant value of the wave function. It can be calculated in dependence on the X-value as 

54


30 

30 

0 

20 

0 

20 

10 

X 

10 

X 

T 

10 

T 

10 

20 

20 

30 

1 

0.5 

0 

-0.5 

-1 

0 

30 

1 

0 

0 

-0.5 

I/I0 

0.5 

U/U0 


line with a lossless plasma for a slowly oscillating input current ( =0:2). 

55

30 

30 


0 

20 

0 

20 

10 

X 

10 

X 

T 

10 

T 

10 

20 

20 

30 

1 

0.5 

0 

-0.5 

-1 

0 

30 

1 

0 

-1 

-2 

0 

I/I0 

U/U0 


line with a lossless plasma for an input current with high frequency ( = 0:8). Note the high voltage 

amplitude at the entrance. 

56


1 

0.8 

0.6 

0.4 

0.2 

9.9 9.95 10.05 10.1 

1 

0.8 

0.6 

0.4 

0.2 

99.99 99.995 100.005 100.01 

1 

0.8 

0.6 

0.4 

0.2 

999.999 1000 1000. 

Figure 3.16: Wave front of the current for = 0:5 depending on the spatial coordinate. 

1 

0.8 

0.6 

0.4 

0.2 

9.9 9.95 10.05 10.1 

1 

0.8 

0.6 

0.4 

0.2 

99.99 99.995 100.005 100.01 

1 

0.8 

0.6 

0.4 

0.2 

999.999 1000 1000. 

Figure 3.17: Wave front of the voltage for = 0:5 depending on the spatial coordinate. 

T = X ,1=X, so that an arbitrary initial point (X0;T0) follows a phenomenological trajectory 

given by 

X , T 

X0 , T0 

= T0 

T 

: (3.84) 

This again is due to dispersion. The high frequencies move ahead, the lower ones more and 

more lag behind, such that the high-pass lter e ect becomes more marked. The behaviour 

of the wave is therefore locally similar to the output of a high-pass lter whose characteristic 

frequency is tuned higher and higher, i. e. the needle of the output pulse becomes sharper. 

As last item of the examination, we discuss the propagation of a rectangular pulse through 

the plasma. Owing to the linearity of the medium, we already stated in (3.68) that such a 

pulse can be constructed from two delayed step functions. If the duration of the pulse is an 

integer multiple n of the signal period, we can use identical step responses for switch-on and 

switch-o and need not take care of the additional phase shift otherwise introduced by an 

arbitrary pulse length. In our scaled variables, the pulse duration then becomes 

T t= = 2n : (3.85) 

We expect that both the leading and the trailing edge of the pulse cause a sharp wave front 

travelling at the velocity of light. The contour plots for short pulses with low ( g. 3.18) 

and high ( g. 3.19) input frequencies con rm this suspicion. They also show how the phase 

trajectories of the peaks and troughs of the wave asymptotically approach the wave front. 

Note that if we trace the evolution of such a peak, we nd that it travels always at a velocity 

57

120 

100 

80 

60 

40 

20 

0 

120 

100 

80 

60 

40 

20 

0 


0 20 40 60 80 100 120 

0 20 40 60 80 100 120 

Figure 3.18: Evolution of the current (upper graph) and voltage (lower graph) for a short pulse with 

=0:2 spanning two signal periods (abscissa: X, ordinate: T ). 

58


80 

60 

40 

20 

0 

80 

60 

40 

20 

0 

0 20 40 60 80 

0 20 40 60 80 

Figure 3.19: Evolution of the current (upper graph) and voltage (lower graph) for a short pulse with 

=0:8 lasting for two signal periods (abscissa: X, ordinate: T ). 

59

0.5 

-0.5 

I/I0 

-1 


100 125 150 175 200 225 250 275 

Figure 3.20: Current due to a short pulse spanning two signal periods (n = 2) with = 0:2 far away 

from the signal source (at X = 100). 

greater than that of light. Its speed is a maximum immediately behind the entrance and 

decreases as it approaches the wave front. This mirrors in fact the phase velocity of a locally 

dominating frequency. In an experiment where no sharp wave front is present, however, a 

su ciently narrow-band wave packet could likely be mistaken to move at a superluminal 

velocity. 

Fig. 3.20 nally shows what a short current pulse looks like at a considerable distance from the 

tunnel entrance. The twowave fronts are clearly marked by sharp needle-like pulses. Note that 

after switch-o , the residuary oscillations following the two wave fronts have slightly di erent 

temporary frequencies and consequently form wave packets due to interference. Dispersion, 

however, has rendered the original pulse nearly unrecognisable. 

3.8 Turn-on e ects in a wave guide 

As we have seen in the last section, the wave front of a TEM wave in a lossless plasma 

propagates at the velocity of light. With a view to the microwave experiments carried out in 

search of superluminality, we stress that the results obtained for the unbounded plasma also 

hold for the TE modes of a rectangular wave guide because of the identical dispersion relation. 

While the strictly causal wave front motion may seem plausible for a purely transverse wave, 

there is still the open question as to how a more general mode reacts to a disturbance. In 

this context, it is of particular interest how a eld component in the direction of propagation 

evolves. We will thus explore the turn-on of such a mode. 

The TE01 or H01 mode in a rectangular wave guide ( g. 3.21) is characterised by a longitudinal 

component of the magnetic eld. The eld components perpendicular to that direction are 

60 

T


b 

y 

z 

x 

Figure 3.21: Rectangular wave guide. 

constant along one axis and sinusoidal along the other, the sine spanning only a half period. 

The cuto frequency then is 

!p = c 

; 

b 

(3.86) 

b being the width of the wave guide, which is then half the transverse wavelength of the eld. 

Remark (Plasma and cuto frequencies) To emphasise the similarity to the plasma 

example, we retain the symbol !p for the cuto frequency, although it is normally denoted 

by !c in the literature. 

The steady-state components of this mode are well known (Piefke [16], Harrington [93]). For 

frequencies above cuto , !>!p,wehave 

Ez=Esin y x 

e,j 

b 

Hx = j b! E cos y 

b 

e,j x 

Hy = , ! E sin y 

b e,j x ; 

(3.87) 

with the dispersion relation 

= 1q 

! 2 , !p 

c 

2 : (3.88) 

Note that we haveomitted the explicit time dependence of the elds. In the evanescent region, 

the propagation constant becomes imaginary, 

j = = 1q 

!p 

c 

2 , ! 2 ; (3.89) 

and the eld components change to 

Ez = E sin y x 

e, 

b 

Hx = j b! E cos y 

b 

e, x 

Hy = j ! E sin y 

b e, x : 

61 

(3.90)


What we want to study is the evolution of the elds if we switch on the transverse component 

of the magnetic eld at t = 0 in such a way that only the TE01 mode is excited. This 

assumption is quite reasonable since this mode has the lowest cuto frequency and is thus 

the dominating mode in the wave guide [15, 93]. So the boundary condition reads 

Hy(0;t)= (t)H0cos !0t sin y 

b 

: (3.91) 

With the substitution H0 = j 0 

! E and taking the real parts of the complex functions, we 

nally obtain the steady-state elds 

H0 

Ez;s 

Hx;s 

H0 

Hy;s 

H0 

= 

p =" = 

r 

1 

1 , !0 

!p 

= sin y 

b e, 0x cos !0t 

r !p 

!0 

1 

2 

2 

, 1 

cos y 

b e, 0x cos !0t 

sin y 

b e, 0x sin !0t ; 

(3.92) 

where 0 = 1p 

!p c 

2 , !0 2 is the attenuation constant at the signal frequency. Comparing 

these solutions with the ones we obtained for the current (3.69) and voltage (3.71) in the 

plasma transmission line, we notice a striking similarity: the transverse component of the 

magnetic eld, Hx, corresponds to the current in the transmission line, and the electric eld 

to the negative voltage. As the initial conditions are also the same, we can apply the same 

method we used successfully in the last section to obtain the transient parts of the 

With the familiar scaled variables 

elds. 

= c 

!p 

; T = !pt; X= !px 

c 

; = !0 

!p 

; Y = y 

b 

we can immediately write down the complete solutions for the transverse elds, 

H0 

Hy 

H0 

= sin Y 

Ez p = sin Y 

=" 

h e ,X p 1, 2 

, 2 Z 1 

0 

cos T , 

sin ( X) cos T p 1+ 2 

1+ 2 , 2 

h 

,X 

e p 1, 2 1 

p sin T , 

1= 2 , 1 

, 2 Z 1 

0 

p 1+ 2 cos ( X) sin T p 1+ 2 

62 

1+ 2 , 2 

d 

i 

; (3.93) 

d 

i : 

(3.94) 

(3.95)


Hx 

4 

2 

0 

-2 

10 

-4 

0 

7.5 

2.5 

X 

5 

5 

T 

2.5 

0 

7.5 

Figure 3.22: Evolution of the longitudinal component (3.96) of the magnetic eld for aTE01 wave. 

The parameters are = 0:8 and Y =1=4. 

The longitudinal magnetic component is best calculated from the condition that the magnetic 

eld be source-free, @Hx=@x = ,@Hy=@y, which yields after some simple manipulations 

h Hx 

,X 

= cos Y e 

H0 

p 1, 2 1 

p cos T , 

1 , 2 

, 2 Z 1 cos ( X) cos T p 1+ 2 

1+ 2 , 2 

(3.96) 

i 

d : 

0 

We know already how the transverse components behave, so we need not reproduce the results 

of the last section. The longitudinal component of the magnetic eld, however, is new. A 

numerical evaluation is shown in g. 3.22 . We see that the wave front ofHx, too, propagates 

with c. Furthermore, at t = x=c, the longitudinal component vanishes. Thus the front 

of the wave travelling along the wave guide after turn-on consists exclusively of transverse 

components, i. e. the mode acts exactly like a TEM mode at this instant. This becomes even 

more impressive if we look at the magnetic eld lines of the wave ( g. 3.23). The example 

demonstrates clearly that even waves with eld components in propagation direction do not 

propagate faster than light. 

Remark (Optical cuto frequencies) According to (3.86), the cuto frequency of a 

wave guide is inversely proportional to its width or | in the case of a cylindrical guide 

| its diameter. This imposes an inconvenient limit on the miniaturisation of integrated 

optical devices. There is, however, a way to circumvent the problem. Takahara et al. [99] 

proposed to use metallic wave guides at optical frequencies. At these high frequencies, the 

metal actually exhibits a negative dielectric constant, so that alow-pass TM mode can 

63 

10

0.8 

0.6 

0.4 

0.2 

Y 

1 

0 


0 1 2 3 4 5 

Figure 3.23: Snapshot of the magnetic eld of a TE01 wave. The parameters are = 0:8 and T =4:87 . 

propagate without a lower cuto frequency. In fact, this is a surface wave at the interface 

between metal and `normal' dielectric, and therefore the geometrical limitations are no 

longer relevant. Unfortunately, this mode is associated with a rather high attenuation, 

but for small-distance applications, this might be tolerable (see also Paschke [100]). 

3.9 A Gaussian pulse in plasma 

To round o the results of the previous sections, we now investigate the behaviour of a 

Gaussian pulse in a lossless plasma. The model we use is still the transmission line of section 

3.5 with the dispersion relation 

k(!) = !p 

c 

s ! 

!p 

2 

,1: (3.97) 

The boundary condition, i. e. the pulse applied at the interface x = 0, is given by 

u(0;t)=U0e , t,t 0 2 

Re e j!0t ; (3.98) 

where t0 denotes the position of the peak and is the standard deviation of the exponential 

distribution. Owing to the linearity of the system, we mayuse the complex notation and take 

the real part of the expressions if necessary. The voltage at any point inside the plasma is 

then given in the well-known manner as the Fourier integral 

u(x; t) =Re 

Z 1 

,1 

A(!) e j(!t,k(!)x) d! : (3.99) 

Note that since the pulse has an in nite duration, there is no initial condition to comply 

with. Consequently, we cannot apply a trick as in the turn-on case of section 3.7 to make the 

evaluation easier, and we have to compute the wave integral directly. 

The spectrum A(!) of the initial pulse can easily be calculated, which gives 

A(!) = 1 

2 

Z 1 

,1 

u(0;t) e ,j!t dt = p e 

2 , (!,! 0 ) 

2 

64 

2 

e ,jt0(!,!0) : (3.100) 

X


Like always, we introduce a normalised integration variable 

and normalised parameters 

= ! 

!p 

T = !pt ; T0=!pt0; =!p ; X= !px 

c 

; = !0 

!p 

(3.101) 

: (3.102) 

With these new variables, we can write the voltage at any position along the line as 

U(X; T )=U0 2 p 

Z 1 

,1 

( , ) 

, 2 e 

2 

e ,jT0( , ) e j( T,K( )X) d ; (3.103) 

with the normalised dispersion relation K( )= p 2 ,1. As for the sign of the square root, 

we must keep in mind that since we regard only right-going waves, the negative branch applies 

to negative frequencies. In the evanescent region, we must also choose the negative solution 

of the root in order to obtain attenuation along the line. So in the three parts of the range of 

integration, the dispersion relation is given by 

K( )= 

8 

>< 

>: 

, p 2 ,1 if 1 

: (3.104) 

Remark (Transfer function of the plasma) Looking at the wave integral from a 

system theoretical point of view, we obtain still another reasoning for the choice of the 

sign of the dispersion relation. We can think of the wave integral (3.99) as the response 

of a system to a signal u(0;t), 

u(x; t) = 

Z 1 

where the transfer function of the system is 

A(!)H(!) e 

,1 

j!t d! ; (3.105) 

H(!) =e ,jk(!)x : (3.106) 

It is well known that a physically meaningful system must give a purely real response 

when excited with a purely real signal. According to the basic properties of the Fourier 

transform, this can only be guaranteed if the real and the imaginary parts of the transfer 

function are even and odd functions in !, respectively. It is easy to see that the de nition 

of the dispersion relation given above satis es these requirements. Within the pass 

band, the imaginary part of H(!) is essentially the sine of an odd argument function 

and thus also odd, whereas the real part is the cosine of an odd function and thus even. 

In the evanescent region, the imaginary part of the transfer function vanishes, and the 

exponentially decaying real part remains even. 

The current in the transmission line could be calculated by inserting the voltage into the 

di erential equations (3.75) and solving for I, which gives 

i(x; t) =L 0 

Z 1 

A(!) 

,1 

k(!) 

! ej(!t,k(!)x) d! : (3.107) 

65

0.75 

0.5 

0.25 

-0.25 

-0.5 

-0.75 

U/U0 

1 

50 100 150 200 T 


|A| 

4 

3 

2 

1 

-2 -1 1 2 η 

Figure 3.24: Initial pulse and its spectrum jA( )j for the parameter values T0 = 100 and = 17. The 

carrier frequency is = 0:5 in these particular graphs. 

Unfortunately, a pole turns up at ! =0,preventing straightforward integration. Taking the 

Cauchy principal value of the integral would solve the problem, but as we do not need the 

current to gain some general insight into the propagation of a Gaussian wave, we leave it 

aside. 

The expression for the voltage (3.103) is valid in both the stop and pass bands. We can thus 

compute the voltage inside the plasma for a given pulse with varying carrier frequencies. The 

initial shape and its spectrum are shown in g. 3.24. If the cuto frequency of the plasma 

is assumed to be !p = 10 10 9 s ,1 (a value roughly corresponding to Nimtz' microwave 

experiments), then the pulse width is about 6 ns, which is fairly short. Accordingly, the 

spectrum is quite wide, and so we must expect a small portion of the pulse to be classically 

transmitted even though the major part of the eld originates from the evanescent region. 

The rst case we examine is the base-band signal with =0. The result given in g. 3.25 

is not very spectacular since the spectrum is practically con ned to the range below cuto . 

Thus there are no propagating waves, and all we can see is the exponential decay along the 

spatial coordinate. In contrast to the following gures, this one shows no modulation, and 

so the vanishing voltage outside the actual pulse also denotes the least possible value of the 

eld. Therefore the area surrounding the contour of the pulse is shaded in black. 

When we use a carrier with = 0:5, the behaviour of the wave is much more interesting. 

Fig. 3.26 shows rst of all a dominating evanescent signal that resembles the unmodulated case 

treated before. It is particularly noteworthy that the carrier frequency increases gradually 

with X. This is manifested in the graph by the narrowing of the black and white stripes 

and can be explained as a pulse reshaping e ect. Since the dispersion in the plasma causes 

higher frequency components to be less attenuated than lower ones, the centre frequency of 

the spectrum is slightly raised. Note that although it looks as if the pulse became shorter with 

increasing X, this is not the case. It is only the attenuation in combination with the limited 

plotting range that creates this deceptive impression. The overall shape of the pulse remains 

more or less the same, albeit with reduced amplitude. Direct comparison of the unmodulated 

and the present results reveal another detail: the plot ranges of the two graphs are identical, 

66


250 

200 

150 

100 

50 

0 

0 10 20 30 40 50 

Figure 3.25: Evolution of the voltage for = 0. The abscissa denotes X, the ordinate T . 

250 

200 

150 

100 

50 

0 

0 10 20 30 40 50 

Figure 3.26: Evolution of the voltage for = 0:5. The abscissa denotes X, the ordinate T . 

67


which enables one to see that the modulated pulse penetrates deeper into the medium than 

the base-band signal. This is in accordance with the dispersion relation and the respective 

attenuation coe cient. 

The bulk of the signal energy is concentrated within the parabola-shaped region in the timespace 

graph. But we also recognise a small fraction of the pulse running away down the 

line when the major part of the signal is already diminishing. The high oscillation frequency 

of this low but very broad pulse (twice the carrier frequency) shows that it consists of the 

spectral components just above cuto . In g. 3.24, which shows exactly the spectrum of this 

particular case, the spectrum seems to be su ciently below cuto , but evidently it is not. We 

can roughly calculate the frequency that dominates this part of the signal by measuring the 

slope X= T parallel to the peaks and troughs visible in the graph. They give an estimate 

for the phase velocity, while the lines connecting the outer ends of the ripples are related to 

the group velocity. Using the de nition of the dispersion relation (3.97), we readily nd the 

normalised expressions for the phase velocity, 

and the group velocity, 

vp 

c = 

r 2 

vg 

c = 

r 

1 , 1 

2 

2 , 1 ; (3.108) 

; (3.109) 

respectively. With these two equations, which directly correspond to the appropriately measured 

slope X= T , we nally obtain an interval for the carrier frequency, 2 [1; 1:02]. 

The left bound of the interval is obtained from the left end of the propagating wave close 

to X = 0 where the tangents to the ripples are horizontal and the ends of the valleys lie 

on a vertical line. The same consideration applied to the right end of the wave gives the 

upper limit of the interval. It is clear, though, that the data we can gather from the graph 

to nd this limit is directly dependent on the resolution we choose for the voltage amplitude. 

In reality, the interval is as unlimited as the spectrum of the pulse, and only the numerical 

accuracy imposes constraints on the reliability of the results (which indeed prevented us from 

discovering the same e ect in the base-band pulse). One must of course not overlook the fact 

that the amplitude of the propagating part is nine orders of magnitude smaller than the main 

pulse. Therefore it is likely to be missed in an experiment. 

The third example ( g. 3.27) shows a pulse where the carrier frequency equals the cuto frequency 

( = 1). Here, half the spectrum lies in the pass band. Thus the pulse is propagated, 

but it is severely distorted. The reason for this behaviour is evidently the dispersion relation. 

A moderate pulse distortion requires k(!) to be approximately linear in the frequency range 

that makes up the dominating part of the spectrum. About cuto , this condition is de - 

nitely not met, and we end up with a mixture of spectral components having grossly di erent 

propagation velocities. 

In g. 3.28, = 1:5 was chosen for the carrier. As we could have expected, this gives a 

pulse propagating straight through the medium. This result can now be used to verify the 

68


250 

200 

150 

100 

50 

0 

0 10 20 30 40 50 

Figure 3.27: Evolution of the voltage for = 1. The abscissa denotes X, the ordinate T . 

250 

200 

150 

100 

50 

0 

0 10 20 30 40 50 

Figure 3.28: Evolution of the voltage for = 1:5. The abscissa denotes X, the ordinate T . 

69


classical understanding of the phase and group velocities. The slope of the stripes yields the 

phase velocity, and the contour of the entire pulse (most easily traced at the edge where they 

fade into the grey surroundings) de nes the group velocity. Obviously, these two velocities 

are di erent, which empirically a rms the theoretical ndings of section 1.1: in a dispersive 

medium, the envelope of a pulse propagates at a distinct velocity with the train of carrier 

oscillations moving indepently underneath its overall shape. As for the determination of 

from the graph, both (3.108) and (3.109) give the correct answers. 

Remark (Dispersion e ects) In a dispersive medium, we should of course be able 

to observe a broadening of the pulse. We are, however, at a loss to do so in this case 

because the initial pulse is too wide and the distance too short for this e ect to become 

visible. But if we su ciently reduced the plotted voltage range (by some nine orders of 

magnitude), we would encounter an e ect similar to that in g. 3.26: namely, a small 

portion of the wave travelling, if at all, at very low speed. Again, this is caused by the 

small frequency range immediately above cuto where the group velocity is hardly larger 

than zero. In contrast, the e ect of the evanescent components (which are present also in 

this case) is not noticeable at all. 

In accordance with the theory, the examples show that if the spectrum of a pulse is bundled 

below cuto , there is no phase shift, and the pulse decays exponentially without allowing for 

wave propagation. The amount of attenuation is determined by the centre frequency of the 

pulse, !0, so that we can estimate the voltage amplitude by 

U(X) / e ,Xp 1, 2 

: (3.110) 

If in a data transmission system we decided, for the sake of proper signal detection, to 

tolerate an attenuation of 10 ,4 (80 dB), we could calculate the admissible length of the 

line. For = 0:5 this would give X = 10:64, whereas with = 0:8, we could cover a 

normalised distance of X =15:35. The actual length of the transmission line can be obtained 

from the de nition of the scaled space coordinate, x = Xc=!p. In a plasma with a cuto 

frequency of 10 GHz, X = 10 corresponds to an actual distance of x = 4:77 cm. So if we 

were to use signals with superluminal group velocities (as they are sometimes called) for data 

transmission, we could cover but a few centimetres. This, however, makes their usefulness for 

practical applications rather questionable. 

The maximum of a band-limited pulse can be transmitted at a velocity greater than that of 

light, there is no doubt about it. But does this mean that information can also be transmitted 

superluminally? The response of a linear system to an input signal can be written as a 

convolution integral, 

u(x; t) = 

Z 1 

,1 

of the input u(0;t) and the transfer function 

g(x; t) = 1 

2 

u(0; ) g(x; t , ) d ; (3.111) 

Z 1 

e 

,1 

,jk(!)x e j!t d! : (3.112) 

70


Note that while communication engineers talk of transfer functions (or, likewise, impulse 

responses), theoretical physicists frequently call g(x; t) a Green's function. In either case, it 

is said to be causal if it is identically zero for t

Chapter 4 

One-dimensional quantum 

tunnelling 

4 One-dimensional quantum tunnelling 

Das Bestreben, der Dispersion Herr zu werden, leuchtet mir fur langsame 

Wellen ein, das Ubrige ist mir aus den Andeutungen nicht klar geworden. 

Das Au allende ist, wie viel man mit klassischer Mechanik befriedigend 

machen kann. Wenn es nur einmal gelange, das Prinzipielle an den Quanten 

einigermassen zu erklaren! Aber meine Ho nung, das zu erleben, wird 

immer kleiner. 

Albert Einstein in a letter to Arnold Sommerfeld (1. 2. 1918 [103]) 

Having exhausted the question of electromagnetic wave propagation we now turn to the 

second area where tunnelling occurs and where, after all, the term `tunnelling' was coined. 

To start with, however, we shall restrict our investigation to the comparatively simple step 

potential barrier, where we have to cope with only one interface. In a second step, we regard 

tunnelling in its original meaning as the motion of a particle through a classically impenetrable 

barrier. In the simplest form of a rectangular potential wall, such an investigation requires 

the inclusion of two interfaces, which renders the treatment more complicated albeit by no 

means impossible. In both cases, we study how the particle evolves in time and space upon 

impinging on the barrier. 

The chapter begins with the well-known formulation of the scattering process consisting of an 

incident free particle being partly re ected and partly penetrating the obstacle. The solutions 

will be given as Fourier integrals and primarily be independent of the actual initial condition. 

We shall then explore several possibilities of initial wave forms or particle shapes that nally 

are used to obtain numerical results of the wave propagation both inside and outside the 

barrier. The square potential barrier will be considered next, and we shall brie y examine the 

variety of tunnelling time de nitions known from the literature for monochromatic waves. The 

concluding numerical evaluation of tunnelling events is restricted to Gaussian wave packets. 

72

4.1 The potential step 


Being interested chie y in the dynamical evolution of a wave inside an evanescent barrier, we 

assume the barrier to be in nitely extended into the right half space. The potential function 

is therefore 

( 

0 if x


spectrum in wave number space, A( ), each of the monochromatic components is itself a 

steady-state solution of Schrodinger's equation, and the superposition of all partial waves 

then yields the desired evolution of a single wave. The general form of this solution is 

(x; t) = 

Z 1 

C A( ) e 

,1 

j( x,!( )t) d ; (4.7) 

where C and also have tobechosen according to the respective region. The incident wave 

then reads 

inc = 

Z 1 

,1 

j x,j 

~ 

A( ) e 2m 2t d : (4.8) 

The formulation of the re ected part is not so straightforward and requires a moment's 

consideration. The re ection coe cient contains the propagation constants of both the region 

outside the tunnel and the tunnel itself. The link between the respective dispersion relations, 

however, is provided by the frequency ! that alone remains invariant as the monochromatic 

wavelets penetrate into the barrier, whereas the propagation constant changes. Therefore we 

must not mix up 1 and 2. Instead, we retain 1 as and express 2 by means of (4.6) 

as 2 2 = 2 , 2m!p=~. Consequently, the integration range falls into three distinct intervals 

determined by 

2 = 

8 

>< 

>: 

, 

q 2 , 2m!p 

q 2m!p 

~ if 

~ 

q 

2m!p 

~ 

q 

2m!p 

~ 

q 

2m!p 

~ 

q 2m!p 

~ 

: (4.9) 

Remark (Transformation of ) The use of the negative rootinthetransformation 

becomes obvious if we call to mind that the dispersion relations are even functions in and 

separated in ordinate direction only by the cuto frequency !p. Hence the transformation 

rule must be even, too, which gives 2 = sign p 2 , 2m!p=~ outside the evanescent 

region. 

With (4.9) and (4.4), we nally obtain the three parts of the re ected wave, 

ref = 

Z , 

+ 

,1 

q 2m!p 

~ 

Z q 2m!p 

~ 

q 2m!p 

, ~ 

Z 1 

+ q 2m!p 

~ 

A( ) + 

A( ) 

, 

q 

2 2m!p 

, ~ q 

2 2m!p 

, ~ 

q 

2m!p 

, j ~ , 2 

q 

2m!p 

+ j ~ , 2 

A( ) , 

q 

2 2m!p 

, ~ 

+ 

q 

2 2m!p 

, ~ 

74 

,j x,j 

~ 

e 2m 2t d + 

,j x,j 

~ 

e 2m 2t d + 

,j x,j 

~ 

e 2m 2t d : 

(4.10)


Inside the barrier, we must again apply (4.9), although one might perhaps argue that in this 

region the original 2 would have been more appropriate. This would in fact be true if the 

spectrum of the initial wave, A( ), had been given in terms of the wave numbers inside the 

tunnel. But as it is, the outset of our investigation is a wave strictly con ned to the outside of 

the barrier, and so we cannot but describe it in that wave number space. Thus the transmitted 

part of the wave becomes 

tun = 

Z , 

+ 

,1 

q 2m!p 

~ 

Z q 2m!p 

~ 

q 2m!p 

, ~ 

A( ) 

A( ) 

Z 1 

+ q A( ) 

2m!p 

~ 

, 

+ 

2 

q 2 , 2m!p 

~ 

+ j 

2 

q 2m!p 

~ , 2 

2 

q 2 , 2m!p 

~ 

e ,jx 

e jx 

q 

2, 2m!p 

~ ,j ~ 

2m 2 t d + 

e ,x 

q 2m!p 

~ , 2 ,j ~ 

2m 2t d + 

q 

2, 2m!p 

~ ,j ~ 

2m 2t d : 

(4.11) 

As usual, we introduce normalised variables to ease the numerical evaluation, 

r 

~ !p 

c = 

2m ; T = !p t; X= !p 

x; (4.12) 

c 

as well as a new integration variable 

= 

s ~ 

2m!p 

= c 

!p 

: (4.13) 

Note that in this context, c has nothing at all to do with the velocity of light. It is nothing 

butavariable that incidentally has the physical dimension of a velocity. The individual wave 

functions can be rewritten as 

inc = !p 

c 

ref = !p 

c 

+ !p 

c 

+ !p 

c 

Z 1 

A( ) e 

,1 

jX ,j 2T d ; (4.14) 

Z ,1 

,1 

Z 1 

,1 

Z 1 

1 

A( ) + p 2 , 1 

, p ,jX ,jT 2 

e d + 

2 , 1 

A( ) , jp1 , 2 

+ j p ,jT 2 

e,jX d + 

1 , 2 

A( ) , p 2 , 1 

+ p ,jX ,jT 2 

e d ; 

2 , 1 

75 

(4.15)

tun = !p 

c 

+ !p 

c 

+ !p 

c 

Z ,1 

,1 

Z 1 

,1 

Z 1 

1 

A( ) 

A( ) 

A( ) 


2 

, p e 

2 , 1 ,jX 

p 

2,1,jT 2 

d + 

2 

+ j p 1 , 

2 e,X 

p 1, 2 ,jT 2 

d + 

2 

+ p e 

2 , 1 jX 

p 

2,1,jT 2 

d : 

(4.16) 

These equations describe the complete scattering process independently of the initial wave. 

Remark (Energy and momentum space) From electromagnetic waves we are used to 

decomposing wave packets either into a frequency or wave number spectrum. Throughout 

the remainder of this chapter, we shall adhere to these terms also for quantum mechanical 

wave packets. It should be noted, however, that quantum physicists more commonly talk 

about energy and momentum spectra. This seems a bit confusing, but in fact is equivalent 

to our terminology. The energy space corresponds to the frequency spectrum because of 

the relation E = ~!. The momentum in quantum mechanics is de ned as p = ~ and 

therefore corresponds to our usual wave number space. So, apart from the factor ~, both 

views are identical. 

4.2 Initial wave forms 

There are two mutually exclusive choices for the initial conditions. First, we can restrict the 

location of the wave packet to the space outside the barrier. In this case, the spectrum will 

inevitably contain components with an energy above the barrier. Conversely, we can as well 

limit the bandwidth of the wave packet to energies below the barrier, but then the wave will 

not be bounded in space and therefore extend into the barrier from the beginning. We deem 

this more undesirable and thus choose the rst possibility. 

We now calculate the spectra of three di erent wave forms to be used as initial conditions 

in a scattering process. The position of the wave packets is a snapshot of the very moment 

when they reach the barrier. 

The rst and simplest one is a rectangular pulse. It is given by 

( 

0 if x


-60 -50 -40 -30 -20 -10 

1 

0.5 

-0.5 

-1 

Ψ0 

X 

0.15 

0.125 

0.1 

0.075 

0.05 

0.025 

A 

-2 -1 1 2 ξ 

-2 -1 1 2 ξ 

Figure 4.1: Wave function and spectrum of a rectangular pulse for k = 6 and = 0:8. The left graph 

shows the real part of the wave function (thin line) and the corresponding probability density (thick 

line). The right graph gives the absolute value jA( ) l ,1=2 j of the wave number spectrum (4.22). The 

evanescent part is the dark grey area, and the peak of the spectrum lies at = 1=2 . 

which immediately gives 

Arect( )= p l 1 

2 

sin( 0 , ) l 

2 

( 0 , ) l 

e 

2 

,j( 0, ) l 2 : (4.19) 

We prefer, however, to write the spectrum with the normalised variable (4.13). To this end, 

we add the additional de nitions 

= !0 

!p 

; 2 k = 0 l: (4.20) 

The length l of the original wave packet is thus given in a multiple k of the wavelength of 

the modulation frequency. From the dispersion relation (4.6) and (4.12) we nd the identity 

!p=c = 0= p , which together with (4.13) yields 

l =2 kp : (4.21) 

If we furthermore write the sine function in its Euler form, we obtain the normalised spectrum 

of the rectangular wave packet, 

A rect( ) 

p l = j 

4 2 k 

1 

1 , 1 

p 

,j2 k 1, p 

1 

e 

, 1 : (4.22) 

Fig. 4.1 shows both the rectangular wave packet and its spectrum (4.22). Note that in the 

scaled notation, the length l changes to 2 k= p and the phase function 0 x becomes p X. 

The second initial wave packet we consider has a triangular or tent shape, 

0 (x) = 

8 

>< 

>: 

0 if x

-60 -50 -40 -30 -20 -10 

3 

2 

1 

-1 

Ψ0 

X 


A 

0.14 

0.12 

0.1 

0.08 

0.06 

0.04 

0.02 

-2 -1 1 2 ξ 

-2 -1 1 2 ξ 

Figure 4.2: Triangular pulse with k = 6 and = 0:8. The left graph shows the real part of the 

wave function (thin line) and the corresponding probability density (thick line). The right graph gives 

the absolute value jA( ) l ,1=2 j of the wave number spectrum (4.25), peaked about = 1=2 . The 

evanescent region is the interval [,1; 1]. 

Like before, the factor p 3=l is due to the normalisation R 1 

easily found to be 

or, in scaled variables, 

A tria ( ) 

p l 

p 

3 

= 

4 3k2 Atria( )= p p 

3 

l 

4 

1 

1 , 1 

p 

,1 0 0 

dx =1. The spectrum is 

sin( 0 , ) l 

4 

( 0 , ) l 

! 2 

e 

4 

,j( 0, ) l 2 (4.24) 

,j k 1, p 

1 

2e 2 

The wave packet and its spectrum are shown in g. 4.2 . 

,j2 k 1, p 

1 

, e 

, 1 : (4.25) 

The last example is also the most popular in the literature: the Gaussian wave packet. If we 

start with a normal distribution N( ; 2 ) for the probability density and choose = ,l=2, 

= l=(2n), we get the wave function 

0 (x) = 

s 

2n 

l p 2 e,n2 ( x l +1 2) 2 

e j 0x 

: (4.26) 

As the envelope function is not truncated at x =0,care must be taken to set n su ciently 

large so that the wave function almost vanishes inside the barrier. The spectrum of this wave 

packet is simply 

Agauss ( )= p 1 

lp 

p e 

2 n 2 , ( 0 

78 

, )l 

2n 

2 

,j( 0, ) l 2 (4.27)


-60 -50 -40 -30 -20 -10 

3 

2 

1 

-1 

Ψ0 

X 

0.12 

0.1 

0.08 

0.06 

0.04 

0.02 

A 

-2 -1 1 2 ξ 

-2 -1 1 2 ξ 

Figure 4.3: Gaussian pulse with k =6, =0:8 , and n =4. The left graph shows the real part of the 

wave function (thin line) and the corresponding probability density (thick line). The right graph gives 

the absolute value jA( ) l ,1=2 j of the wave number spectrum (4.28). The maximum of the spectrum 

is at = 1=2 , and the evanescent part is coloured dark grey. 

and with the use of our scaled variables, respectively, 

A gauss ( ) 

p l 

= 

1 

p 2 n p 2 e 

, 

0 

@ 2 k p 1 

,1 

2n 

1 

A 2 

,j k 1, 1 

p 

Fig. 4.3 shows again the initial wave packet and the corresponding spectrum. 

: (4.28) 

The gures shown above demonstrate impressively how large the portion of spectral components 

outside the evanescent region can be even though the modulation frequency is selected 

from the forbidden band. The reason is that the pulses are all very short, i. e. they span only 

very few wavelengths of the carrier wave and therefore have a broad spectrum. This applies 

even to the Gaussian wave that otherwise exhibits a spectrum rapidly decreasing at either 

side of the centre frequency or wave number. 

The inspection of the spectra naturally raises the question as to what percentage of the whole 

wave probability density actually lies in the evanescent region. This quantity can easily be 

calculated from the spectral probability density jA( )j 2 in the interval [,1; 1] referred to the 

total probability as given by R 1 

,1 jA( )j2 d . Not surprisingly, the total probability is the same 

for all three waveforms, 

Pt = lp 

4k 

2 ; (4.29) 

which of course stems from the normalisation of the probability density. 

Remark (Total probability) In principle, there is an alternative and more convenient 

way to calculate the probability of the wave packet. Instead of integrating the absolute 

values of the spectra (4.22), (4.25), and (4.28), we can as well use the spatial waveform. 

79

0.1 

0.001 

0.00001 

-7 

1. 10 

-9 

1. 10 

0.1 

0.001 

0.00001 

-7 

1. 10 

-9 

1. 10 

ΔΡ 

ΔΡ 

0.2 0.4 0.6 0.8 1 Ω 

0.2 0.4 0.6 0.8 1 Ω 

0.1 

0.001 

0.00001 

-7 

1. 10 

-9 

1. 10 

0.1 

0.001 

0.00001 

-7 

1. 10 

-9 

1. 10 

ΔΡ 

ΔΡ 


0.2 0.4 0.6 0.8 1 Ω 

0.2 0.4 0.6 0.8 1 Ω 

Figure 4.4: Fraction of the probability contribution from the pass-band spectral components (4.32) 

for rectangular (solid line), triangular (dotted line), and Gaussian (dashed line) initial wave forms and 

short waves. The parameters are k = 6 (upper left diagram), k = 20 (upper right diagram), k = 100 

(lower left diagram), and k = 1000 (lower right diagram), respectively. The Gaussian wave packet was 

computed for a wide (n = 4, right dashed line in each picture) and a narrow (n= 10, left dashed line) 

example. 

For the Fourier transform pair (4.7) and (4.18), Parseval's theorem reads 

Z 1 

,1 

j (X)j 2 dX =2 

Z 1 

,1 

jA( )j 2 d : (4.30) 

Owing to normalisation, these integrals should give unity. The reason why (4.29) still 

di ers from this value is simply the use of the scaled variables, where the factor p l could 

not be transformed. If we had calculated the probability from the original spectra A( ), 

we would have obtained unity as expected. For the ratio of evanescent contribution and 

total probability we are interested in, the constant factor is of course irrelevant and cancels 

out. 

We now calculate the probability contribution from the spectral components in the evanescent 

region, 

Pe = 

Z 1 

,1 

jA( )j 2 d ; (4.31) 

80

4.3 Examples of scattering processes 

and determine the fraction Pe=Pt for the various inital waveforms. To plot this ratio, however, 

it is more suitable to regard the complementary quantity, i. e. the part of the probability that 

still originates from the pass band, 

P =1, Pe 

Pt 

: (4.32) 

Evaluations of this relation are depicted in g. 4.4 in dependence on the centre frequency and 

for di erent lengths of the wave packet. 

The plots reveal that even for waves comprising a large number of carrier wavelengths, the 

pass band still contributes a signi cant part of the total probability ifwe start with a square 

wave. The situation is much better for triangular waves that even surpass narrow Gaussian 

wave packets for frequencies slighty below cuto . 


In this section we nally examine several examples of scattering events of electrons impinging 

on a step potential barrier. The partial waves outside and inside the barrier are de ned by 

the Fourier integrals (4.14) { (4.16). The appropriate wave number spectra of the initial wave 

forms are taken from section 4.2, and the integrals are evaluated numerically. The plots always 

show the probability density function P = j j depending on the normalised variables of 

space and time, X and T . 

Let us begin with an initially rectangular wave like in g. 4.1. As we have already seen, a large 

portion of its spectral components lies in the pass band, which makes such a wave a rather 

unfortunate choice for an investigation of the tunnel e ect. It is, however, useful to study the 

e ects of partial transmission of the high-frequency components. But before we do so, let us 

brie y consider the wave packet as such. Owing to its wide spectrum, we must expect that it 

will soon lose the original shape and take on a smoother form. This suspicion can easily be 

con rmed by evaluating the incident wave(4.14) not only to the left of the barrier, but also 

in the right half space, where it corresponds to a particle moving freely without obstacle. The 

re ected and transmitted parts of the wave are therefore not of concern in this case. Fig. 4.5 

shows indeed that dispersion quickly changes the shape of the wave packet from the initial 

rectangle to a broad pulse with a marked peak and a number of small sidelobes. 

Remark (Plot ranges) In the sequel, the three-dimensional plots of the probability 

density outside the barrier are always accompanied by a contour plot of the same function. 

To allow the perception of as many details as possible, the plots often deliberately do not 

show the full range of function values. This is particularly important for the contour plots 

to avoid an exceedingly large number of contour lines. The bright white holes in these 

plots thus indicate the areas that have been clipped. 

Now let this particle collide with the step potential. The plot of the probability density 

outside the barrier ( g. 4.6) shows the incident wave packet moving towards the barrier while 

already losing its shape. The actual scattering event is characterised by strong interference 

81

30 

25 

20 

15 

10 

5 

0 

-20 

0 

30 

X 


20 

20 

T 

10 

-20 -10 0 10 20 30 40 

0 

0 

40 

Figure 4.5: Evolution of the probability density P = j j of a freely moving wave packet (4.14) with 

initially rectangular shape for the parameters k =3and =0:8. 

82 

1.5 

1 

0.5 

P


40 

30 

20 

10 

-40 

-30 

40 

30 

-20 

X 

T 

20 

-10 

10 

0 

-40 -30 -20 -10 0 

0 

2 

1 

3 

0 

0 

Figure 4.6: Evolution of the probability density P = j j of a scattering wave packet with initially 

rectangular shape outside the barrier. The parameters are k = 3 and = 0:8. 

83 

4 

P

30 

1 

0.88 

0.6 .6 

P 0.4 .4 

0.2 .2 

0 

0 

20 

10 

X 

10 

20 

T 

0 

30 


Figure 4.7: Evolution of the probability density P = j j of a scattering wave packet with initially 

rectangular shape within the barrier. The parameters are k =3and =0:8. The spectral components 

with higher energy than the edge of the barrier propagate further. 

of the emerging re ected wave and the trailing tail of the still arriving wave. The ultimately 

re ected part exhibits a smooth shape bearing no resemblance to the original rectangle. 

Inside the barrier ( g. 4.7), we notice the rapid decay of the wave function originating from 

the evanescent spectral components. The crests and troughs running not in parallel to the 

axes, however, are patterns characteristic for propagating waves. We also notice very small 

but irregular ripples for times immediately after the wave begins to penetrate the barrier. 

These are in fact the spectral components of the pass band that are being transmitted. For 

about T < 10, the computation grid is too coarse to resolve the high-frequency oscillations. 

We shall have a closer look at them later. 

Remark (Actual physical dimensions) At this point it is sensible give some realworld 

values for the normalised variables X = !px=c and T = !pt we use to scale the 

pictures. In terms of the barrier energy V = !p~ and with c = p !p~=(2m), they read 

T = V 

~ t; X= 

r 

2mV 

~ 2 

: (4.33) 

If we assume the e ective mass of the impinging electron to be 0:063 m0 with the rest 

mass m0 =9:12 10 ,31 kg, we obtain for two widely used barrier energies [83] the values 

given in table 4.1. 

84 

40


P 

30 

2 

1.55 

1 

0.5 .5 

0 

0 

20 

10 

X 

10 

20 

T 

0 

30 

Figure 4.8: Evolution of the probability density P = j j of a triangular wave packet within the 

barrier. The parameters are k =3and =0:8. 

40 

barrier energy T =1b= X=1b= 

0:3eV=4:8 10 ,20 J t =2:194 fs x =1:42 nm = 14:2A 

0:1eV=1:6 10 ,20 J t =6:582 fs x =2:46 nm = 24:6A 

Table 4.1: Actual dimensions for typical barriers. 

We now move on to triangular electrons, the spectrum of which is far more well-behaved 

than that of a rectangular particle. For the rst example we use the same parameters as 

in the previous case. The behaviour outside the barrier ( g. 4.9) is essentially the same as 

before, but the wave forms are much smoother now. The sharp peak of the initial wave 

packet naturally disappears after a short time, and the re ected wave already resembles a 

Gaussian distribution. The same observation can be made for the part that penetrates the 

barrier ( g. 4.8). Since the parameters were chosen such that a fairly large contribution still 

comes from the pass band, we recognise a soft crest moving ahead. Actually, the smoothlooking 

wave function is superposed with small high-frequent ripples that, like in the previous 

example, cannot be resolved in this plot. 

The wave packets considered so far were extremely short (i. e. comprising very few wavelengths 

of the carrier), which consequently resulted in a broad spectrum. Let us now turn to the more 

85

P 

0 

6 

4 

2 

0 

-40 

10 

40 

30 

20 

10 

T 

20 

-30 

30 

40 

-20 

X 

-10 

0 

-40 -30 -20 -10 0 


Figure 4.9: Evolution of the probability density P = j j of a scattering triangular wave packet 

outside the barrier. The parameters are k =3and =0:8. 

86 

0


P 

8 

6 

4 

0 

2 

0 

-150 

50 

175 

150 

125 

100 

75 

50 

25 

0 

T 

100 

-100 

150 

X 

-50 

-140 -120 -100 -80 -60 -40 -20 0 

Figure 4.10: Evolution of the probability density P = j j of a longer triangular wave packet outside 

the barrier. The parameters are k =10and =0:2. 

87 

0

P 

0 

2 

1.5 

1 

0.5 

0 

0 

1 

50 

2 

X 

T 

100 

3 

150 

4 


Figure 4.11: Evolution of the probability density P = j j of a longer triangular wave packet within 

the barrier. The parameters are k = 10 and = 0:2. 

realistic case of a longer electron with triangular shape. In addition, we choose a carrier wave 

with lower energy. The contribution of the pass band to the total probability density inthe 

initial wave form is then low enough that we may expect `proper' penetration of the wave 

into the barrier. This reduces the disturbing e ects of the transmitted components, although 

we will of course never really get rid of them. Since the wave spans more wavelengths of the 

carrier, interference between incident and re ected part generates more peaks, as can be seen 

from g. 4.10 . 

Inside the barrier, the probability density is largely determined by evanescent components, 

so that the overall shape of the function decays exponentially ( g. 4.11). However, a closer 

inspection of the probability would again reveal high-frequency ripples covering the surface at 

least for small values of T . Owing to the choice parameters, they are very small in comparison 

with the evanescent part, but nonetheless become prominent for large values of X, aswe shall 

see shortly. 

Remark (Evolution of the peak) The contour plots of the scattering events give 

a bright illustration of the di culties associated with the phase time concept. In this 

approach, the trajectory of the peak of the wave packet determines the tunnelling time. 

But as the plots show, it is impossible to identify the peak in front of the barrier because 

of the interference of incident and re ected wave. Hence the trajectory can only be 

extrapolated from the evolution of the wave packet far away from the barrier where 

interference has not yet distorted its shape (see also the respective remark in section 2.2). 

The last example shows a Gaussian wave packet with a narrow spectrum. The parameters 

are the same as those used in g. 4.4 , so that the fraction of the pass-band components is 

given by the left graph in g. 4.4 . The plots of the scattering process in front ofthebarrier 

88


P 

10 

7.5 

5 

2.5 

0 

-100 

140 

120 

100 

80 

60 

40 

20 

-75 

-50 

X 

-25 

0 

-100 -80 -60 -40 -20 0 

Figure 4.12: Evolution of the probability density P = j j of a scattering Gaussian wave packet 

outside the barrier. The parameters are k =6,n=4,and =0:2. 

89 

0 

0 

50 

100 

150 

T

P 

2 

1.5 

1 

0.5 

0 

0 

2 

X 

4 

0 


Figure 4.13: Evolution of the probability density P = j j of a Gaussian wave packet within the 

barrier. The parameters are k =6,n=4,and =0:2. 

( g. 4.12) provide no new or unexpected information, except that this time the shape of the 

wave packet seems to remain una ected throughout the whole event. Also inside the barrier 

( g. 4.13), the shape of the probability density looks Gaussian, and like before, there are no 

over-the-edge components that keep on propagating. 

Remark (Trajectory of the maximum) Suppose we would, at every point where 

X > 0, determine when the maximum of the wave arrives. From g. 4.13, we could 

expect this instant to be the same everywhere independent of the position along the 

barrier. Astonishingly enough, this is not the case, and we nd that the farther we shift 

the point of observation into the barrier, the earlier the peak is detected. Accordingly, 

the entire pulse becomes narrower, apart of course from the exponentially diminishing 

amplitude. While this result looks rather weird at rst sight, there seems to be a plausible 

explanation: the front ofthe pulse, as mentioned already in section 2.1, consists of the 

higher-frequency components of the spectrum, i. e. those with a high E = !~. If they are 

evanescent, which we presume here, they experience no phase shift as they penetrate into 

the barrier. Their stationary penetration depth, however, is given by the reciprocal of the 

propagation constant = j 1p 

~ 2m(V , E) and tends to in nity as the E approaches the 

barrier energy V . Hence the higher-frequency components reach deeper into the barrier 

than the low-frequency components that make up the tail of the pulse, and this is why 

the pulse seems to be shorter in a greater distance. 

The grid chosen for the computation of the preceding three-dimensional pictures does of course 

not allow for the resolution of the ne structures caused by the high-frequency components 

of the waves. It is therefore worthwhile to re-examine the evolution of the waves inside the 

barrier at selected positions, namely directly at the interface (X = 0) and su ciently far 

down the line (X = 10) where the evanescent parts of the spectra should have vanished 

compared with the transmitted components. In contrast to the previous examples, however, 

90 

50 

T 

100 

150


2.5 

2 

1.5 

1 

0.5 

P 

0 20 40 60 80 100 120 140 T 

2.5 

2 

1.5 

1 

0.5 

P 

0 20 40 60 80 100 120 140 T 

Figure 4.14: Pulse shapes produced by several short wave packets at the interface of the barrier. The 

parameter are k = 6 and = 0:2. The left graph shows the rectangular (solid line) and triangular 

(dotted line) packet. On the right side, the results of a narrow (n= 10, solid line) and a wide (n =4, 

dotted line) Gaussian packet are given. 

0.014 

0.012 

0.01 

0.008 

0.006 

0.004 

0.002 

P 

-6 

8. 10 

-6 

6. 10 

-6 

4. 10 

-6 

2. 10 

P 

20 40 60 80 100 120 140 T 

20 40 60 80 100 120 140 T 

0.0001 

0.00008 

0.00006 

0.00004 

0.00002 

-8 

5. 10 

-8 

4. 10 

-8 

3. 10 

-8 

2. 10 

-8 

1. 10 

P 

P 

20 40 60 80 100 120 140 T 

20 40 60 80 100 120 140 T 

Figure 4.15: Pulse shapes produced by several short wave packets at X = 10 inside the barrier. The 

parameter are k =6and =0:2. The graphs show the rectangular (upper left), triangular (upper 

right), narrow Gaussian (n = 10, lower left), and wide Gaussian (n =4,lower right) packet. 

91

5 

4 

3 

2 

1 

P 

0 100 200 300 400 T 

5 

4 

3 

2 

1 

P 


0 100 200 300 400 T 

Figure 4.16: Pulse shapes produced by several long wave packets at the interface of the barrier. The 

parameter are k = 20 and = 0:2. The left graph shows the rectangular (solid line) and triangular 

(dotted line) packet. In the right graph, the results of a narrow (n= 10, solid line) and a wide (n =4, 

dotted line) Gaussian packet are given. 

0.012 

0.01 

0.008 

0.006 

0.004 

0.002 

P 

100 200 300 400 T 

-6 

6. 10 

-6 

5. 10 

-6 

4. 10 

-6 

3. 10 

-6 

2. 10 

-6 

1. 10 

P 

100 200 300 400 T 

P 

-7 

1. 10 

-8 

8. 10 

-8 

6. 10 

-8 

4. 10 

-8 

2. 10 

100 200 300 400 T 

Figure 4.17: Pulse shapes produced by several long wave packets at X = 10 inside the barrier. The 

parameter are k =20and =0:2. The graphs show the rectangular (left), triangular (middle), and 

Gaussian (right) packet. In the latter, the results for a narrow (n= 10, solid line) and wide (n =4, 

dotted line) packet are displayed. 

we nowchoose the parameters so as to obtain as narrow a spectrum as possible. To permit a 

direct comparison of the spectra, we again take the parameter values of g. 4.4 together with 

alow-frequency carrier ( = 0:2). 

Fig. 4.14 shows the pulses of a short electron (k = 6) at the interface of the barrier. Not 

surprisingly, the rectangular wave packet exhibits strong oscillations. As a direct consequence 

of dispersion, the temporal frequency of these oscillations decreases rapidly in the course of 

time. On the other hand the triangular pulse, though having also spectral components in the 

pass band, shows no such oscillations. The graph also reveals that the Gaussian wave packets 

do not produce Gaussian pulses, but wave forms with a remarkable skewness. This e ect is 

more marked if the wave packet is narrow and therefore has a broad spectrum, as explained 

above. 

Inside the barrier ( g. 4.15), the wave forms undergo signi cant changes. For the rectangular 

wave packet, the high-frequency components that can propagate into the barrier become 

more pronounced. This also goes for the triangular packet, which had looked so smooth 

at the entrance. There the evanescent parts had obviously dominated the shape, but now 

92

4.4 The square barrier 

they have vanished, and only the over-the-barrier contributions remain. Even the narrow 

Gaussian pulse breaks up into distinct oscillations. Only the originally wide Gaussian wave 

packet retains its shape, of course with a dramatically reduced amplitude. So it is just this 

last case where we can assume the spectrum to have been narrow enough and su ciently 

below the top of the barrier. Note also that the amplitude of the narrow pulse is about 

two orders of magnitude larger than that of the wide pulse, which undeniably identi es the 

high-frequency components. 

The last examples explore a longer wave packet (k = 20) at the same positions. At the 

interface ( g. 4.16), the situation has not changed very much. There are still oscillations, 

although with a lower amplitude, in the rectangular pulse, and its shape can now be identi ed 

much easier than before. The skewness of the Gaussian pulses is much smaller now, according 

to the narrower spectrum. Within the barrier ( g. 4.17), it is interesting to notice that the 

rst peak of the rectangular wave packet has about the same height as for the short electron 

in g. 4.15. This demonstrates once more that the number of carrier wavelengths the particle 

embraces has no great in uence on the fraction of pass-band spectral components, as can 

also be seen from g. 4.4 . The oscillations that remain from the triangular wave packet, 

however, have become smaller as compared with the short particle before. But the most 

striking di erence is the behaviour of the narrow Gaussian pulse, whose spectrum this time 

is also entirely below the barrier level. Consequently, the relation between the amplitudes 

inside the barrier is roughly the same as at the interface. 


The investigation of the penetration of an electron into a step potential barrier provided 

already some insight into the behaviour of evanescent waves in quantum mechanics. It did 

not, however, give an answer to the question of the tunnelling time of a particle, because an 

in nitely extended step barrier is of course always opaque and permits no tunnelling through. 

Neither can the results of the step potential be easily extrapolated to a square barrier due 

to the presence of a second interface that will inevitably cause re ections and interference. 

There is thus no choice other than to examine also the case of a square potential barrier 

V (x) = 

8 

>< 

>: 

0 if xd 

: (4.34) 

Since there are now three distinct regions, we have three waves to determine. To the left of 

the barrier, there is the incident and a re ected part, 

1 = C 1 e ,j!t e j 1x + C2 e ,j!t e ,j 1x : (4.35) 

Inside the barrier, also a right- and a left-going solution will exist, 

2 = C 3 e ,j!t e j 2x + C4 e ,j!t e ,j 2x ; (4.36) 

93


whereas behind the barrier, only the transmitted part of the wave will remain, 

3 = C 5 e ,j!t e j 1x : (4.37) 

To obtain the coe cients, we set C 1 = 1 and establish the continuity conditions of the wave 

functions and their spatial derivatives at the interfaces x = 0 and x = d, respectively, 

C 1 + C 2 = C 3 + C 4 

1C 1 , 1C 2 = 2C 3 , 2C 4 

C 3 e j 2d + C4 e ,j 2d = C5 e j 1d 

2C 3 e j 2d , 2C 4 e ,j 2d = 1C 5 e j 1d : 

Solving this set of equations, we nd the coe cients 

j( 2 

C2 = 

2 , 1 2 ) sin 2d 

2 1 2 cos 2d , j( 1 2 + 2 2 ) sin 2d 

( 1 2 + 1 2 )e ,j 2d 

C3 = 


( 1 2 , 1 

C4 = 

2 )ej 2d 


C 5 = 

2 1 2e ,j 1d 

2 1 2 cos 2d , j( 1 2 + 2 2 ) sin 2d : 

(4.38) 

(4.39) 

If we furthermore express the propagation constant inside the barrier, 2, bythe one outside as 

we did in (4.9) and use our normalised variables (4.12) and (4.13) together with the normalised 

barrier thickness D = !p 

d, the coe c cients become 

C 2 = , 

C 3 = 

C 4 = 

C 5 = 

j sin D p 2 , 1 

2 p 2 , 1 cos D p 2 , 1 , j(2 2 , 1) sin D p 2 , 1 

( p p 

2 2 ,jD 2,1 

, 1+ )e 


( p p 

2 2 jD 2,1 

, 1 , )e 


2 p 2 , 1e ,j D 

2 p 2 , 1 cos D p 2 , 1 , j(2 2 , 1) sin D p 2 , 1 : 

94 

(4.40)


1 

0.8 

0.6 

0.4 

0.2 

|C5| 

0.5 1 1.5 2 

Figure 4.18: Transmission coe cient jC 5j depending on the normalised propagation constant for 

several values of the barrier thickness D (dotted line: 1, dashed line: 5, solid line: 10). Note the 

relation 2 = !=!p. 

As for the signs of the square roots, the same considerations as in section 4.1 apply, so 

that for < ,1, the negative value of the square roots must be taken, whereas > 1 

corresponds to the positive solution. Note that the wave functions inside the barrier were 

de ned for the transmissive case. If the energy of the monochromatic wave is below V0, then 

p 2 , 1 7! j p 1 , 2 , and hyperbolic functions take the places of the circular functions. 

The transmission coe cient, jC 5j, is shown in g. 4.18 in dependence of the barrier thickness. 

Not surprisingly, the thicker the barrier, the closer the frequency of an evanescent wave, 

!, must be to the cuto frequency, !p, in order to allow a noticeable fraction to tunnel 

through. Above the top of the barrier ( >1), the transmission coe cient exhibits oscillations 

depending on as well as D, and only very high-frequency waves are transmitted entirely. 

With the coe cients for the individual parts of the wave, we can formulate the wave function 

in the three regions as Fourier integrals with an arbitrary spectrum of the initial wave, A( ), 

1 = !p 

c 

Z 1 

A( ) e 

,1 

jX ,j 2T d , 

Z 1 

!p 

c ,1 

A( ) j sin Dp 2 , 1 

N( ) 

95 

e ,jX ,j 2 T d ; 

ξ 

(4.41)

2 = !p 

c 

3 = !p 

c 

!p 

c 

Z 1 

,1 

Z 1 

Z 1 

,1 

,1 

A( ) 

A( ) 

p 2 , 1+ 2 


e 

N( ) 

j(X,D) 

p 

2,1,jT 2 

d + 

p 

2 2 , 1 , 

N( ) 

A( ) 2 p 2 , 1 

N( ) 

e ,j(X,D) 

p 

2,1,jT 2 

d ; 

(4.42) 

2 

j(X,D) ,jT 

e d ; (4.43) 

where we set N( )=2 p 2 ,1 cos D p 2 , 1 , j(2 2 , 1) sin D p 2 , 1. 

4.5 Tunnelling time de nitions for a square barrier 

Before looking at the time-dependent evolution of wave packets impinging on the square 

barrier of the previous section, we brie y discuss some well-known approaches to compute 

the tunnelling time through such anobstacle. They all apply to monochromatic waves, but 

have also been used to describe the behaviour of small-band pulses, in particular Gaussian 

wave packets. 

For a start, we consider the time it takes for a free particle to travel a distance d. This can 

easily be found from the dwell time de nition (2.17), which with the monochromatic wave 

= C1ej( x,!t) and the propagation constant = p 2m~!=~ yields 

f = d 

2 

r 

2m 

: (4.44) 

~! 

We could obtain exactly the same result with the application of the group velocity de nition 

and the `classical' argument that the propagation time is the distance d divided by the group 

velocity 1=vg = p m=(2~!). With the normalised variables (4.12), this time becomes 

f = d 

p : (4.45) 

2c 

In the sequel, we shall use this undisturbed propagation delay of a free electron as a reference 

for other concepts. 

With a consideration similar to the aforementioned, we can derive a simple rst-cut approximation 

to the barrier traversal time. Supposing that the particle moves freely inside a 

potential barrier with the residual energy E , V0, we replace ! in the above expression by 

! , !p and get the so-called semi-classical time [83, 86], 

d 

s = 

2c p : (4.46) 

, 1 

Note that for waves with energy below the top of the barrier, this time is imaginary, a fact 

that has widely been critisised. Referred to the free particle, the semi-classical time reads 

s 

f 

= 

r 

96 

: (4.47) 

, 1


The de nition of the dwell time has already been given in (2.17). For a rectangular barrier, 

this time is found to be [83, 88] 

D = mk 

~ 

2 d( 2 , k2 )+k0 2 sinh 2 d 

4k2 2 + k0 4 sinh 2 ; (4.48) 

d 

where in our notation k 2 =2m!=~, k0 2 =2m!p=~, and 2 = k0 2 , k 2 . Using scaled variables 

and D = d!p=c, the expression can be written as 

D 

f 

=2 

1,2 + sinh 2Dp 1, 

2D p 1, 

4 (1 , ) + sinh 2 D p 1 , 

: (4.49) 

Buttiker [88] derived a traversal time based on the idea of a quantum clock. This traversal 

time is determined by the precession of the spin a particle experiences in a small magnetic 

eld con ned to the barrier. This Buttiker time (or Buttiker-Landauer time, as it was called 

in the survey of Hauge and St vneng [85]), consists of two components, 

with the dwell time d given by (4.48) and the Larmor time 

z = 

mk0 2 

~ 2 

In normalised notation, this reads 

z 

f 

= 

r 

1 , 

B = p d 2 + z 2 ; (4.50) 

( 2 , k2 ) sinh 2 d + 1 

2 dk0 2 sinh 2 d 

4k2 2 + k0 4 sinh 2 : (4.51) 

d 

(1 , 2 ) sinh2 D p 1, 

D p 1, 

+ 1 

2 sinh 2Dp 1 , 


: (4.52) 

The last one of the tunnelling time de nitions considered here is the phase time, which 

essentially describes the movement of the peak of a wave packet. For our square barrier, it is 

given in the literature [83] as 

p = m 

~k 

which can nally be rewritten as 

p 

f 

=2 

2 dk2 ( 2 , k2 )+k0 2 sinh 2 d 

4k2 2 + k0 4 sinh 2 ; (4.53) 

d 

(1 , 2 )+ sinh 2Dp 1, 

2D p 1, 


: (4.54) 

Prior to comparing these four traversal times for two selected cases, we add a fth based 

on the dwell time de nition. We recall that the dwell time was de ned as the ratio of the 

probability density in the barrier and the incident ux, integrated over the barrier region. 

The incident ux, however, contains also a portion that will be re ected from the barrier, and 

it is by no means evident why this re ected wave should be taken into account when a time 

97

τ 

8 

7 

6 

5 

4 

3 

2 

1 


0 0.5 1 1.5 2 Ω 

Figure 4.19: Tunnelling times referred to the propagation time of a free particle, f , for a thick barrier 

(D = 10). The curves denote the phase time p (solid line), dwell time d (dotted line), Buttiker- 

Landauer time B (dashed line), the absolute value of the semi-classical time j sj (dot-dashed line), 

and the e ective dwell time e (thick solid line). 

measure for the traversal through the barrier is sought. Therefore we try a di erent approach 

and use only the wave function inside the barrier for the computation of the ux. 

In principle, we follow the same path as in the calculation of the electromagnetic energy 

velocity in section 3.5 . The local velocity of the wave then, is the ratio between the ux 

J = j~ , 

r , r : (4.55) 

2m 

and the probability density averaged over a period in time, 

v = J 

(4.56) 

Inside the barrier, we must use 2 from (4.36). After a lengthy calculation, we end up with 

the reciprocal of the local velocity, a local transmission time 

1 

v = 

s 2m 

~(!p , !) 

2 + 2 

8 

e ,2 (x,d) + e 2 (x,d) +2 

2 , 2 

2 + 2 

; (4.57) 

with = 1 = p p 2m!=~ as the propagation constant outside the barrier and = ,j 2 = 

2m(!p , !)=~ as the one within, respectively. To obtain the e ective overall traversal time, 

98


τ 

8 

7 

6 

5 

4 

3 

2 

1 

0 0.5 1 1.5 2 Ω 

Figure 4.20: Tunnelling times referred to the propagation time of a free particle, f , for a thin barrier 

(D = 3). The curves denote the phase time p (solid line), dwell time d (dotted line), Buttiker- 

Landauer time B (dashed line), the absolute value of the semi-classical time j sj (dot-dashed line), 

and the e ective dwell time e (thick solid line). 

we integrate (4.57) over the barrier, 

Integration is straightforward, and with our scaled variables we nd 

e 

f 

= 

1 

2(1 , ) 

e = 

Z d 

0 

1 

dx : (4.58) 

v 

sinh 2D p 1 , 

2D p 1 , 

+1,2 : (4.59) 

Fig. 4.19 shows a comparison of the individual de nitions for a thick barrier. For waves 

above the top of the barrier, > 1, all approaches more or less give the same results. The 

dwell time, phase time, and the Buttiker-Landauer time are nearly identical and exhibit an 

oscillatory behaviour, whereas the semi-classical time is something like the mean value of 

the oscillations and our e ective dwell time gives an upper bound. For evanescent waves, all 

de nitions except e result in traversal times that are shorter than the propagation time of a 

free particle ( = f < 1) at least for low-frequency waves. The e ective dwell time reaches its 

maximum at = 0, which seems reasonable from the particle point of view: if the energy is 

low, it takes longer to penetrate the barrier. Note also the pole of the semi-classical time at 

=1. For a thin barrier ( g. 4.20), the di erence between evanescent and transmitted waves 

is less marked, which is of course due to the fact that the transmission coe cient for thin 

barriers is su ciently high also in the region below the barrier top. Likewise, the maximum 

of our e ective dwell time is much smaller. 

99


Remark (Absolute scaling) In the literature, diagrams like in g. 4.19 and g. 4.20 

often appear to compare various approaches. Unlike our gures, however, they give the 

tunnelling time usually directly in seconds or scaled to some xed value. Buttiker, for 

example, used f = d 

q 

2m 

instead of (4.45) and furthermore took 2 ~!p 

p !=!p as abscissa 

variable. 

The considerations leading to the de nition of (4.59) are not entirely new. Moura and Albuquerque 

[104] described the same idea in a short note in 1990, and Leavens and Aers [91] 

obtained the result by investigating Bohm trajectories. They also pointed out that this approach, 

too, is associated with a combination of a transmitted and re ected wave | this time 

arising from the rear interface of the barrier. This was reason enough for Landauer [86] to 

call into question the physical usefulness of the Bohm trajectories. 

4.6 Examples of tunnelling events 

The barrier traversal times discussed in the last section are valid in a strict sense only for 

monochromatic waves, which isnotavery practical case. Hence we follow the same procedure 

we successfully applied in previous chapters and investigate a few examples of wave packets 

impinging on rectangular barriers of di erent height and width. The goal is to determine 

the behaviour of the transmitted part of the wave, if there is any, and possibly establish a 

connection to the theoretically found tunnelling times. As we have seen in the scattering 

experiments with the step potential, discontinuities like edges or sharp peaks in the original 

wave function are smoothed out by dispersion after a short time. Therefore we limit the 

analysis to Gaussian wave packets, which o er the additional advantage of being easier to treat 

numerically. Even more important is the fact that if we want to study pure tunnelling without 

the disturbing e ects of over-the-barrier components, we must ensure a proper bandwidth 

limitation. Out of the three possible initial wave forms we proposed in section 4.2, the 

Gaussian wave is the only one to comply su ciently with this requirement. 

The departure point for the calculations are the wave functions in the three regions (4.41) 

{ (4.43) together with the initial spectrum (4.28). We evaluate the integrals numerically 

and plot the results. Like in the investigation of scattering, the probability density will be 

depicted in a three-dimensional picture as well as a corresponding contour plot that allows a 

better perception of the paths the individual waves move along. In addition, we determine 

the trajectories of the peak of the wave packets in order to gain insight into the controversial 

question whether they can be ascribed a physical meaning to or not. 

At rst, we consider a high-energy electron interacting with a wide barrier. The parameters 

of the initial wave packet are the same as in the rst scattering example of section 4.3, and 

g. 4.21 shows the results. The edges of the barrier are marked with lines on the faces of the 

plotted area. In addition, the three-dimensional picture was rotated so that the region outside 

the barrier is on the right side of the plot. The behaviour in front of the barrier is similar 

to the scattering case, there is the well-known interference between incoming and re ected 

waves. Behind the barrier, we notice a faint wave emerging from the tunnel. More interesting, 

however, is the wave function inside the barrier itself. As the incident wavepenetrates into 

100


P 

2 

40 

1 

0 

40 

40 

30 

20 

10 

30 

20 

T 

20 

10 

0 

X 

0 

-20 

-40 

0 

-40 -20 0 20 40 

Figure 4.21: Evolution of the probability density P = j j of a Gaussian wave packet with the 

parameters k =6,n= 4, and = 0:8 impinging on a barrier of thickness D = 10. 

101


the tunnel and bounces back again, a fraction of it seems to get caught in the barrier, running 

back and forth between the interfaces while decaying not too quickly. This peculiar e ect 

has nothing to do with evanescent waves but stems from the ordinarily transmitted parts 

of the spectrum. It is just the common observation made for propagating waves that every 

change in the medium inevitably causes re ections. Looking carefully, we even discover a 

small after-runner of the re ected wave (at T = 40 and X ,20 in the three-dimensional 

plot), which can be traced back to the instant when the wave inside the barrier hits the left 

interface (at X = 0) for the rst time. 

To investigate this behaviour in greater detail, we regard an even thicker barrier and the same 

electron as before (as in the close-up of g. 4.22). In this case, the bouncing of the wave inside 

the barrier is more marked, and each time it reaches an interface, a small wave is released to 

the outside. This way, apart from the widening of the pulse due to dispersion, the inner wave 

gradually fades away. 

Returning to the original example, we examine the trajectory of the peak of the wave packet 

( g. 4.23). To determine this path, we havetwo possibilities to identify the maximum (see also 

the discussion in section 1.5): a temporal and a spatial one. Here we consider the temporal 

maximum, i. e. the moment in time for a given position in space when the observed eld reaches 

its maximum amplitude. We recognise the linear motion of the incident and transmitted wave, 

as well as the not surprising fact that a de nition of the peak's trajectory is not meaningful 

in the interference zone in front of the barrier. Interestingly enough, the trajectory of the 

transmitted wave is more strongly inclined towards the axis than the incident wave packet, 

meaning that the packet to the right of the barrier runs faster. Again, this is largely due to 

the spectral components that can propagate over the barrier, whereas the evanescent parts are 

entirely attenuated by the thick obstacle. Hence the low-frequency components are ltered 

out, and the wave packet emanating from the far side of the barrier now has a spectrum that 

is narrower, but centred about a higher frequency than before. Here and in the subsequent 

examples, the actual centre frequencies for both the incoming and transmitted wave packets 

are computed from the trajectories and given in the captions of the respective gures. In the 

current example, this centre frequency is even above the cuto frequency of the barrier. The 

propagation of the high-frequency part is also responsible for the s-like shape of the trajectory 

within the barrier and the o set between the extrapolated trajectories of the incident and 

transmitted waves. 

It is not di cult to calculate the actual centre frequency from the trajectories. Since the 

wave packets are smooth and well-behaved, we may safely take the view of the phase time 

supporters and assume that the wave proceeds at the group velocity, which wehavefound to 

be 

r 

2~! 

vg = : 

m 

(4.60) 

It is easy to t a straight line into the space-time points of the trajectory outside the barrier 

and away from the interference region, and from the normalised variables, we obtain directly 

102


200 

150 

100 

50 

30 

20 

0 

50 

10 

X 

T 

100 

0 

150 

0 

-10 0 10 20 30 

200 

0 

-10 

0.1 

0.075 

P 

0.05 

0.025 

Figure 4.22: Probability density P = j j ofawave packet with the parameters k =6,n= 4, and 

=0:8 inside a barrier of thickness D = 20. 

103

30 

25 

20 

15 

10 

5 

T 


-10 0 10 20 30 

Figure 4.23: Trajectory of the temporal peak of a wave packet with the parameters k =6,n= 4, and 

=0:8interacting with a barrier of thickness D = 10. The e ective centre frequency of the incident 

wave is ci 0:8181, for the transmitted wave itis ct 1:11. 

a relation to the carrier frequency !c=!p = c, 

T 

X 

c t 1 

= = 

vg t 2 p : (4.61) 

c 

Remark (Calculation of the centre frequency) For the incident wave, the e ective 

centre frequency c should of course be equal to the carrier frequency selected for the 

initial wave. The value obtained from the trajectory, however, is slightly di erent, and 

experiments show that for a trajectory consisting of spatial maxima, the correspondence 

would have been better. This is not surprising as the initial wave packet was de ned in 

space rather than in time. Though being fairly close together, the spatial and temporal 

maxima are still slightly di erent. 

It should also be noted that the centre frequency calculated from the trajectory is not the 

peak frequency of the spectrum, as one could suspect. According to (4.43), the spectrum 

of the initial pulse is ltered through the transmission coe cient, which introduces a 

skewness to the originally symmetric distribution. In the particular case of = 0:8 , the 

spectrum is even oscillatory, because of the resonance peaks of the transmission coe cient 

above cuto (see also g. 4.18). 

Figs. 4.24 and 4.25 show the situation for the unaltered initial wave packet, but with a decreasing 

width of the barrier. The e ects the obstacle exerts on the wave remain in principle 

the same as before, with one great exception: there is no residual wave inside the barrier after 

104 

X


-40 

40 

30 

20 

10 

-20 

0 

X 

40 

30 

20 

T 

20 

0 

-40 -20 0 20 40 

10 

0 

1 

0 

40 


=0:8interacting with a medium-sized barrier (D = 5). 

105 

2 

P

P 

0 

3 

2 

1 

10 

0 

-40 

T 

40 

30 

20 

10 

20 

30 

-20 

40 

0 

X 

20 


40 

0 

-40 -20 0 20 40 

Figure 4.25: Probability density P = j j of a wave packet with the parameters k =6,n= 4, and 

=0:8interacting with a thin barrier (D = 1). 

106


the interaction took place. Seemingly, the barrier is too thin for a fully developed wave packet 

to evolve. As we could have expected, the transmitted wave grows stronger as the barrier size 

is reduced, and for the thin obstacle even surpasses the re ected wave in magnitude. Not at 

all unexpected comes the form of the trajectories of the two wave packets ( gs. 4.26 and 4.27, 

respectively). In the example of the medium-sized obstacle, the trajectory of the transmitted 

wave packet is still more inclined than that of the impinging wave, albeit not to such an extent 

as with the thick barrier above. For the thin barrier, on the other hand, this e ect is hardly 

noticeable at all, which con rms the intuitive suspicion that a thin barrier does less harm to 

the spectrum of a pulse since the evanescent components have a better chance of tunnelling 

through. 

Apart from the di erent inclinations of the trajectories, there is another instructive detail 

the pictures reveal. Suppose we regard only the interfaces of the barrier in search of a 

plausible way to de ne a tunnelling time. Intuitively, we could measure the delay between 

the appearance of the wave packet's peak at the front and rear edge. Such an approach is, 

however, deceptive in that it disregards the way the peak at the front interface is generated. 

In fact, it is not the peak of the incident wave packet but it originates rather incidentally 

from the superposition of incoming and re ected wave. Actually, it reaches the barrier later 

than the free wave packet would without the presence of the obstacle. For the spectator at 

the front edge of the barrier, it looks as though the approach of the packet is slowed down by 

interference. Therefore the trajectory, if at all, is meaningful only in an asymptotic sense if 

extrapolated from the undisturbed region to the barrier. 

We now explore a di erent initial wave packet, namely one with a spectrum su ciently 

con ned to the evanescent region. Since we can expect genuine tunnelling this time, we must 

choose a rather thin barrier in order to obtain a recognisable wave packet behind the tunnel. 

Indeed, g. 4.28 shows that a medium-sized obstacle already re ects almost the complete 

incident wave, and practically nothing reaches the far side of the barrier. The situation is 

naturally better for a thin barrier, as g. 4.29 demonstrates. It is clear, though, that the lower 

the centre frequency of the wave is, the narrower the barrier must be in order to obtain a 

noticeable tunnel e ect. 

The inspection of the trajectories of the wave packets outside the barrier also yields nothing 

unexpected. Like before, the path of the wave leaving the far side of the barrier is more 

inclined towards the axis, which emphasises once more the high-pass behaviour of the barrier. 

Consequently, this e ect is more pronounced if the barrier is wide. So far, the results qualitatively 

agree with those obtained for the higher-frequency wave packet treated before. There 

is, however, a signi cant di erence: from a comparatively thick barrier (like in g. 4.30), the 

peak of the tunnelling wave packet emanates before the peak of a freely moving packet would 

have reached the front of the barrier. Accordingly, the trajectory even runs backwards in time 

inside the barrier. This phenomenon is still more impressive for thicker barriers ( gs. 4.32 and 

4.33). At the rst super cial examination, this could be taken for a violation of causality, 

because it indeed looks like turning the clock back while the electron is tunnelling through 

the wall. Taking one step further, we could as well see this result as a proof for a wave packet 

travelling faster than the speed of light | although the velocity of light actually never showed 

up throughout the entire derivation of the tunnel e ect. 

107

25 

20 

15 

10 

5 

T 


-10 0 10 20 30 


=0:8interacting with a medium-sized barrier (D = 5). The incident wave is actually centred about 

ci 0:8181, the transmitted wave about ct 0:996. 

25 

20 

15 

10 

5 

T 

-10 0 10 20 30 


=0:8interacting with a thin barrier (D = 1). The e ective centre frequencies are ci 0:8181 for 

the incident and ct 0:8293 for the transmitted wave. 

108 

X 

X


P 

2 

0 

1 

0 

140 

120 

100 

80 

60 

40 

20 

50 

T 

-50 

100 

X 

0 

50 

0 

-80 -60 -40 -20 0 20 40 60 


=0:2interacting with a medium-sized barrier (D = 3). 

109

P 

2 

0 

1 

0 

140 

120 

100 

80 

60 

40 

20 

50 

T 

-50 

100 

X 

0 


50 

0 

-80 -60 -40 -20 0 20 40 60 

Figure 4.29: Probability density P = j j of a wave packet with the parameters k =6,n= 4, and 

=0:2interacting with a thin barrier (D = 1). 

110


80 

60 

40 

20 

T 

-30 -20 -10 0 10 20 30 

Figure 4.30: Trajectory of the temporal peak of a wave packet with the parameters k = 6, n = 4, 

and =0:2interacting with a medium-sized barrier (D = 3). The e ective centre frequencies are 

ci 0:2045 for the incident and ct 0:217 for the transmitted wave packet. 

80 

60 

40 

20 

T 

-30 -20 -10 0 10 20 30 

Figure 4.31: Trajectory of the temporal peak of a wave packet with the parameters k = 6, n = 4, 

and = 0:2 interacting with a thin barrier (D = 1). The incident wave isactually centred about 

ci 0:2045, the transmitted wave about ct 0:2103. 

111 

X 

X


Thinking the results over again, we recognise that the aforementioned speculations have no 

foundation. The wave packet seems to be advanced, but as a matter of fact, it is not. Rather, 

we have one more example of pulse reshaping, an e ect we encountered already in section 

2.1. It is the high-frequency components in the wave packet that arrive earlier at the barrier 

than the rest and in addition have a greater tunnelling probability (or penetration depth). So 

it is these spectral components that constitute the packet behind the barrier, which wehave 

seen from the increased group velocity. Hence the peak is not conserved in the tunnelling 

process. The fact that we could draw a trajectory inside the barrier stems from the de nition 

of the maximum we decided to observe. Had we not taken the temporal maximum of the 

wave function, but the spatial counterpart, we would not have been able to nd a peak within 

the barrier at all, due to the exponential decay. 

There is still another aspect of the trajectory interpretation: if the barrier is wide enough that 

the e ects mentioned above are signi cant, the maximum of the probability density beyond 

the wall becomes vanishingly small. So the trajectory covers the important fact that we are 

trading propagation velocity for tunnelling probability or, in other words and from a practical 

point of view, signal speed for signal energy. 

Though being con ned to the evanescent region, the spectrum of the initial pulse in the 

previous examples still had a considerable width originating from the fact that we investigated 

very `short' electrons with k = 6. This is of course no reason to raise doubts about the 

correctness and validity of the results, in particular the nding that pulse reshaping would 

allow for negative tunnelling times, if the peak of the wave packet is taken as a reference point 

for measuring the propagation. Yet it is sensible to consider the behaviour of electrons with 

a narrower spectrum. We can expect that in such a case, the acceleration of the transmitted 

wave packet will be less prominent because the spectrum rapidly decreases to both sides of 

the original centre, leaving little room for a frequency shift. Consequently, the backward-shift 

in time of the trajectory inside the barrier will occur to a lesser extent, so that eventually 

the tunnelling time will reach a small but positive value. Indeed, g. 4.34 satis es these 

expectations. The extrapolated trajectory of the incident wave packet is nearly parallel to 

that of the transmitted wave, and the small velocity di erence is noticeable only in the 

numerical determination of the trajectory ascent. 

Let us now try to establish a connection to the various tunnelling time de nition of the last 

section. Remaining with the trajectory approach, we must nd unambiguous de nitions of the 

paths the wave packets move along. Behind the barrier, this is easy because the trajecory is 

linear. Immediately in front of the barrier, however, interference impairs a clear perception of 

the maximum. Obviously we must x the trajectory to points far away from the barrier where 

the incident wave is still undisturbed. We have seen already that the velocity of the wave 

packet can safely be assumed to be the group velocity of its carrier frequency, and we know 

of course the exact initial position x0 of the peak at t =0. Wecan then calculate the time 

the wave packet takes to arrive at the barrier | it is simply the propagation time required 

to cover the distance x0 if no barrier is present at all. The moment when the transmitted 

peak emanates from the rear interface can be computed from the trajectory like before, and 

subtracting the two times we obtain what can be regarded as the tunnelling time. This is the 

approach pursued also by Collins et al. [83]. 

112


80 

60 

40 

20 

T 

-30 -20 -10 0 10 20 30 


=0:2impinging on a thick barrier (D = 10). The e ective centre frequencies are 

0:23385. 

ci 0:2045 and 

ct 

80 

60 

40 

20 

T 

-20 0 20 40 


=0:2 impinging on a very wide barrier (D = 20). The e ective centre frequencies are ci 0:2045 

and ct 0:2617. 

113 

X 

X

200 

150 

100 

50 

T 


-125 -100 -75 -50 -25 0 25 

Figure 4.34: Trajectory of the temporal peak of a wave packet with the parameters k = 20, n =4, 

and = 0:2 impinging on a thick barrier (D = 10). The e ective centre frequencies are ci 0:2004 

and ct 0:2028. 

From the wave form de nitions in section 4.2, we nd the initial position of the peak, 

X0 = k 

p : (4.62) 

The time the wave packet needs to travel this distance is readily obtained from the propagation 

velocity of the free electron (4.45) 

Tinc = X0 k 

p = 

2 2 

: (4.63) 

Since we want to explore the dependence of the tunnelling time on the carrier frequency, we 

must ensure that the spectrum has equal width for all values of . The dominating factor in 

the wave number spectrum (4.28) is e ,( k=n ( =p ,1)) 2 

, whichs yields the relation 

k 

p = ; (4.64) 

n 

with some arbitrary, non-negative constant . The width of the initial wave packet, n, is held 

constant, is our independent variable, so we must set k = n p = in the spectrum and 

get the nal expression for the time of free motion, 

Tinc = n 

p : (4.65) 

2 

114 

X


7 

6 

5 

4 

3 

2 

1 

τ 

0.5 1 1.5 2 2.5 3 Ω 

8 

6 

4 

τ 

0.5 1 1.5 2 2.5 3 Ω 

Figure 4.35: Numerically determined tunnelling times compared with the phase time theory for thin 

(D = 1, left graph) and medium-sized (D = 5, right graph) barriers and a small-band Gaussian 

electron ( = 100, n = 4). 

20 

15 

10 

5 

-5 

-10 

τ 

0.5 1 1.5 2 2.5 3 Ω 

20 

15 

10 

5 

τ 

0.5 1 1.5 2 2.5 3 Ω 

Figure 4.36: Numerically determined tunnelling times compared with the phase time theory for a thick 

barrier (D = 10), but electrons of di erent bandwidths (left graph: =10:54 b= k = 6 at = 0:2, 

right graph: = 35:12 b= k = 20 at = 0:2, n = 4 in both cases). The left plot also shows the 

semi-classical time (dashed line). 

Fig. 4.35 shows the results for a small-band electron and two di erent barrier widths. Out of all 

theoretical tunnelling time approaches in section 4.5, the phase time suits best the numerical 

results. This is not really surprising, because the way we set up the numerical computation of 

the tunnelling time mimics exactly the basic idea of the phase time. We follow the motion of 

the wave packet's maximum, and furthermore the spectrum is sharply peaked about a single 

wave number. Note that for low incident energies, the phase time prediction and the numerical 

results disagree. Collins et al. attributed this behaviour to the transmission coe cient they 

thought to be exponentially varying at low energies | an argument we cannot follow after 

inspection of g. 4.18, which shows the transmission coe cient to be particularly smooth for 

small = p !=!p. Rather, we can identify the inevitable pulse reshaping phenomenon as 

the reason of this discrepancy. Owing to our assumptions, the spectral width of the wave 

packet is held constant irrespective of its centre frequency. Hence the extent to which the 

frequency can be shifted during the tunnelling event is also xed or at least upper-bounded. 

115

17.5 

15 

12.5 

10 

7.5 

5 

2.5 

τ 

0.5 1 1.5 2 2.5 3 Ω 

35 

30 

25 

20 

15 

10 

5 

τ 


0.5 1 1.5 2 2.5 3 Ω 

Figure 4.37: Numerically determined tunnelling times compared with the phase time theory for a thick 

barrier (left graph: D = 10, right graph: D = 20) and small-band electrons ( = 100, n = 4). 

The frequency shift may be small, but it always occurs. The relative variation of the group 

velocity, on the other hand, is much larger for very small energies because the group velocity 

approaches zero in the DC-limit, and so the acceleration e ect the wave packet experiences 

is larger, too. 

Remark (Trajectories inside the barrier) The above explanation becomes even 

more elucidating if we consider how the trajectory inside the barrier emerges. We already 

pointed out (see section 4.3) that it is the result of the successive arrival of the wave 

packet's components, all travelling at distinct velocities. The rst to arrive are the highenergy 

parts that also have a higher tunnelling probability. Hence comes the inclination 

of the trajectory towards the axis. The steady-state tunnelling probability corresponds to 

the stationary penetration depth, which naturally is very small for low energies, so that 

its relative increase due to a xed frequency shift is much larger for low than for high 

energies. 

The considerations given above also anticipate what will happen if we choose a broader 

spectrum for the incident wavepacket. Fig. 4.36 shows the results for a thick barrier, where 

the e ect is even more pronounced. The left graph was drawn with the same parameters we 

chose for calculating the trajectories of the low-energy electrons earlier in this section. We 

recognise that for . 0:5, the numerically determined tunnelling time becomes negative, 

which is in line with the observations we made when computing the trajectories. Likewise, 

there is no resemblance at all between the numerical results and the phase time theory | 

on the contrary, they are rather close to the simple semi-classical time in that they do not 

exhibit any resonance peaks above cuto and have a maximum exactly at = 1, in contrast 

to all other de nitions. 

The situation changes if the bandwidth of the impinging electron is reduced like in the right 

picture in g. 4.36, where the parameters of the trajectory in g. 4.34 were used. Finally, in 

g. 4.37 we selected again an incident wavepacket with a `properly' narrow bandwidth and 

obtain the expected excellent agreement between numerics and the phase time prediction. 

116


The two graphs also reveal that the deviation between the two curves deteriorates as the 

barrier thickness is increased. 

The conclusions to be drawn from the above are: 

The barrier acts like a high-pass lter favouring the high-energy components of the 

incident wave packet. Thus the transmitted portion of the wave experiences a frequency 

skewing and a shift of its e ective centre frequency to higher values. This pulse reshaping 

e ect occurs always, no matter how narrow-band the wave packet originally was, and 

it is more marked for low-energy waves. 

The phase time theory is in good agreement with numerically found tunnelling times, 

provided that the wave packet exhibits a dominating peak that can be traced and 

taken as a measure for the propagation. Furthermore, the wave packet must also be 

very narrow-band so that the group velocity approximation for the propagation velocity 

holds. The spectrum need not be con ned to the evanescent region, and it may also 

extend into the pass band. 

The phase time approximation becomes worse for very low energies of the incident 

electron due to the relatively severer e ect of pulse reshaping in this region. 

If the initial wave packet is not small-band, the phase time gives no appropriate description 

of the tunnelling time except for very large energies in the pass band. If the 

tunnelling time is still based on the evolution of the peak of the wave packet, even 

negative delay times may be obtained. It is questionable whether the de nition of the 

tunnelling time is still sensible then. 

As for the faster-than-light debate, we stress that the foregoing examination was based 

on the non-relativistic Schrodinger equation. In the entire chapter, the velocity of light 

appeared not a single time. So in order to judge how fast the particles or waves actually 

travel, we would need to take the theory of relativity into account. Note also that in 

contrast to the electromagnetic case, the quantum mechanical wave function has no nite 

wave front velocity. The reason for this di erence is the quadratic dispersion relation 

of quantum waves, whereas those of wave guides and lossless plasmas are hyperbolic. 

This exposes the limitation of the analogy between undersized wave guides and square 

barriers: it is valid as long as the respective di erential equations are regarded. As one 

looks into their solutions, however, it breaks down. 

117

118 

4 One-dimensional quantum tunnelling

Interlude 

Wave functions in graphical 

representation 

Wave functions are complicated things. They are most easily formulated as Fourier integrals 

| provided, of course, we can nd the Fourier transform of some initial state of the wave 

and have a closed expression for the dispersion relation. In many cases, these are only 

minor hurdles, and the formulation of the wave integral is readily found. But this is already 

where simplicity ends and troubles start, for the evaluation of the integral is under most 

circumstances anything but straightforward. There are rare cases where the initial wave or 

the dispersion relation are so well-behaved that the inverse Fourier transform can be calculated 

right away or after application of some sophisticated mathematical tricks. Sometimes, special 

values for x or t also make integration feasible. However, normally the wave functions balk 

at analytical evaluation. 

On the other hand, it is clearly desirable and valuable to be able to calculate the wave 

integrals in order to depict at least snapshots of the wave and to underline analytical ndings 

in a graphical way. The early investigators of wave propagation were hard-pressed to tackle 

this problem and had no choice other than to resort to approximation techniques. For the 

examination of the wave front and the rst precursor in a Lorentz medium, Sommerfeld [12] 

used a high-frequency expansion of the dispersion relation, which then led to integratable 

expressions. This approximation, however, was valid only about the wave front where highfrequency 

components are dominating. Brillouin [19] and Baerwald [21, 22], who looked into 

the evaluation of the signal behind the wave front, employed the method of saddle point 

integration where they made use of two approximations: the rst one being the fact that 

the relevant contributions to the wave integral come from the vicinities of the saddle points, 

and the second one being the approximation of the integrand about these saddle points. By 

using these techniques, the authors could reduce the integrals to combinations of well-known 

functions and eventually obtain graphical representations. 

With the advent of electronic computing machines, the problems with the evaluation of wave 

integrals were alleviated. First, computers allow us to make more accurate plots of given 

functions since they can calculate function values at a much larger number of points much 

faster than it was possible with the tedious and error-prone look-up methods using mathematical 

tables. Second and more importantly, numerical methods are available to evaluate 

integrals that would be impossible be by hand calculations. Still, numerics are no general 

remedy for all problems raised by the investigation of wave functions. Actually, the di culty 

of nding a closed form for the integral at all is only transformed into a question of computing 

119

Interlude 

power and numerical accuracy, both of which can in turn be hard to cope with. Consequently, 

particularly in the early days of computers, approximation and expansion methods were still 

widely employed to obtain reasonable results within reasonable time. 

Scanning the literature, we nd two principal approaches to treat the wave propagation problem 

by means of numerical computing. The rst is to formulate the wave integral and then 

apply numerics to evaluate the expression. As mentioned before, the integrand is often simplied 

or transformed via series expansion to facilitate integration either in terms of convergence 

or accuracy. This way the normally in nite integrals may frequently be truncated at some 

nite value without introducing too large an error. The second approach is to dispense with 

the wave integral itself and start from the di erential equation together with the appropriate 

initial and boundary conditions describing the propagation phenomenon. This usually 

involves some sort of discretisation of the di erential equation on a properly chosen grid in 

order to nd a numerical solution. Such a strategy was for example pursued by Weiland [105], 

who used it calculate electromagnetic elds in complex geometric structures. He divided the 

area of interest into small elementary cells of essentially arbitrary shape. Out of the many 

possible discretisations, he selected one that uses the integral form of the Maxwell equations 

such that at the boundaries between the cells, the tangential component oftheelectric eld 

and the perpendicular component of the magnetic eld are continuous (which is not necessarily 

the case if other discretisation schemes are used). Solving Maxwell's equations is then 

reduced to the easier task of solving matrix equations. 

The alternative approach of evaluating the wave integrals numerically has been used by a 

larger number of authors | at least as far as electromagnetic wave propagation is concerned. 

At the beginning of the computer era, Haskell and Case [97] did not really evaluate the integral 

directly, but used the familiar method of saddle-point integration to obtain an approximation 

for large values of x. This strategy was stimulated by the observation that most solutions at 

that time were suitable only for small propagation distances, and they ended up with a solution 

consisting of Fresnel integrals that could be computed numerically. As for the question which 

part of the integrand is to be expanded, a variety of aswers were attempted. Knop [96], for 

example, expanded the sine function de ning the input signal of the transmission line into a 

series of Bessel functions and then applied the inverse Laplace transform. Trizna and Weber 

[23], on the contrary, expanded the entire integrand also into a series of Bessel functions 

that could be integrated numerically. More recently, Wyns et al. [25] used inverse Laplace 

transform, but prior to that, approximated the factor e st by a function which then could 

be treated by series expansion. As a result, they could integrate the simpli ed functions 

analytically and employ a truncated summation to achieve the desired accuracy. Albanese et 

al. [106] cleverly sidestepped the entire problem of Fourier transform by regarding a repetitive 

input pulse so that the Fourier integral degenerated to the much simpler Fourier sum. Bolda 

et al. [74], as the last and most recent example, used the modern method of Fast Fourier 

Transform (FFT) to compute the propagation of a Gaussian pulse in an inverted medium. 

The term `wave propagation' is deeply linked to a strong and colourful impression of a lm-like 

sequence of individual pictures showing the motion of a wave packet. It should thus not come 

as a surprise if some authors took up this idea and attempted to generate such lms in order 

to bring the term `motion' to optical life. Indeed, there have been such approaches in the 

120

Wave functions in graphical representation 

past. It is interesting to notice that they were chie y restricted to quantum mechanical wave 

packets. Seemingly, the imagination of quantum particles moving like waves was much more 

demanding and called for a visualisation rather than the analogous problem of electromagnetic 

wave propagation. There are quite a number of examples of computer animation of wave 

packets, the rst and nowadays almost legendary one being that of Goldberg et al. [107], 

who analysed the scattering of a Gaussian wave packet from potential barriers and wells 

as early as 1967. To obtain the wave functions they did not formulate the wave integrals, 

but instead solved the Schrodinger equation numerically. The method they used is the wellknown 

Crank-Nicholson algorithm that essentially is based on an equidistant discretisation 

of the di erential equation. This in turn leads to a set of recursive equations, and to get 

initial values for the recursion, the entire system is placed in a box with in nitely high walls. 

Clearly the walls must be su ciently far away from the region of interest to avoid re ections 

during the considered time span. In addition, the resolution must be chosen ne enough to 

follow the oscillations of the wave function. 

The Crank-Nicholson scheme has become the method of choice for many other authors, too. 

Collins et al. [108, 83] did not shoot movies with it, but used it to simulate the evolution of 

a Gaussian wave packet in order to compare their review of tunnelling time de nitions with 

numerical results. They reported that the discretisation of the discontinuous square barrier 

potential pro le was crucial for the success of the method and had a signi cant in uence on the 

results. In the recent past, computers have become powerful enough that this method can even 

be employed to generate on-line animations of wave motion, like in the Mathematica-based 

example of Robb [109], who combined the convenience of a modern mathematical software 

environment with the computational power of a `workhorse' written in C. A software package 

solely dedicated to quantum physics is the program of Hiller et al. [110], which besides many 

other functions to explore quantum mechanics, also contains a section where the motion of a 

wave packet through a given potential structure can be observed. 

Somehow di erent to the aforementioned examples is that by Merrill [111], who in the early 

days of computers presented a simple BASIC program to study the propagation of waves in a 

dispersive medium. Intended | like all other animation approaches | primarily for students, 

it required the initial wave form and the dispersion relation to be supplied by the user. By 

use of the trapezoidal rule, it then computed the spectrum of the initial wave form as well as 

the inverse Fourier transform of the wave at some later time. 

Trying to present moving pictures in a written work is a practically futile attempt due to 

the obvious lack of motion. There is only one conceivable situation where something like a 

contiuously owing time enters an otherwise static book | when a reader quickly thumbs 

through the pages without taking the time to read through them. While clearly being undesirable 

from the author's point of view, this opens a subtle possibility to still convey some 

information, even information a thorough reader would hardly ever recognise. Hence there 

are two `thumb movies' included in the outer top corners of the pages. The sequence on the 

odd pages shows a Gaussian wave packet tunnelling through a square barrier like in section 

4.6. The parameters | for the sake of completeness | are n =4,k=6, =0:8, and D =4. 

The displayed spatial range is [,40; 40], and time runs from T = 0 to approximately T = 42. 

The barrier has been portrayed as a rectangle, however, its height has no physical signi cance 

121

Interlude 

whatsoever. Consequently the interference peaks in front of the interface can grow higher 

than its top without swashing over it. 

The even pages depict the evolution of an initially triangular wave packet impinging on a 

step barrier like in section 4.3. The parameters of this wave packet are k =3and =0:8. 

The spatial region of interest is [,40; 40], and like before, the time interval is [0; 42]. Here, 

too, the height of the barrier is shown only for graphical reasons and has no actual meaning. 

In order to provide a spectacular result, the wave number spectrum was chosen so as to 

comprise considerable pass-band components as well, which is why we can see a small and 

rapidly widening wave running further into the barrier. 

122

Part II 

Numerical aspects of wave 

equations 

In the rst part, we focused on the physical meanings of energy velocity and wave propagation. 

To this end, we took the numerical solutions of the wave equations for granted and left aside 

the particular problems associated with these computations. It is now time to make up for this 

de ciency and demonstrate how we obtained the numerical answers to our physical questions. 

The second part of this work is therefore dedicated to the computation of in nite integrals 

by means of numerical algorithms. 

In the following chapter we shall rst give anoverview of the basics of this branch ofnumerical 

mathematics and various approaches that have been published over the years. Unfortunately, 

we shall discover that none of the commonly available software algorithms is suited to help 

us solve our problems. Consequently, the next chapters deal with general thoughts on the 

particular di culties and nally the implementation of a suitable quadrature strategy to 

compute the type of wave integrals we are interested in. Although the implementation of the 

quadrature routine itself is application-oriented, it is still independent of the actual examples 

we examined in the rst part. Therefore, we shall also present how the general functions are 

applied to the problem of the tunnelling particle. 

The programming language used for the actual implementation of the functions was Mathematica 

| despite FORTRAN's still dominating role in the world of numerical mathematics. 

There were several reasons for this choice: rstly, Mathematica o ers a high-level functional 

programming environment which makes it ideally suited to the prototyping of algorithms. 

Secondly, it can perform symbolic as well as arbitrary-precision numerical calculations. Although 

quadrature functions naturally place the main emphasis on numerical computation, 

the symbolic capabilities are also needed for auxiliary functions and the testing of the implementation. 

An additional advantage is that | in contrast to FORTRAN | Mathematica 

can cope with complex-valued integrands in arbitrary precision, which allows a very convenient 

programming of certain wave integrals. Finally, the visualisation of the results requires 

no extra e orts and is readily accomplished from within Mathematica. The cumbersome 

and time-consuming computations could of course be implemented in the signi cantly faster 

FORTRAN at a later date, but this was beyond the scope of this work. 

123

Chapter 5 

Numerical quadrature and 

extrapolation 

5 Numerical quadrature and extrapolation 

Someone has recently de ned an applied mathematician as an individual 

enclosed in a small o ce and engaged in the study of mathematical problems 

which interest him personally; he waits for someone to stick his head 

in the door and introduce himself by saying, \I've got a problem." Usually 

the person coming for help may beaphycicist, engineer, meteorologist, 

statistician, or chemist who has suddenly reached a point in his investigation 

where he encounters a mathematical problem calling for an unusual 

or nonstandard technique for its solution. [...] The range of mathematical 

topics from which queries may arise is all-inclusive. However, a topic which 

arises frequently enough to merit some discussion is one which particularizes 

the statement \I've got a problem," to \I've got an integral." 

Milton Abramowitz [112] 

In the course of the case studies in the rst part, we encountered exclusively solutions in the 

form of in nite wave integrals. Unfortunately, these functions cannot be integrated analytically, 

and therefore we must resort to numerical quadrature. The term numerical quadrature 

is generally used in numerical mathematics to distinguish the process of nding a numerical 

value for a de nite integral from the numerical solution of di erential equations, which is then 

called numerical integration. 

The objective of this chapter is to give an overview of relevant aspects of this part of numerics. 

In particular, these are univariate quadrature and the problem of convergence or 

series acceleration, which is almost inevitable if one tries to compute integrals over an in nite 

range. Many of the presented algorithms have been incorporated in specialised computer routines 

that provide ready-to-use solutions. Particular emphasis is placed upon these computer 

routines and their applicability to the class of integrals we are interested in. 

124

5.1 Univariate numerical quadrature 


All e cient quadrature methods approximate the integrand by elementary functions (polynomials) 

whose integrals can be determined analytically. This is fairly simple if the integrand is 

smooth. Oscillatory functions, however, are di cult to approximate and require either higher 

order polynomials or a subdivision of the integration interval. Things get even worse if the 

interval is in nite, like it is for our wave integrals. In this section we shall thus give a brief 

survey of both published algorithms and available computer routines that tackle the problem 

of univariate quadrature of in nitely oscillating functions. 

5.1.1 Algorithms 

Many authors have considered nite integrals of the form 

I = 

Z b 

a 

f(x) e jq(x) dx : (5.1) 

Such an approach may be applicable to in nite integration intervals if the integrand decays 

fast enough (exponentially) that a truncation is justi ed, so a look at some of the methods 

is reasonable (see also [113]). Levin [114] pursues the idea that if f were of the form f(x) = 

jq 0 (x)p(x)+p 0 (x), then the integral could be evaluated directly. Thus the integration problem 

is transformed into the problem of solving a di erential equation. 

Evans [115] subdivides the integral 

I = 

NX 

i=1 

Z a+ih 

a+(i,1)h 

f(x) sin 

cos 

!q(x)dx (5.2) 

and approximates both f(x) and q(x) in each subinterval by linear or quadratic forms, respectively. 

These can be integrated analytically, which is equivalent to the trapezoidal and 

Simpson's rules for general quadrature. The quadratic approximation in fact dates back to 

a remarkably early idea of Filon [116]. An entirely di erent approach is that of Ehrenmark 

[117] who proposes a three-point formula with weights chosen such that the formula is exact 

for the functions x, sin kx, and cos kx. This method is fairly easy to implement and, according 

to Evans [115], lends itself to an extension to a higher number of points. 

Xu and Mal [118] compute wave number integrals of the form R b 

f(x) cos(rx) dx by a modi ed 

a 

Clenshaw-Curtis scheme where they approximate only f(x) by Chebyshev polynomials. To 

achieve the desired accuracy, they turn to adaptive interval subdivision if the given maximum 

order polynomial approximation is not su cient. For certain integrands, specialised complexplane 

techniques as descibed by Davies [119] may be employed. By transforming the contour 

of integration, the integral can be evaluated easily. 

If the integration interval is in nite and the decay of the integrand is such that the integral 

converges too slowly, a common technique is to partition the integral and to extrapolate from 

a nite series of subintervals to its limit. All proposed methods require some knowledge of 

125


the integrand, especially about the distribution of its zeros. Some of the following methods 

are also discussed in the classic book of Davis and Rabinowitz [113] and the survey paper of 

Rabinowitz [120]. 

Lyness [121] consideres integrals of the type R 1 

f(x) dx with f(x) slowly decaying and oscil- 

a 

lating in sign. The zeros are required to be asymptotically equidistant like inf(x)=g(x)j(x) 

where j(x) is a circular or Bessel function and g(x) is positive over the whole integration 

interval. Under these circumstances, the integral can be re-written as 

I = 

1X 

j=0 

Z xj+1 

xj 

f(x) dx = 

1X 

j=0 

wj ; (5.3) 

where the subinterval length xj+1,xj equals the asymptotic distance between successive zeros 

of the integrand. The resulting series is then accelerated by simply calculating the average 

of two adjacent elements, which isvery e cient as long as the sequence wj is alternating in 

sign. This idea was later used by Lyness and Hines [122] as the basis of a quadrature routine, 

and it was further extended by Espelid and Overholt [123]. They suppose the oscillating 

behaviour of the integrand f(x) stems from a periodic function that is antisymmetric and 

posesses a constant period. Like Lyness, they partition the interval into half periods to create 

an oscillating series amenable to extrapolation. However, they also observe the quadrature 

error in the subintervals and re ne the partition if necessary. 

It is exactly the Lyness partition that Haider and Liu use to calculate Fourier or Bessel 

transforms over nite integrals [124]. They call it `partitioned' Gaussian method, however, 

they make no attempt to adapt it to in nite integrals, since the integrands they consider 

decay exponentially and therefore may be safely truncated at some large value of x. For 

functions with an in nite number of zeros, they admit that the method will fail. 

The methods discussed so far are useful only if the integrand is at least asymptotically periodic. 

If, however, the distance between subsequent zeros vanishes as the integration variable 

approaches in nity, other algorithms are required. Probably the most versatile ideas that are 

applicable to a wide range of even irregularly oscillating integrands have been developed by 

Sidi in a number of papers. 

In [125] Sidi introduces a set of extrapolation methods (namely, the D- and D-transformations) 

e 

to compute integrals of the form R 1 

a f(x) dx by evaluating a number of integrals R xl f(x) dx, 

a 

where the xl are chosen to be subsequent zeros of the f(x) with a


Poincare-type asymptotic expansions 

(x) x 

1X 

i=0 

i 

: (5.5) 

xi Speci cally, (x) may be written as (x) = (x)+ (x), where its polynomial part (x) isof 

degree and (x) P 1 i=0 +i=x i as x !1. The xl are taken to be the zeros of u , (x) ,so 

they approach the zeros of f(x) only asymptotically. The W -algorithm in combination with 

a Clenshaw-Curtis rule was used by Hasegawa and Torii [127] to compute in nite integrals 

with circular functions as oscillating factors. 

R xl 

a f(x) dx by (xl) = R xl+1 

xl 

In an even later work [128], Sidi returns to the ideas of Lyness and replaces the integrals 

f(x) dx, thus forming a sequence of integrals between successive 

zeros of u , (x) , whose members are at least for large xl alternating in sign. Nonetheless, he 

uses a modi cation of his W -algorithm to accelerate the convergence of the resulting series 

P1i=,1 (xi), where (x,1) = Rx0 f(x)dx denotes the integral from the lower interval limit 

a 

to the rst zero that is greater than a. In contrast to the original W -transformation, the 

modi ed version needs only analysis of the phase (x) of oscillations, whereas information 

on the amplitude is not required. The choice of the partition points xl is simple, too, as it 

involves nothing more than the determination of the largest zero of a given polynomial. 

As a variant to Sidi's method, Ehrenmark [129] proposes not to use subdivision, but again 

the sequence of truncated integrals F (xn) = Rxn f(x)dx. However, he chooses not the zeros 

a 

of f(x) as xn, but seeks those values of x at which the truncated integral equals its limit 

limx!1 F (x). 

Another approach by Khanh [130] is based on the direct evaluation of the tail of the integral. 

It is applicable to integrands that satisfy either the linear di erential equation of rst order 

f(x) =p(x)f 0 (x)orthat of second order f(x) =p(x)f 0 (x)+q(x)f 00 (x) with f(x), p(x), and 

q(x) having asymptotic expansions at in nity. Then the tail R 1 

f(x) dx may be approximated 

x 

by a linear combination of these expansions and their derivatives. The error can be controlled 

by choosing the point where the tail is to be evaluated and the number of involved terms 

su ciently large. The remaining nite integral R x 

f(x) dx has to be calculated using some 

a 

other method. 

5.1.2 Computer routines 

Compared to the large number of scienti c papers that have addressed the problem of in nitely 

oscillating integrals, there are astonishingly few ready-to-use computer programs available. 

Many of them are public domain software and are contained in the comprehensive library 

Netlib that is accessible via the World-Wide-Web under the address http://www.netlib.org. 

Another entry point for seeking computer routines is the GAMS (Guide to Available Mathematical 

Software) that is maintained by the US National Institute of Standards and Technology 

at http://gams.nist.gov and provides a classi cation scheme allowing fast retrieval 

of appropriate routines. For other software resources on the Internet, see also the book by 

Uberhuber [131]. 

127


Searching these indices, one nds essentially three di erent quadrature routines, all written 

in FORTRAN: 

OSCINT is part of the sublibrary Toms (toms/639) in Netlib and was written by 

Lyness and Hines [122]. It is aimed at integrands that may be separated into an oscillating 

factor with an at least asymptotically constant period (i. e. circular or Bessel 

functions) and an ultimately positive factor. The user has to specify the function to 

be integrated, its (ultimate) period, and a starting point from which onwards the extrapolation 

is performed. The algorithm then carries out a sequence of nite interval 

quadratures, each over an integral having the length of a half period, and accelerates 

the resulting sequence of partial sums by means of the Euler transformation [113]. By 

default the quadrature is performed using the trapezoidal rule with a number of sampling 

points given by the user. The quadrature error is not controlled, and the user 

may as well replace the trapezoidal by a more sophisticated algorithm. The routine 

uses double precision arithmetic. 

DQAINF by Espelid and Overholt [123] can be found in the sublibrary Numeralgo 

(numeralgo/na6). They extend the approach of Lyness and Hines in that the integrand 

may be a linear combination of periodic functions p(x), all with the same (ultimate) 

period, thus f(x) = Pk i=1 pi(x)gi(x). The gi(x) are supposed to have similar asymptotic 

expansions as x ! 1, gi(x) = x , P1 j=0 c (i) 

j =xj , with > 0. The user must specify 

basically the period of the functions, the rst subdivision point, and of course an error 

tolerance. Then the algorithm computes integral approximations over half-period 

intervals using a 21-node Gauss rule and estimates the true integral value by means 

of extrapolation. In addition, the quadrature error in the intervals is estimated and 

if necessary, the interval is subdivided into three equal parts. There are both single 

(SQAINF) and double precision versions available. 

DQAWF is a routine of the Quadpack package [132], which is included in many 

sublibraries of Netlib (Slatec, Cmlib). This double precision routine has also a single 

precision counterpart (QAWF) and is designed for calculating Fourier transforms. Thus 

the integrand needs to be of the form f(x) sin !x or f(x) cos !x. With f(x) and ! given 

by the user, the algorithm integrates between the zeros of the oscillating factor. The 

quadrature is based on a Clenshaw-Curtis rule in combination with a globally adaptive 

subdivision strategy, so there is virtually no limitation imposed on the function to 

be integrated. The convergence of the resulting series is then accelerated using the 

-algorithm [113, 133]. 

Variants of this routine with slight modi cations may also be found in proprietary 

libraries. In the NAG library (The Numerical Algorithms Group) it is called D01ASF 

[134], in the IMSL (International Mathematical and Statistical Libraries) [135] it appears 

as DQDAWF. 

All these subroutines are best suited to integrands with a constant period as x !1. If this is 

not the case, like for sin (f(x)) with f(x) having a polynomial part with degree greater than 

128

5.2 Convergence acceleration 

one, these routines cannot be applied directly. It may be possible to extract an oscillatory 

factor with constant period in order to obtain, say, sin( ~ f(x)) cos !x such that integration 

can be carried out over individual `wave packets'. It is obvious that this requires the use 

of a routine with an adaptive subdivision strategy like those from the Quadpack package. 

Depending on the parameters, however, the number of zeros between the zeros of the periodic 

envelope may be very high, which in turn increases the number of subdivisions needed and 

the execution time of the program. For comparison, the example of wave propagation in the 

transmission line lled with plasma (section 3.7) was computed with the DQAWF subroutine. 

Although it worked very well (and fast) in general, it failed to produce correct results for 

special cases like the boundary. 

As pointed out before, Sidi investigated irregularly oscillating functions [128], which would 

suit our purposes exactly. Unfortunately, there is no implementation of his work available in 

a standard subroutine library. Thus the conclusions of this section are: 

For the computation of in nitely oscillating integrals, a combination of subdividing the 

integration range and extrapolating the sequence of partial sums is the method of choice. 

Although many authors have addressed this topic, there is no ready-to-use software 

package available for the case of irregular oscillations. 


One of the basic ideas in computing the value of an in nite integral is to de ne a sequence that 

converges to this value. What remains is the problem of nding this limit numerically with 

su cient accuracy. This section will introduce both the concept of series acceleration and 

the notation used throughout subsequent chapters. Furthermore, we shall present two widely 

used algorithms that are relevant for the implementation of our own quadrature strategy in 

Mathematica. 

Suppose that a sequence fSng is known to converge and only a nite set of its members is 

available, then any member of it (usually the last one) may be taken as a rough estimate for 

the limit S. If, however, the convergence is very slow, the computational e ort to determine 

enough sequence members to achieve a reasonably accurate result may be too high. In such 

cases, alternatives are sought to accelerate the convergence of the sequence. A very simple 

possibility is to form the average of two consecutive sequence members 

Tn = Sn + Sn+1 

2 

; (5.6) 

which is particularly suitable for alternating sequences (see, for example, Lyness [121] or Boyd 

[136] for an extension of this scheme). Such a sequence transformation T : fSng 7!fTngmust 

have two properties in order to be of practical use [137, 138]: 

It must be regular, that is, fTng must converge to the same limit as fSng. 

fTng must converge faster than fSng, which is expressed by limn!1 Tn,S 

Sn,S =0. 

129


Apart from linear transformations (like the Euler transformation [139]), nonlinear transformations 

are widely used for convergence acceleration. One of them is Aitken's 2 process 

with the transformation rule [113, 137] 

Tn = SnSn+2 , S 2 n+1 

Sn+2 , 2 Sn+1 + Sn 

(Sn+1 , Sn) 2 

= Sn , 

; (5.7) 

Sn+2 , 2 Sn+1 + Sn 

the second expression being preferable for implementation due to its better numerical stability. 

This transformation is known to accelerate sequences with linear convergence, that is [138, 

139], 

Sn+1 , S 

lim 

n!1 Sn , S 

and is exact (i. e. Tn = S 8n) for sequences of the form 

= ; 62 f1; 0g (5.8) 

Sn = S + a n ; a 6= 0; j j


(1) 

,1 =0 

(2) 

,1 =0 

(3) 

,1 =0 

(4) 

,1 =0 

(5) 

,1 =0 

. 

(0) 

0 

(1) 

0 

(2) 

0 

(3) 

0 

(4) 

0 

= S0 

= S1 

= S2 

= S3 

= S4 

. 

(0) 

1 

(1) 

1 

(2) 

1 

(3) 

1 

. 

(0) 

2 = S e(2) 0 

(1) 

2 = S e(2) 1 

(2) 

2 = S e(2) 2 

. 

(0) 

3 

(1) 

3 

. 

(0) 

4 = S e(4) 0 

Figure 5.1: The -array. Only the values (n) 

2k in the even columns are valid estimates for the sequence 

limit S. 

then the -algorithm gives the exact result (0) 

2p 

of the original sequence are required. 

. 

. .. 

= S [113], which means that 2p + 1 elements 

In any case, a sequence transformation may be seen as a functional relation between a member 

of the transformed sequence and a certain subset of the original sequence. Since our goal is 

to obtain a better estimate for the limit of the sequence, we normally desire only one result 

from the function and not a transformed sequence. Thus we can write 

Tn = e S (k) 

n = F (Sn;Sn+1 :::Sn+k) ; (5.13) 

with k +1 being the number of sequence members involved in the transformation and n 

the starting index. If the original sequence comprised N members, then we would typically 

compute S e(k) N,k or S e(N) 0 as the (hopefully) best approximation to the exact limit S. 

131

Chapter 6 

Towards a quadrature routine 

6Towards a quadrature routine 

\Can you do Addition?" the White Queen asked. \What's one and one 

and one and one and one and one and one and one and one and one?" 

\I don't know," said Alice. \I lost count." 

\She can't do Addition," the Red Queen interrupted. \Can you do Subtraction? 

Take nine from eight." 

\Nine from eight I can't, you know," Alice replied very readily: \but| " 

\She can't do Subtraction," said the White Queen. 

Lewis Carroll, Through the looking glass 

With the discouraging result of the search for an o -the-shelf program to solve our problems, 

there is nothing left but to write a dedicated quadrature routine that suits our needs. This 

is, however, not at all a straightforward task, and we have to avoid carefully a number of 

pitfalls associated with the in nitely oscillating integrals we wish to compute. This chapter is 

thus devoted to some practical considerations that emerge in the course of the development 

of such a quadrature routine. While it seems obvious that a combination of partitioning and 

extrapolation is best suited to this purpose, there are some detailed questions yet to solve. 

The most interesting one is certainly the choice of the partition itself, and we shall therefore 

discuss which points are most suitable for this subdivision procedure | the zeros, extrema, 

or any other points of the integrand. A rather implementation-speci c question is how the 

partial integrals are to be computed e ciently. We shall also investigate a strategy how to 

control the accuracy of the computation. 

Throughout the subsequent sections, we shall present many examples programmed in Mathematica 

to illustrate the various di culties that need to be considered. To understand these 

pieces of code, a certain familiarity with Mathematica or any other functional programming 

language is favourable, but not imperative. Since most of the used functions are selfexplaining, 

we shall give further explanations only where they are deemed necessary. 

132

6.1 Partitioning the integration interval 


Once we know that an in nite integral exists, we can theoretically choose any arbitrary 

partition of the interval to calculate the sequence of partial sums. In practice, however, we 

are better o to select the subdivision points such that the resulting sequence is amenable 

to convergence acceleration. A careful choice is already half the guarantee for a trustworthy 

result. Let an example illustrate this pitfall. 

We consider the well-known integral 

Z 1 

0 

sin x 

x dx = : (6.1) 

2 

We subdivide the interval at the zeros of the sine and calculate the rst 100 elements of 

the sequence of partial sums. To this end, we rst evaluate the contributions of the integrals 

Ii = R xi+1 

f(x) dx between the partition points and then accumulate them to yield the partial 

xi 

sums Si = Pi k=0 Ii. We nally plot this alternating sequence. 

In[1]:= sequ = Table[NIntegrate[Sin[x]/x,fx,i Pi,(i+1) Pig], 

fi,0,100g]; 

partial = FoldList[Plus,0,sequ]; 

1.59 

1.58 

1.57 

1.56 

1.55 

1.54 

z1 = ListPlot[partial,PlotStyle->fPointSize[0.012]g]; 

20 40 60 80 100 

In[2]:= SequenceLimit[partial] - N[Pi/2] 

Out[2]= -15 

-1.77636 10 

133


Remark (Cumulative sums) Using the function FoldList is a very elegant and fast 

way to compute cumulative sums of a given list. FoldList[Plus,x,fa,b,cg] returns 

fx,x+a,x+a+b,x+a+b+cg [140]. Normally, x will be set to zero. If the leading zero element 

of the result might disturb subsequent calculations, it can be omitted with Drop[list,1]. 

The numerically found limit of this sequence is exact within the machine precision of 16 digits. 

Now we inadvertently skip every other zero in the calculation of the series and integrate 

over full instead of half periods. Consequently the series no longer alternates, which has a 

catastrophic e ect on the determination of the limit. 

In[3]:= sequ = Table[NIntegrate[Sin[x]/x,fx,2 i Pi,2 (i+1) Pig], 

fi,0,100g]; 


1.5675 

1.565 

1.5625 

1.56 

1.5575 

1.555 

1.5525 

z2 = ListPlot[partial,PlotStyle->fPointSize[0.012]g]; 

20 40 60 80 100 


Out[4]= -0.00025589 

Remark (Improvement of the result) The result may neither be improved by increasing 

the precision of the numerical computations nor by including a larger number of 

sequence members in the extrapolation. 

The example may be considered rather pathological, but we can con rm the di erences between 

half- and full-period intervals also for other logarithmically converging integrals like 

134


R 1 

sin x 0 2 dx = 

functions like 

p =2 

2 

or R 1 

0 

sin x2 x dx = . For exponentially decaying integrands or even 

4 


(1+x) p ; p>2, the di erence in the extrapolation error vanishes. We conclude 

that the -algorithm (which is actually behind SequenceLimit) yields far more reliable results 

if it is applied to alternating series. 

This experience is in line with a number of articles that have been published over the years. 

Smith and Ford [141] found this algorithm to be particularly useful for such series. Bell and 

Phillips [142] proved that Aitken's 2 process is well-conditionend for all cases where the 

errors en = Sn , S oscillate in sign. Wynn [143] showed the stability of the -algorithm 

for oscillating sequences. Tailored to his (forward averaging) acceleration algorithm, Lyness 

[122] also mentions the problems associated with interval subdivision like a wrong asymptotic 

distance between the consecutive zeros. As for monotonic sequences, Smith and Ford [141] 

tested several convergence acceleration algorithms on logarithmically converging series like 

P 11 1=n 2 and found that the -algorithm failed on all of them. 

Remark (Possible alternative) An alternative would have been to use the function 

EulerSum contained in the Standard Package NumericalMath`NLimit` that uses an iterated 

Euler transformation [144]. However, the results are practically the same as with 

the use of SequenceLimit. In addition, the option Terms that speci es how may terms 

are explicitly summed up before extrapolation has to be set to a higher value than the default 

to achieve the desired accuracy even for the favourable case. Increasing ExtraTerms, 

which is the number of sequence members used in the extrapolation process, can result in 

complete garbage in the case of full-period intervals. 

The second possibility for the choice of the partition points is to take the extrema of the 

oscillating integrand. Lyness discovered this to be superior to subdivision at the zeros, but 

only by a relatively small margin. Espelid and Overholt [123] found that it improves the 

convergence by an order of magnitude. In fact it reduces the possible truncation error as 

the positive and negative contributions within a subinterval tend to cancel each other and 

the partial integrals therefore are much smaller. This e ect is somehow comparable to the 

transformation of an alternating series by simply calculating the mean value of two adjacent 

members. We apply this strategy to our example by simply shifting the integration limits by 

] separately. 

2 and including the integral over [0; 2 

In[5]:= sequ = Table[NIntegrate[Sin[x]/x,fx,i Pi + Pi/2,(i+1) Pi + Pi/2g], 

fi,0,100g]; 

partial = FoldList[Plus,NIntegrate[Sin[x]/x,fx,0,Pi/2g],sequ]; 

SequenceLimit[partial] - N[Pi/2] 

Out[5]= -14 

-5.79536 10 

It is not surprising that the integration over half periods is as accurate as before. However, 

the experiment with full periods now gives a better result. 

135

1.575 

1.57 

1.565 


20 40 60 80 100 

Figure 6.1: Sequences of partial sums for R 1 sin x 

0 x dx. The partition points are the zeros ( ) or extrema 

( ) of the integrand. The inner sequences are obtained when the integration range for the subintervals 

is a full period. 

In[6]:= sequ = Table[NIntegrate[Sin[x]/x, 

fx,2 i Pi + Pi/2,2 (i+1) Pi + Pi/2g], 

fi,0,100g]; 

partial = FoldList[Plus,NIntegrate[Sin[x]/x,fx,0,Pi/2g],sequ]; 

SequenceLimit[partial] - N[Pi/2] 

Out[6]= -7 

-2.13857 10 

Fig. 6.1 shows the sequences of partial sums calculated for this example and reveals that the 

extrema are indeed more suitable as partition points because the errors jSn , Sj decrease like 

1=n 2 in this case. When the zeros are used to subdivide the integration range, the errors 

behave like 1=n, which gives only half the speed of convergence. 

Proof We shall rst derive an estimate for the magnitude of the partial integrals. The second meanvalue 

theorem states that if two functions f(x) and g(x) are continuous on [a; b] and g(x) is also 

monotonic, then there exists a point 2 [a; b] such that 

Z b 

a 

Z 

f(x) g(x) dx = g(a+) 

a 

Z b 

f(x) dx + g(b,) f(x) dx : (6.2) 

We set f(x) = sin x, g(x) =1=x and choose the integration interval [(n + ) ; (n +1+ ) ], where 

136


0

6.2 Choosing the rst partition point 


When using an extrapolation method to determine the value of an integral, we face the 

problem that we must conclude from a nite sequence of partial sums to its limit. While it 

is normally easy to judge the existence of an inde nite integral, it takes some considerations 

to nd the right starting point for the sequence being subject to the extrapolation algorithm. 

If the choice was wrong, the algorithm may in some cases return a warning message if it 

encounters convergence problems. However, the checking capabilities are limited and strongly 

data dependent, and most extrapolation algorithms even nd nite results for divergent series. 

Hence the responsibility to be careful always rests with the user. 

To illustrate the possible di culties, we regard a simple example. Consider the integral 

I = 

Z 1 

a 

1 

x +1 sin (x , xm) 2 dx : (6.8) 

We want toevaluate it through extrapolation and choose the partition points to be the zeros 

of the sin-function. For the sake of simplicity, we assume that the lower integration limit a 

be the smallest positive zero. As we need the sequence of partial sums, we must de ne a 

function that returns all zeros of the integrand in ascending order, starting with index 0. To 

that end, it is essential to distinguish the two branches of the parabolic argument (x,xm) 2 . 

It is easy to see that there are o = bxm 2 = c zeros between the lower limit of integration and 

the minimum of the argument function at x = xm, so this value can be used as an index o set 

for the calculation of the zeros. 

In Mathematica notation, the integrand and the function returning the k-th positive zero 

xk are given below. The position of the extremum xm and the index o set o are global 

parameters. 

In[7]:= f[x_] := 1/(x+1) Sin[(x-xm)^2]; 

zero[k_] := If[k >= o, xm + Sqrt[(k-o) Pi], 

xm - Sqrt[(o-k) Pi]]; 

Now we calculate the rst 100 elements of the sequence of partial sums with the zeros used 

as subdivision points. 

In[8]:= xm = 15; 

o = N[Floor[xm^2/Pi]]; 

zmax = 100; 

sequ = Table[NIntegrate[f[x],fx,zero[i],zero[i+1]g, 

Method->GaussKronrod],fi,0,zmaxg]; 


ListPlot[partial,PlotStyle->fPointSize[0.006]g]; 

138

6.2 Choosing the rst partition point 

0.12 

0.1 

0.08 

0.06 

0.04 

0.02 

20 40 60 80 100 

The surprising step in the sequence stems from the minimum of the argument function that 

causes the distance of successive zeros to be enlarged in its neighbourhood, which in turn 

results in exceedingly large partial integrals. For the rst 50 elements, this e ect is hidden 

1 

by the dominating factor in the integrand and the sequence seems to converge. Indeed, 

x+1 

the extrapolation algorithm returns a valid limit without any warning when we feed it with 

the sequence elements 20 to 30. 

In[9]:= SequenceLimit[Take[partial,f20,30g]] 

Out[9]= 0.0313787 

We get a result even if we use the portion of the sequence that seems to diverge as we approach 

the extremum of the argument function. 


Out[10]= 0.0313787 

Only if we start the extrapolation behind the step, we obtain the correct limit and the real 

value of the integral. 


Out[11]= 0.109863 

Remark (Possible warnings) If we had replaced the factor x+1 by x2 , we would 

+1 

have received at least a warning that the algorithm suspected the results to be incorrect in 

the rst two cases (the returned values would have been `correct' anyhow). However, one 

cannot rely completely on such messages as there are other integrands where the results 

are de nitely right despite these warnings. 

139 

1 

x


The conclusion to be drawn from this example is that the rst partition point must be beyond 

any extremum or even point of in exion of the argument function. The most rigorous way 

to ensure this is to require that the argument function and its derivatives be monotonic for 

x>x0. From this point of view, many authors seem to be rather sloppy in the choice of the 

partition. 

Sidi [128], for example, takes x0 to be the rst zero of the circular function sin ( (x)) that is 

greater than the lower integration limit a. Thus x0 satis es the polynomial equation (x) =q 

for some integer q. The consecutive partition points xi ; i>0 are then determined to be the 

largest positive solution of (x) =(q+l) . As can be seen from our example for large values 

of xm, this results in a large second partial integral I1. A solution would be to determine 

also x0 as largest root of (x) =q and not only as the zero closest to a. But then again we 

would obtain an unnecessarily large partial interval (this time I0). 

Even worse is the fact that although the proposed method guarantees that the xi are of 

ascending order, it does by no means ensure that problems like the aforementioned are avoided. 

If there are, for example, extrema at large ordinate values, they might not be comprised in 

the sequence of the partial integrals because the series seems to converge earlier and the 

evaluation is truncated consequently. 

Espelid and Overholt [123] also require the rst partition point to be supplied by the user. 

They can of course provide no information other than to choose it small enough to keep the 

computational costs for the rst interval low and large enough that the asymptotic properties 

of the functions are satis ed. 

There is still another side to the story: the non-oscillating factor of the integrand ought tobe 

smooth compared to the oscillating part, otherwise the sequence of partial sums may again 

show discontinuities. Not only do discontinuous factors like the step function provoke such 

problems, also rather innocent functions like e ,x2 or tanh(x) can hamper the straightforward 

use of extrapolation. Therefore these parts of the integrand, too, should be monotonic for 

x>x0, as should their derivatives. This aspect is also rarely addressed in the literature. 

Sidi [128] does not consider the properties of the modulating factor at all, he just states 

that it must be non-oscillating at x ! 1 and have an asymptotic expansion such that 

the entire function is integrable at in nity. Lyness [121] brie y discusses the problems of 

unexpected peaks in the integrand that are too far out to be included in the sequence used 

for extrapolation. Consequently, for his quadrature routine [122], he lets the user decide where 

to start the extrapolation. 

Remark (Connection to physics) We have already encountered the fact that wave 

integrals are dominated by a rather small region of the integration range. In fact, the 

method of stationary phase is based upon this nding. The mathematical formulation 

that an extremum of the argument function yields a large partial integral is equivalent 

to the concept of local wave number and frequency, both of which are determined by the 

extrema of the wave's phase. 

140

6.3 How to compute the rst integral 


While the integrals Ii = R xi+1 

f(x) dx to the right of the rst partition point x0 are computed 

xi 

between successive zeros and therefore have smooth integrands, the integration range of the 

rst integral I0 = R x0 

f(x) dx can comprise more than just two zeros of the integrand. In 

a 

fact, the integrand can be either smooth or strongly oscillating, depending on the parameters 

of the integrand. For an automatic quadrature routine, however, we need a strategy that can 

cope with a broad variety of functions. 

For the numerical evaluation of de nite integrals, Mathematica o ers three di erent methods 

that can be selected with the respective choice for the option Method of the function 

NIntegrate. The algorithms may be found in any textbook (for example Davis [113] or 

Uberhuber [131]), the Mathematica-speci c details are treated by Keiper [145]. 

GaussKronrod refers to an adaptive Gauss-Kronrod quadrature and is the default value 

for Method. Owing to its simplicity, itiswell suited to smooth integrands. 

DoubleExponential selects an algorithm (DE) that transforms the integrand prior to 

quadrature. This method gives particularly good results with oscillating integrands. 

Trapezoidal uses the well-known trapezoidal rule. It is useful for periodic integrands 

where the quadrature interval is exactly one period. Apart from this case its performance 

is rather poor. 

All methods estimate the error in the approximation. For the trapezoidal and the DE rule, 

which e ectively uses the trapezoidal rule after the variable transformation, the stepsize is 

halved if the error is too large. This way the previous evaluations of the integrand may 

be incorporated in the re ned computation. The Gauss-Kronrod rule, on the other hand, 

is adaptive in that it recursively subdivides the integration interval if necessary and is thus 

able to concentrate its e orts on the regions where the error is really high. There are two 

important parameters that control these subdivision procedures: 

PrecisionGoal speci es how many digits of precision the result of the computation 

should have. E ectively, this sets the limit for the relative error. The default value 

is Automatic, which yields a precision goal with ten digits less than the current setting 

of WorkingPrecision (which in turn defaults to the machine-dependent constant 

$MachinePrecision). The setting of PrecisionGoal is by no means a guarantee that 

the result will satisfy this requirement. 

AccuracyGoal sets the number of digits to the right of the decimal point that should 

be correct in the result and thus speci es the allowed absolute error in the numerical 

routine. The default setting is Infinity, which means that this criterion shall not be 

used. As with PrecisionGoal, the actual result may have less digits of accuracy than 

desired. 

The terms precision and accuracy have very distinct meanings in Mathematica, other than in 

common use. The di erence is best illustrated by an example. 

141


Example 6.3.1 The function Precision returns the number of digits a oating-point number 

posesses. For exact numbers, such as integers or fractions of integers, it returns Infinity. For 

ordinary machine precision numbers, the result is $MachinePrecision, even if the number has less 

digits [146]. 

In[12]:= Precision[1234.5678901234567890] 

Out[12]= 19 

In[13]:= Precision[1234.56] 

Out[13]= 16 

The function Accuracy, on the other hand, refers to the digits to the right of the decimal point. Like 

before, it returns Infinity if the number is exact, and assumes machine precision numbers if the 

argument has less than | in our case | 16 digits. 

In[14]:= Accuracy[1234.5678901234567890] 

Out[14]= 16 

In[15]:= Accuracy[1234.56] 

Out[15]= 13 

In fact, both functions signify the error bounds of a oating-point number in Mathematica. The 

accuracy of a number x then is the absolute error bound log j j while the precision means the 

relative error bound log jx= j [144]. The maximum error clearly depends on the working precision. 

Both values are rounded to the nearest integer and thus look like numbers of signi cant digits although 

they can be incorrect by one digit due to the rounding. 

All other parameters of NIntegrate are not of primary importance for our purpose here and 

will be explained as needed in the following chapters. 

We will now explore the two di erent quadrature algorithms as they are applied to the rst 

partial integral over the interval [a; x0] between the left integration limit and the rst partition 

point. Let us return to the example (6.8) of the previous section. If we set xm = 40, we obtain 

an index o set for the step in the sequence of partial sums o = 509 such that we select our rst 

partition point to be the 510-th zero. We then want to know how long it takes to compute 

this integral in one. 

In[16]:= Timing[ 

NIntegrate[f[x],fx,zero[0],zero[510]g,MaxRecursion->10, 

Method->GaussKronrod] 

] 

142


GaussKronrod DoubleExponential Trapezoidal 

xm =40 

R z510 

0 f(x)dx 103.81 20.59 2968.07 

P 509 

k=0 

R z6 

0 

xm =4 

P 5 

k=0 

R zk+1 

z k 

f(x)dx 20.05 169.56 3119.32 

f(x)dx 0.61 0.82 47.78 

R zk+1 

z k 

f(x)dx 0.38 1.59 36.63 

Table 6.1: Computing times in seconds for di erent quadrature algorithms. 

Out[16]= f103.81 Second, 0.0491901g 

Remark (Recursion depth) The function Timing returns the evaluation time as well 

as the result. Note that because the integration interval spans many zeros of f(x), we 

must increase the maximum number of recursive subdivisions Mathematica tries. The 

according option is MaxRecursion. Otherwise the evaluation would terminate with an 

error message and most likely a wrong result. 

Another possibility istosubdivide the integration range at the zeros of the integrand | as 

we would do for the interval to the right of the rst partition point | and to compute the 

sum of these partial integrals. The computation is now much faster because the individual 

integrals are smooth and we thus bene t from the simplicity of the Gauss-Kronrod formula. 

In[17]:= Timing[ 

Sum[NIntegrate[f[x],fx,zero[i],zero[i+1]g, 

Method->GaussKronrod],fi,0,509g] 

] 

Out[17]= f20.05 Second, 0.0491901g 

We can carry out the same computations for the double exponential quadrature formula and 

for a di erent value of xm = 4, which yields o = 5. The various results are summarised in 

tab. 6.1 . We recognise rst of all that the trapezoidal rule indeed has a poor performance. 

The double exponential algorithm seems to be a suitable choice, even if the transformation of 

the integration variable takes some time, which slows down the computation in comparison 

to the Gauss-Kronrod method if there are only few zeros in the integration interval. The 

DE rule, on the other hand, needs no evaluation of any zeros, which can be a signi cant 

advantage if the zeros cannot be found explicitly. In such a case, a subdivision strategy using 

Gauss-Kronrod quadrature would be inferior as well. To demonstrate this, we assume that 

we have no information about the zeros of the integrand in (6.8). Instead, we use equidistant 

points to partition the interval and determine the total computing time depending on the 

number of subintervals (again with xm = 40). 

143

100 

90 

80 

70 

60 

50 

40 


200 400 600 800 1000 1200 1400 

Figure 6.2: Computing time for a Gauss-Kronrod rule depending on the number of subintervals for 

the integrand in (6.8) with xm = 40. 

In[18]:= timtab = Table[fsteps,First[ 

Timing[ 

min = zero[0]; 

max = zero[510]; 

Sum[NIntegrate[f[x],fx,min + i (max-min)/steps, 

min + (i+1) (max-min)/stepsg, 

Method->GaussKronrod,MaxRecursion->15], 

fi,0,steps-1g] 

] 

]/Secondg,fsteps,1,1401,10g]; 

Remark (Implementation) To speed up the computation, we evaluate the interval 

bounds only once and save them as constants. The list is made up of the number of 

intervals and the computing time for each trial, which is obtained by taking the rst 

element of the list returned by the function Timing. To be able to plot this list, we 

must rst divide each evaluation time by the constant Second. The actual results of the 

individual computations are not stored separately as they are all correct. 

The results in g. 6.2 show that with only a few subdivisions, the overall computing time 

can be dramatically decreased. The minimum, however, is still above 30 seconds, which is 

signi cantly higher than the values in tab. 6.1 . The equidistant partition is also the reason 

why it takes about 800 subintervals to achieve the minimum although there are only 510 

zeros in the integration range. The linear slope beyond the minimum stems from the time- 

144

6.4 Asymptotic partition 

consuming preparation of the internal sampling points and weights needed for the Gauss- 

Kronrod formula. 

There are two conclusions to be drawn from these experiments with respect to the implementation 

of numerical quadrature for an automatic algorithm: 

The rst integral between the lower integration bound and the rst partition point is 

best evaluated with the setting Method->DoubleExponential, which yields an optimum 

performance in case of a strongly oscillating integrand in the interval and is still 

reasonable in all other cases. 

The subsequent integrals that are then used for extrapolation ought to be computed 

with Method->GaussKronrod, because here the integrand is smooth enough that the 

simplicity of the Gauss-Kronrod rule will be an advantage. 


The zeros or extrema of the integrand are a perfect choice for the de nition of the subintervals 

and the best one as far as the suitability for series acceleration is concerned. In terms of 

computing time, this is true only as long as they are easily computable. If the oscillating 

factor is su ciently complex so that these points can only be determined numerically, itmay 

be more e cient tolook for a di erent partitioning strategy. A good idea is to consider the 

integrand's behaviour at in nity and choose a partition that approaches the zeros as x !1. 

This way we make sure that the resulting series is at least ultimately alternating and can be 

treated by a convergence acceleration algorithm. More formally, ifwehaveanintegral of the 

kind 

I = 

Z 1 

a 

(x) e j (x) dx ; (6.9) 

where the argument function (x) has an asymptotic expansion as x !1 

(x)=x p 

with p being a non-negative integer, we can take its polynomial part 

(x) =x p 

1X 

i=0 

pX 

i=0 

ai 

; (6.10) 

xi ai 

x i 

(6.11) 

to calculate the subdivision points, which maybemuch easier because we need to solve only 

a polynomial equation. This strategy has been pursued by Sidi [128]. 

Again we present an example to make the application of the basic idea clear. The integral 

Z 1 

x 

I = 

x2 p 

sin x2 , a2 dx = (6.12) 

, a2 2 

a 

145

is most easily obtained from R 1 

0 


sin y 

y dy with the substitution y = p x 2 , a 2 . For the purpose 

of demonstration, we assume that we cannot calculate the zeros of the integrand directly. 

Instead, we know that for x a, p x 2 , a 2 behaves like x, which is exactly the polynomial 

part (6.11) of its expansion (6.10). So we compute the partition points as if the argument 

of the sin-function was x, and we plot the sequence of partial sums for the parameter value 

a =50 . Owing to the singularity of the integrand at x = a, we determine the value of the 

rst subinterval analytically and include it explicitly in the sequence of the partial sums. 

In[19]:= a = 50 Pi; 

firstval = Integrate[Sin[Sqrt[x^2 - a^2]]/(x^2 - a^2) x, 

fx,50 Pi,51 Pig] 

Out[19]= SinIntegral[Sqrt[101] Pi] 

In[20]:= sequ = Table[NIntegrate[Sin[Sqrt[x^2 - a^2]] x/(x^2 - a^2), 

fx,i Pi,(i+1) Pig], 

fi,51,300g]; 

partial = FoldList[Plus,N[firstval],sequ]; 

1.576 

1.574 

1.572 

1.568 

1.566 

z1 = ListPlot[partial,PlotStyle->PointSize[0.006]]; 

50 100 150 200 250 


Out[21]= -14 

1.59872 10 

We emphasise that in this case the computation of the extrema instead of the zeros yields no 

better result since the partial integrals in either case cover both negative and positive portions 

146


of the integrand. The terms `zero' and èxtremum' lose their meanings as they now refer to 

the approximation rather than to the real integrand. Thus we would actually bene t from 

the e ect we discussed in the last section only for x !1. 

In the computation of the limit we used the complete sequence of partial sums with almost 250 

members. Normally, wewould not begin with that large a number because their determination 

could be time consuming. As the rst few sequence members look fairly irregular, the burning 

question is how many wemust take in order to obtain a reliable result for the limit and how 

we can improve the accuracy if needed. This obviously requires that more sequence members 

be taken into account, however, there are two possibilities to do so: 

1. We can increase the number of sequence members used in the extrapolation, that is, we 

compute limk!1 e S (k) 

n with n = 0 if we start with the very rst member. 

2. We leave the length k of the subset subject to extrapolation xed and start the extrapolation 

at a member with a higher index, which islimn!1 e S (k) 

n . 

Both limiting processes have not received much attention in the literature. Sidi [126] found 

that for his W -transformation, method 1 gives better results and is more e cient. 

We now examine these two possibilities brie y for our example. For the rst one, we take 

the rst k members of the sequence to determine the limit and plot the absolute value of the 

approximation error e S (k) 

n , S depending on k. 

In[22]:= approxerr1 = Table[Abs[SequenceLimit[Take[partial,f1,kg]] - N[Pi/2]], 

fk,10,240g]; 

LogListPlot[approxerr1,PlotStyle->PointSize[0.006],PlotRange->All]; 

0.001 

-6 

1. 10 

-9 

1. 10 

-12 

1. 10 

0 50 100 150 200 

147


For the second version we take only twelve consecutive sequence members to compute the 

limit but start at di erent positions n. Again we plot the approximation error as a function 

of the index value where extrapolation starts. 

In[23]:= approxerr2 = Table[Abs[SequenceLimit[Take[partial,fn,n+12g]] - N[Pi/2]], 

fn,230g]; 

LogListPlot[approxerr2,PlotStyle->PointSize[0.006],PlotRange->All]; 

0.001 

-6 

1. 10 

-9 

1. 10 

-12 

1. 10 

0 50 100 150 200 

Remark (Logarithmic plots) The function LogListPlot is part of the standard 

package Graphics`Graphics` that has to be loaded prior to evaluating the previous 

commands. The reason why we take exactly twelve members of the sequence to determine 

the limit in the second trial is simply that this is the default value of the option 

NSumExtraTerms used in the function NSum when in nite sums are computed by extrapolation. 

We notice that the results of the rst experiment look more reliable for high numbers of 

included sequence members, whereas there are singular points with surprisingly high errors in 

the second trial even for large n. On the other hand, the second process is signi cantly faster 

since it uses fewer values for the extrapolation. By combining these two processes one could 

nd a reasonable compromise between speed and accuracy. Other possibilities for improving 

the results are discussed in the following section. 

Since we normally do not know the actual limit of the sequence of partial sums, we would 

like toderive some criterion how tochoose the parameters of the extrapolation. We quickly 

see that the approximation error (x) , (x) is not very useful for this purpose because 

it neglects the periodicity of the sin-function. A better approach is to regard the angular 

velocities 

@ (x) 

@x 

@ (x) 

and @x , respectively. The di erence between the argument function and its 

148

6.5 Considerations for a Mathematica implementation 

polynomial approximation causes the sequence of partial sums to take on the form of distinct 

`wave packets' in the rhythm of the beat frequency @ , 

@x (x) , (x) . A reasonable criterion 

could thus be to seek the abscissa value where such awavepacket comprises a given number 

of k elements, that is 

0 (x) 

0 (x) , 0 (x) = k: (6.13) 

There is, unfortunately, no general guideline how to choose k | the larger it is, the better 

the result will be. 


Let us now put the considerations of the last section into a context that is closer to an 

implementation in Mathematica. To evaluate an in nite sum by means of extrapolation, 

we normally use the built-in function NSum with the option Method->SequenceLimit. This 

function internally performs exactly the same operations as we did in the example above | 

rst it computes a sequence of partial sums and then it passes an appropriate portion of this 

list to the function SequenceLimit for extrapolation. The selection of this sublist is controlled 

by the two parameters NSumTerms and NSumExtraTerms, respectively. The extrapolation 

process itself may be in uenced by the option WynnDegree. 

NSumTerms is the number n of terms that are summed up explicitly before the extrapolation 

is performed. 

NSumExtraTerms is the number k of terms that are used in the extrapolation. 

WynnDegree sets something like the `degrees of freedom' for the extrapolation [144]. It is 

also the only option to SequenceLimit. If its value is 1, Aitken's 2 algorithm is used, 

for values greater than one the -algorithm is invoked. Apart from this distinction, the 

actual function of this parameter is not documented further, but it seemingly speci es 

how far the -array ( g. 5.1) is computed. 

Remark (Sequence extrapolation) According to Keiper [144], the basic algorithm of 

SequenceLimit transforms the sequence into a sequence of length n,2. In the triangular 

scheme that is often used to illustrate the data dependencies of the -algorithm, this is 

equivalent to the computation of the two columns to the right of the starting sequence. 

As we have seen in section 5.2, the very rst execution of this algorithm yields the same 

result as the 2 algorithm. The basic transformation is repeated WynnDegree times. If 

the resulting sequence still has more than two elements, the whole process starts again 

until the outcome is too short or | in terms of the -array |until the triangle is completed. 

One could suspect that if the -array iscompleted anyhow, the result ought to 

be independent of the parameter WynnDegree, but this is not true. The di erence is that 

the column to the left of the starting sequence always consists of zeros, so for each time 

the algorithm is restarted in the middle of the -array, the respective column to the left 

of the previous result is replaced with zeros, which naturally a ects the nal outcome. 

149

0.001 

-6 

1. 10 

-9 

1. 10 

-12 

1. 10 

-15 

1. 10 

0 50 100 150 200 


Figure 6.3: Extrapolation error e S (k) 

n , S for n = 0 and varying k, computed with the 2 algorithm 

( ) and the -algorithm ( ). 

There is one exception to the aforementioned operation principle of the sequence transformation: 

If WynnDegree is set to Infinity, as many transformations as possible are carried 

out in a row without restarting the process. This is now the case where the -array is really 

computed from left to right, which matches the descriptions of the -algorithm known 

from the literature. 

It is worthwile to review the results of the previous example (6.12) with a view to the extrapolation 

algorithm used. Being indi erent to this detail in the last chapter, we left the 

settings of SequenceLimit at their defaults, which selected the 2 algorithm. If we choose 

the -algorithm (for example with WynnDegree->Infinity) and carry out the computation 

anew, we can compare both algorithms directly. Fig. 6.3 shows this comparison for the case 

where the number of terms subject to extrapolation are varied, and g. 6.4 the alternative 

with a xed number of extrapolation terms and a di erent starting index. In either case, the 

-algorithm yields better results for a given number of sequence members. 

Apart from the algorithm used, which is likely to be chosen once and for all in a series of 

computations, there are in principle two knobs for the ne tuning of the extrapolation results. 

So far, we only twiddled either one or the other, exploring the boundaries of this parameter 

space. Yet it might be possible to nd a good compromise for the values of n and k, and 

therefore we want to study their e ects in some detail now. We again take the integral (6.12), 

this time with a = 20, and vary both NSumTerms and NSumExtraTerms (here denoted as ns 

and ne). We compute this array for di erent values of WynnDegree and plot the logarithm to 

the base 10 of the approximation error, which is essentially the number of correct digits. 

150


0.001 

-6 

1. 10 

-9 

1. 10 

-12 

1. 10 

-15 

1. 10 

0 50 100 150 200 

Figure 6.4: Extrapolation error e S (k) 

n , S for k = 12 and varying n, computed with the 2 algorithm 

( ) and the -algorithm ( ). 

In[24]:= a = 20 Pi; 

firstval = N[Integrate[Sin[Sqrt[x^2 - a^2]]/(x^2 - a^2) x, 

fx,20 Pi,21 Pig],40]; 

seq = Table[NIntegrate[Sin[Sqrt[x^2 - a^2]]/(x^2 - a^2) x, 

fx,i Pi,(i+1) Pig], 

fi,21,350g]; 

partial = FoldList[Plus,firstval,seq]; 

errtab = Table[fne,ns,Log[10,Abs[SetPrecision[ 

SequenceLimit[Take[partial,fns+1,ns+ne+1g], 

WynnDegree->1],40]-N[Pi/2,40]]]g, 

fne,6,50,1g,fns,0,60,1g]; 

The results for WynnDegree->1 are shown in g. 6.5 . It is interesting to notice that if only a 

small number of terms are used for extrapolation, peaks appear where the extrapolation error 

becomes unexpectedly high. These peaks seem to be related to the areas of the sequence 

where the alternation is not very pronounced | the `nodes' of its wave-like envelope. A 

contour plot of the results ( g. 6.6) reveals that these peaks follow the lines 2 ne + ns = const. 

This is not surprising if we remember that each pass of the sequence transformation reduces 

the length of the sequence by two. In the end there may be either one or two members left 

depending on whether we started with an odd (2m +1) or even (2m +2) number of sequence 

members. Both results | after exactly the same number m of transformations | are not 

supposed to di er very much. If su cient terms are used for the extrapolation or if it is 

started where the wave packets are broad enough (for low ne but high ns), then we obtain 

151


the best results that are in uenced chie y by the implementation details and the associated 

numerical e ects. This region is characterised by extrapolation errors that are constant for 

ne + ns = const and thus depend on the total number of sequence terms regardless of how 

many are actually used for extrapolation. 

The contour plot ( g. 6.6) also shows that it is di cult to decide where to begin the extrapolation 

if not too many terms are to be used. For 40 terms, for example, the optimum lies 

somewhere at ne = 15, ns = 25. 

The results we obtain when choosing the -algorithm with WynnDegree->2 are shown in 

g. 6.7. The inconvenient peaks have disappeared (which is the same for all even values of 

WynnDegree, but not for the odd ones), yet it is hard to nd optimum values for ne and ns. 

The setting WynnDegree->Infinity yields the best extrapolation results ( g. 6.8). It is still 

true that there are pairs of sequences that give the same estimates for the limit S e(2m+1) n and 

eS (2m+2) 

n+1 , respectively, which can be seen from the bands running in parallel to the ns-axis. 

The respective contour plot (6.9) shows that the best way to improve the extrapolation result 

is to increase ne, perhaps with a few terms summed up explicitly. We have thus con rmed 

the suspicion that the limiting process 1 from the last section is indeed superior. 

For comparison we compute the extrapolation error if we do not use approximate partition, 

but take the zeros of the integrand as partition points. Just the determination of the partial 

integrals is di erent. The rest of the evaluation remains the same as with the asymptotic 

partition before. 

In[25]:= a = 20 Pi; 

firstval = N[Integrate[Sin[Sqrt[x^2 - a^2]]/(x^2 - a^2) x, 

fx,a,Sqrt[Pi^2+a^2]g],40]; 

seq = Table[NIntegrate[Sin[Sqrt[x^2 - a^2]]/(x^2 - a^2) x, 

fx,Sqrt[(i Pi)^2+a^2],Sqrt[((i+1) Pi)^2+a^2]g], 

fi,1,300g]; 

We nd that the results are far better and practically independent of the value we choose for 

WynnDegree. 

We thus conclude that when implementing an automatic quadrature routine with Mathematica, 

we ought to consider the following points: 

The setting WynnDegree->Infinity gives the best results for asymptotic partition and 

reasonably good ones for standard (zero) partition, so it is a suitable choice for a default 

value of this option. 

To increase the accuracy of the extrapolation result, the number of terms included in 

the extrapolation (NSumExtraTerms) should be increased, whereas NSumTerms may be 

left unchanged. 

Whenever feasible, the zeros of the integrand should be chosen as the partition points. 

This yields more accurate results with fewer terms involved. 

152


0 

-5 

-10 

-15 

0 

20 

ne 

40 

Figure 6.5: Negative number of correct digits, log e S (k) 

n , S , for varying n (ns) and k (ne), computed 

with the 2 algorithm (WynnDegree->1). 

60 

50 

40 

30 

20 

10 

0 

0 

20 

10 20 30 40 50 60 

Figure 6.6: Contour plot of g. 6.5, abscissa: ne, ordinate: ns. 

153 

40 

ns 

60

0 

-5 

-10 

-15 

0 

20 

ne 

40 

0 




with the -algorithm (WynnDegree->2). 

-5 

-10 

-15 

10 

20 

30 

ne 

40 

50 



with the -algorithm (WynnDegree->Infinity). 

154 

0 

20 

20 

40 

ns 

40 

ns 

60 

60

6.6 Controlling the accuracy of the extrapolation result 

60 

50 

40 

30 

20 

10 

0 

10 20 30 40 50 60 

Figure 6.9: Contour plot of g. 6.8, abscissa: ne, ordinate: ns. 


As we have seen in the last sections, we can improve the result of the extrapolation under 

most circumstances by increasing the number of sequence elements being subject to the 

extrapolation. The severe problem that still remains is that this is only a qualitative nding 

and that we have no guideline whatsoever as to how this parameter is to be chosen. From a 

user's point of view, the exact number of sequence members necessary to achieve a certain 

accuray is entirely irrelevant, provided that the goal is reached. On the contrary, it is much 

more convenient to let the user specify only the desired accuracy and to leave the choice of 

the other parameters to the algorithm, which then has to be somehow adaptive. 

If we want to adapt only the parameters of a basic numerical algorithm, then we in turn need 

a meta algorithm that starts from a given parameter set, calls the basic algorithm, judges the 

results and changes the parameters. This procedure must be repeated iteratively until either 

the outcome meets the requirements or a maximum number of iterations is reached | just 

to prevent the algorithm from being lost in an in nite loop or wasting precious computing 

time. From the previous sections, we can easily conclude what the three major elements of 

such a meta algorithm must look like: 

1. The basic algorithm consists of the computation of the sequence of partial sums for a 

given integrand together with the extrapolation. It returns an estimate of the integral. 

2. The parameters that can be in uenced directly by the meta algorithm are the number 

155

-5 

-7.5 

-10 

-12.5 

10 

20 

30 

ne 

40 

50 

0 




with the 2 algorithm (WynnDegree->1) and the zeros of the integrand as partition points. 

of sequence members used in the extrapolation as well as the precision of the internal 

computations. We have already noticed that it is most promising to use the entire 

sequence for this purpose rather than to shift the starting point for the extrapolation 

to higher index values. 

3. The parameter to be controlled is the accuracy of the result or | in other words 

| its error. We thus need a function that gives us an a posteriori estimate of the 

approximation error. 

A simple way to obtain an error estimate in Mathematica is to use the functions Accuracy 

and Precision on the result of a numerical computation. The former returns the number of 

signi cant digits to the right of the decimal point and therefore is a measure for the absolute 

error. The latter gives the total number of signi cant digits in a oating-point number. For 

very small numbers, the zeros between the decimal point and the rst non-zero digit are not 

counted, thus the precision of a number denotes its relative error (see also the explanations 

in example 6.3.1). 

Unfortunately, these two functions work only with variable-precision arithmetic, which cannot 

make use of the fast hardware-based machine-precision operations. Instead, it is implemented 

in software to keep track of the uncertainty associated with every number in the calculation, 

which makes it much slower than machine-precision arithmetic. For this reason, when Mathematica 

comes across a machine number in the course of the computation, all subsequent 

156 

20 

40 

ns 

60


operations are carried out in machine-precision, since it makes no sense to maintain a high 

precision if one of the operands involves an error of unknown magnitude [144]. Therefore, the 

user must be extremely careful not to inadvertently introduce a machine number somewhere 

in the calculation. 

After this introduction, we examine an example similar to (6.12). The only di erence is a 

constant factor that makes f(x) 1 to demonstrate the di erence between precision and 

accuracy of the extrapolation results. 

Example 6.6.1 We consider 

I = 

Z 1 

a 

10 10 

x 

x2 p 

sin x2 , a2 dx = 10 

, a2 2 10 

with a = 50. The code needed to generate the sequence of partial sums is slightly di erent now. 

In[26]:= a = 50 Pi; 

firstval = N[Integrate[10^10 Sin[Sqrt[x^2 - a^2]]/(x^2 - a^2) x, 

fx,50 Pi,51 Pig],40]; 

seq = Table[NIntegrate[10^10 Sin[Sqrt[x^2 - a^2]]/(x^2 - a^2) x, 

fx,i Pi,(i+1) Pig,WorkingPrecision->27], 

fi,51,300g]; 

partial = FoldList[Plus,firstval,seq]; 

trueval = N[10^10 Pi/2,40]; 

(6.14) 

For the numerical quadrature routine NIntegrate, we set WorkingPrecision->27. Since the default 

setting for PrecisionGoal is ten digits less than WorkingPrecision,we can expect the series members 

to be accurate to about 17 digits or perhaps even more (in fact, Accuracy[seq] returns the worst 

case value 19). All exact expressions are converted to arbitrary-precision numbers with the function 

N[x,prec], in contrast to the ordinary N[x], which yields machine-precision numbers. 

To compare the extrapolation results directly with the respective accuracy estimated by Mathematica, 

we calculate the logarithm of the actual extrapolation error and invert it such that we obtain essentially 

the number of correct decimal digits for each trial. Corresponding to the results of the last chapter, 

we set WynnDegree->Infinity. The following statements compute the tables for the entire sequence 

being subject to extrapolation. The results are shown in g. 6.11. 

In[27]:= err = Table[fne,-Log[10,Abs[SequenceLimit[Take[partial,ne], 

WynnDegree->Infinity]-trueval]]g, 

fne,6,Length[partial]g]; 

acc = Table[fne,Accuracy[SequenceLimit[Take[partial,ne], 

WynnDegree->Infinity]]g, 


Example 6.6.2 For the above integral (6.14) we now compute the relative approximation errors 

and the respective estimates obtained by the function Precision. Fig. 6.12 shows the results. 

157

7.5 

5 

2.5 

-2.5 

-5 

-7.5 

Figure 6.11: Absolute extrapolation error , log e S (k) 

0 

of Accuracy[x] ( ). 


50 100 150 200 250 

, S for varying k ( ) and the respective results 

In[28]:= err = Table[fne,-Log[10,Abs[SequenceLimit[Take[partial,ne], 

WynnDegree->Infinity]/trueval - 1]]g, 


acc = Table[fne,Precision[SequenceLimit[Take[partial,ne], 

WynnDegree->Infinity]]g, 


The examples make su ciently clear that the absolute error is not adequate as the only 

criterion for terminating the iteration. When the true value of the integral is very large, the 

absolute error can safely be large as well, even if it a ects digits to the left of the decimal 

point. On the other hand, for very small integrals, the absolute error must be very small, 

too, in order to obtain reasonable results. Consequently, the selection of the tolerable error 

limit requires knowledge of the a priori unknown value of the integral, which is impractical 

for an automatic quadrature routine that is supposed to be user-friendly. 

A better solution is to use the relative error estimate as criterion for error control. In the 

example, we could have achieved a precision of 12 (i. e. a relative error of 10 ,12 ) irrespective 

of the constant factor of the integrand. The only problem that might emerge is that of an 

integral very close to zero, in which case the desired relative error could not be reached for 

numerical reasons. In this case, the speci cation of an absolute error could be a more feasible 

alternative. 

So far, we always assumed that the parameter to be changed by the meta algorithm is the 

158


17.5 

15 

12.5 

10 

7.5 

2.5 

Figure 6.12: Relative extrapolation error , log e S (k) 

0 ,S 

S 

Precision[x] ( ). 

50 100 150 200 250 

for varying k ( ) and the respective results of 

number of sequence members included in the extrapolation. One might argue that altering the 

setting for WorkingPrecision should have a similar e ect. The following example explores 

this possibility. 

Example 6.6.3 The integral (6.12) 

I = 

Z 1 

a 

x 

x2 p 

sin x2 , a2dx = 

, a2 2 

shall be evaluated for a = 50 with di erent values for WorkingPrecision and with WynnDegree-> 

Infinity. As g. 6.13 shows, a high precision of the internal calculations naturally allows for a higher 

precision of the result, but this is only true if enough sequence members are taken into account. Before 

the respective saturation is reached, the setting of the internal precision has almost no e ect on the 

precision of the result. But there is an even more important aspect to this parameter: If the meta 

algorithm was to change the WorkingPrecision, the entire sequence of partial sums would have tobe 

re-computed for each iteration, which isvery time-consuming. If, on the other hand, the number of 

sequence elements treated with SequenceLimit is altered, then only a few new members need to be 

added in each cycle. For these reasons, the value of the internal precision remains xed throughout 

the computation. However, it must be set such that the required precision can be reached at all. 

159

25 

20 

15 

10 

5 


20 40 60 80 100 120 

Figure 6.13: Absolute extrapolation error , log e S (k) 

0 

, S depending on the number of sequence mem- 

bers being subject to extrapolation for WorkingPrecision->20 ( ), 30 ( ), and 40 ( ). 

The conclusions of this section concerning the precision control of an automatic quadrature 

routine are as follows: 

The parameter that is controlled iteratively by the meta algorithm is the length of the 

sequence of partial sums. 

The criterion for the termination of the iteration loop is the achievement of either a 

given absolute or relative error. This is the same concept that Mathematica uses for 

adaptive numerical quadrature. 

The default criterion must be that of the relative error. A speci cation of a minimum 

absolute error must in any case be based upon a speci c knowledge of the quadrature 

problem. 

The working precision must be high enough that the error requirements can be met. 

160

Chapter 7 

Mathematica implementation of a 

quadrature function 

Les mathematiciens n'etudient pas des objets, mais des relations entre les 

objets; il leur est donc indi erent de remplacer ces objets par d'autres, 

pourvu que les relations ne changent pas. La matiere ne leur importe pas, 

la forme seule les interesse. Henry Poincare, quoted in [1] 

Based on the ndings of the previous sections, this chapter describes how an automatic 

quadrature routine for an extrapolation strategy can be implemented in Mathematica. We 

begin the discussion from a user's point of view with the options that control the operation of 

the function. The internal structure of the function is another topic of interest, and nally we 

shall point out some implementation details. The entire source code of the package is listed 

in the appendix. In the last section of the chapter we shall apply the function to examples 

known from the literature to verify the correct operation. 

7.1 User interface of the function OscInt 

The top level function of the package has the same name as the whole package, OscInt. Its 

basic syntax is 

OscInt[fint, fzero, a, opts], 

and its purpose is to nd the numerical value of the integral 

I = 

Z 1 

a 

(x) u( (x)) dx ; (7.1) 

where u( ) is e j , e ,j or any linear combination of both, in particular sin( ) and cos( ). The 

individual input parameters of OscInt have the following meanings: 

161

7 Mathematica implementation of a quadrature function 

fint is the integrand function (x) u( (x)), which can be de ned in two ways. One is 


to use a so-called pure function, e. g. Sin[#]/#& for x (see also the opening remark 

in section 7.5). If the integrand is declared explicitly, then it must be of the form f[x] 

or, if there are several other parameters, f[a,b,c][x]. In this case, only the name of 

the function together with the additional parameters (f[a,b,c]) has to be passed to 

OscInt. 

fzero is the function being used to determine the partition points fx0;x1;x2;:::g. Depending 

on the option FunctionType it is understood either as an explicit enumeration 

of the partition points x (n); n 0 or as the argument function (x) (or an approximation 

(x)) of the oscillating factor of the integrand, which is then used to compute 

the subdivision points xn numerically. In the latter case, the function must be di erentiable. 

In the former case, it is the user's responsibility to ensure that fzero returns 

points in ascending order that lie exclusively to the right of the rightmost extremum or 

in exion point of the argument function. It is, however, not necessary to see to it that 

x (0) > a, as the routine will seek the rst x (k) > a itself. Like fint, the function 

fzero can be given either as a pure function or in explicit notation. 

a is the lower integration limit. 

opts are options the user can set to take in uence on the computation. They are 

explained below. 

Normally, the partition points are chosen such that they lie in a region where (x) and its 

derivative 0 (x) are monotonic, in order to avoid the problems discussed in section 6.2. If an 

enumeration function is passed to OscInt, this function must be written accordingly. If the 

original (x) is used as fzero, then z0 is selected to be larger than the largest real part of the 

solutions of 0 (x) =0. In some cases, however, it might be more convenient tochoose the rst 

subdivision point according to a di erent criterion | particularly if we use an approximation 

(x). Thus there is a second way to invoke OscInt with an explicit speci cation of a lower 

subdivision limit a0: 

OscInt[fint, fzero, fa, a0g, opts]. 

Apart from the new parameter, the meanings of the other input values remain unchanged. 

The partition points are now chosen such that x (k) >a0; k K(a0), or x0 >a0, depending 

on the interpretation of fzero. 

There is a variety of options indispensable to control the evaluation of OscInt. Some of them 

are standard Mathematica options that are used also for other numerical functions, others 

have been newly introduced. 

WynnDegree is the standard option for the computation of the limit of the sequence of 

partial sums. Its default value is Infinity. 

162

7.1 User interface of the function OscInt 

NSumTerms is taken from NSum although this function is not used within OscInt. Its 

meaning, however, is the same | it speci es how many elements of the sequence of 

partial sums are not included in the extrapolation. The default value is 10. 

NSumExtraTerms is used like in NSum to prescribe the number of sequence members 

subject to extrapolation (default value 12). Together with NSumTerms, it de nes how 

many elements of the sequence of partial sums have to be computed in total. 

MinRecursion is a standard option used for all numerical integrations that speci es 

how many levels of recursion are at least required before tests for convergence start. Its 

default value is 0. 

MaxRecursion sets the maximum depth for recursive algorithms and a ects both the 

numerical quadrature with NIntegrate and the numerical solving of equations with 

FindRoot. For the latter, its value is passed to the option MaxIterations. This 

change of name was necessary for reasons of consistency, because OscInt uses the option 

MaxIterations in a di erent way. The default value is MaxRecursion->20. 

AccuracyGoal is used to specify the absolute error limit for NIntegrate and defaults 

to Infinity (i. e. it is not used as a criterion for the quadrature). 

PrecisionGoal sets the relative error limit for the numerical quadrature and like in 

NIntegrate has the default value Automatic, which means that it is set to ten digits 

less than WorkingPrecision. 

PartitionAccuracy is in fact the setting for AccuracyGoal used in FindRoot to determine 

the partition points. With the change of name this parameter can be set independently 

of the AccuracyGoal of NIntegrate. The default value is PartitionAccuracy 

->Automatic. 

AccuracyControl sets the upper limit for the absolute error in the extrapolation result 

in that is speci es the digits of accuracy required. Its default value is Infinity, which 

means that it is not used as a criterion for terminating the iteration. If it is used, care 

must be taken to set the working precision high enough that the desired accuracy can 

be reached. 

PrecisionControl gives the upper limit for the relative error in the extrapolation result 

and is recommended for controlling the iteration. With its default value Infinity, itis 

normally not used. If precision or accuracy control is enabled, the iterative procedure 

continues to increase the length of the sequence of partial sums until either the absolute 

or relative error goal is achieved or the iteration count reaches MaxIterations. 

MaxIterations speci es how many improvements of the extrapolation are tried before 

OscInt terminates with an error message. The default value is 6. 

IterationLength denotes how many elements are added to the sequence of partial sums 

with each iteration. It defaults to 4. 

163


WorkingPrecision is the standard option to set the precision of the internal calculations, 

with $MachinePrecision as default value. Normally it may be set by the user 

to any arbitrary value (between the bounds $MinPrecision and $MaxPrecision, of 

course). If accuracy or precision control are used, then its treatment is slightly different, 

because these functions need high-precision arithmetic which in turn requires a 

working precision that is higher than the machine precision. Thus WorkingPrecision 

is set to the maximum of $MachinePrecision plus one digit, the precision probably 

speci ed by the user, and the value of PrecisionControl plus 13 digits | provided 

that precision control is used at all. The value of AccuracyControl cannot be used to 

set the working precision since this would require one to know the order of magnitude 

of the integral beforehand. 

FunctionType 2 fSinArgument, CosArgument, ZeroListg selects how the function 

fzero is to be interpreted. ZeroList means that the function returns consecutive 

partition points. The other possibilities specify the circular function whose zeros are 

computed for the use as subdivision points. For SinArgument, which is also the default 

value, the equation (x) = k is solved, for CosArgument, it is the equation 

(x) = k + =2. If the oscillating factor u( ) is a pure circular function, then this 

option additionally allows one to choose the extrema rather than the zeros as partition 

points. 

The default options of OscInt cause the function to run at machine precision and with the 

control loop for the precision of the result disabled. The second argument is in this case 

regarded as the argument (x) of the oscillation. These default settings make up the fastest 

and most user-friendly con guration. 

7.2 Structure of the package 

The function OscInt is actually nothing but a driver that calls other functions depending on 

the settings of various options. Fig. 7.1 shows the respective functions and their dependencies. 

The most important distinction inside the package is that between the use of error control 

for the extrapolation and the plain computation of the integral without any feedback loop. 

Depending on this decision, OscInt invokes the following functions: 

PartitionTable computes a list of subdivision points with a given length, starting at 

a suitable point above the lower integration limit. In fact it only calls the functions 

PartitionOffs and PartitionPoints with a reduced set of input parameters. 

PartInt takes as input a set of subdivision points and computes the integral by extrapolation. 

Like PartitionTable, itis used only for straightforward quadrature without 

error control. 

OscIntControlled is the version of OscInt used with error control. It comprises a 

control loop that monitors the precision of the extrapolation result and increases the 

length of the sequence of partial sums if needed. 

164

7.3 Implementation of OscInt and related functions 

OscInt 

PartInt PartitionTable 

OscIntControlled 

PartitionPoints PartitionOffs 

Figure 7.1: The package structure of OscInt 

Auxiliary 

Functions 

PartitionOffs computes an index o set value for the list of partition points. This 

o set is then used in PartitionPoints. 

PartitionPoints returns a list of subdivision points with given length and the possibility 

to specify a starting index. The index o set obtained from PartitionOffs adjusts 

the numbering such that the point with index zero always is the lowest possible partition 

point, with both the lower integration limit a or a manually set rst partition point a0 

taken into account. Therefore, x (0) > maxfa; a0g or x0 > maxfa; a0g, depending on 

the function being used to compute the subdivision points. 

The aforementioned functions are also directly accessible from outside the package. Their 

syntax and implementation details are dealt with in the next section. Apart from these 

functions that constitute the building blocks for the quadrature routine, a few auxiliary 

function are also included in the package. They have nothing to do with the quadrature 

itself, but are useful for several special applications. The names and dependencies of these 

functions are summarised in g. 7.2. 


This section discusses the submodules of the quadrature routine and provides information on 

the actual implementation. However, as we only want to outline the principles of operation, 

we shall concentrate on the most important details and avoid boring source code listings. 

Also, we shall not concern ourselves with the formal way of de ning a Mathematica package, 

as this is covered extensively in books like those by Wolfram [140] and Maeder [146] or in 

tutorial works like the book by Shaw and Tigg [147]. 

165

7.3.1 OscInt 

ApproxLimGeneric 

AsymptoticExpand 

PolynomialDegree 


Approximation error control Zero computation 

ApproxLimQuad 

ApproxLimHyp 

QuadOffset 

QuadZero 

HypOffset 

HypZero 

Figure 7.2: The auxiliary functions of OscInt 

HypOffsetExact 

HypZeroExact 

Since there are two di erent list of arguments that can be passed to OscInt (see also section 

7.1), we must specify two di erent rules that match these possibilities. The more general set 

of arguments is the one where the rst partition point is preset by the user. The case where 

only the lower integration limit is given can be easily extended to the general rule by setting 

a0 = a. Therefore only the body of the general rule needs to contain the entire code, whereas 

the body of the more special case simply calls OscInt again with the proper parameters, 

which is generally considered good programming style. 

OscInt[fint_, fzero_, fa_, a0_g, opts___Rule] := 

Module[fparttab,lowlim,prec, 

wp = WorkingPrecision/.foptsg/.Options[OscInt], 

nt = NSumTerms/. foptsg/.Options[OscInt], 

ne = NSumExtraTerms/. foptsg/.Options[OscInt], 

ac = AccuracyControl/. foptsg/.Options[OscInt], 

pc = PrecisionControl/.foptsg/.Options[OscInt]g, 

If[ac === Infinity && pc === Infinity, 

lowlim = If[a == a0, a, N[Max[a,a0],2 wp]]; 

parttab = PartitionTable[fzero,lowlim,ne+nt+1,opts]; 

PartInt[fint,parttab,a,opts], 

prec = If[pc === Infinity, 

Max[wp,$MachinePrecision+1], 

Max[wp,$MachinePrecision+1,pc+13]]; 

OscIntControlled[fint,fzero,a,WorkingPrecision->prec,opts]] 

]; 

OscInt[fint_, fzero_, a_, opts___Rule] := 

OscInt[fint,fzero,fa,ag,opts]; 

Remark (Multiple de nitions) Note that the order of the two de nitions within the 

package is essential, because Mathematica searches the list from the top to the bottom 

166


in order to nd a rule that matches a given set of arguments. If the two de nitions were 

interchanged, a list fb,b0g would match the pattern a , and consequently the variable a 

would erroneously be assigned the value fb,b0g. 

The function rst extracts the values of a couple of relevant options and then checks whether 

error control is required. If both AccuracyControl and PrecisionControl equal Infinity, 

then no error control is used, and OscInt performs a straightforward computation. The lower 

limit of the list of subdivision points is set either to a or maxfa; a0g, the latter being calculated 

numerically at a higher precision, because the function Max works only with numbers but not 

with symbols. 

Remark (Options and defaults) The ltering of the options is worth a separate 

mention [146]. It is accomplished by a double replacement rule like for instance in wp 

= WorkingPrecision/.foptsg/.Options[OscInt], which works as follows: If the user 

speci es a value WorkingPrecision->prec for this option, then it is passed to the function 

in the argument opts. The statement is now evaluated from the left to the right, and 

the rst replacement evaluates to prec. The second substitution fails because the list 

of default options obtained by Options[OscInt] contains of course no rule prec->x. If, 

on the other hand, the function is called without any option, then the rst replacement 

cannot be evaluated. In this case, the second rule matches and wp is set to the default 

value. 

The number of partition points is obtained from NSumTerms plus NSumExtraTerms plus one, 

so that the sequence of partial sums has the correct length for extrapolation. With the list 

of subdivision points, PartInt is called to compute the actual integral. 

If error control is to be used, a reasonable value for the WorkingPrecision is calculated 

from the machine precision plus one, the parameter wp that might have been set by the user 

(or equals the default value otherwise), and the PrecisionControl parameter if applicable. 

This working precision, which might di er from the user's speci cation, is then used in the 

call of the function OscIntControlled. Note that this particular option is placed before 

the other options opts, which causes the new setting for WorkingPrecision to override any 

other speci cation for this option, because Mathematica takes the rst value it encounters 

and disregards all subsequent ones. 

7.3.2 PartitionTable 

This function actually only calls PartitionOffs and PartitionPoints, but is nonetheless 

useful for the computation of a list of subdivision points. Its syntax is 

PartitionTable[zerof, a, n, opts] 

with the same input arguments as OscInt except for n, which denotes the number of partition 

points to be computed. The options are the same as for OscInt. 

167

7.3.3 PartitionOffs 


The aim of the function PartitionOffs is to nd an index o set for the subsequent computation 

of the partition points such that we can assume that the point with index zero is the 

smallest possible one | irrespective of the lower integration limit or any particular shape of 

the argument function (x). The function call is 

PartitionOffs[zerof, a, opts], 

with zerof as function de ning either the argument (x) oranenumerating function for the 

partition points x (n). The parameter a is not necessarily the lower integration limit but 

can also be any arbitrarily chosen abscissa point meant asalower bound for the sequence of 

partition points. The options are the same as for OscInt. 

How the function is evaluated chie y depends on the option FunctionType that speci es how 

zerof is to be interpreted. We shall rst discuss the case where FunctionType->ZeroList, 

which means that zerof directly returns the subdivision points x (n); n 0. This function 

has to be de ned following one important rule: the rst point it returns must be larger than 

the position of any extremum of (x) or its derivatives, so both (x) and its derivatives 

must be monotonic for x x (0). Then we must simply nd the smallest index o 0 

where x (o) a. This operation is performed by the following statement, where the equation 

x (na) =ais solved numerically and the result is set to o = dnae. For this purpose, zerof[n] 

must be able to accept input parameters of type real although the value it returns makes sense 

only for integer input values. The option MaxIterations is set to the value of the OscInt 

option MaxRecursion, AccuracyGoal in this particular case equals to PartitionAccuracy. 

If[N[zerof[0]] < N[a], 

Ceiling[x/.FindRoot[zerof[x] == a,fx,0,1g, 

WorkingPrecision->wp, 

AccuracyGoal->pa, 

MaxIterations->ma]], 

0] 

Remark (Implementation) To determine whether x (0)


the complete list of solutions. We then regard only the real part of the possibly conjugate 

complex solutions and return the largest one, provided it is still larger than the lower limit a. 

Thus we haveo= maxfxo;ag. 

extrema = NSolve[zerof'[x] == 0,x,WorkingPrecision->wp]; 

If[Length[extrema] == 0 || extrema === ffgg, a, 

Max[Re[x/.Last[extrema]],a]] 

Remark (Extrema) To nd the extrema of (x), we cannot use FindRoot since we do 

not know where to start the iteration. Hence we cannot ensure that we obtain the largest 

solution. This is why wehave to use the slower NSolve. From the code it is also clear that 

the function zerof must be di erentiable. The odd-looking test for extrema === ffgg 

is nothing but the condition 0 (x) 0, because this structure is what NSolve returns to 

indicate that the equation is satis ed for all x. 

Example 7.3.1 Consider the argument function 

(x) =x(x,1+j)(x,1,j)(x,4+j)(x,4,j) 

=x , x 4 ,10x 3 +35x 2 ,50x +34 ; 

which has no extremum at all, as the plot shows. Consequently, the rst derivative has two pairs of 

conjugate complex zeros, and PartitionOffs indeed returns the larger real part of the two. 

In[1]:= f[x_] := x (x-1+I) (x-1-I) (x-4+I) (x-4-I) 

In[2]:= Plot[f[x],fx,-0.5,4.5g]; 

60 

40 

20 

-20 

1 2 3 4 

In[3]:= NSolve[f'[x]==0,x] 

Out[3]= ffx -> 0.740373 - 0.294381 Ig,fx -> 0.740373 + 0.294381 Ig, 

fx -> 3.25963 - 0.294381 Ig,fx -> 3.25963 + 0.294381 Igg 

In[4]:= PartitionOffs[f,0] 

Out[4]= 3.25963 

169

7.3.4 PartitionPoints 


This function returns a list of subdivision points in ascending order. The way to call it is 

PartitionPoints[zerof, a, n, offs, irel, opts], 

zerof being the well-known function (x) orx (n). The argument a denotes the lower bound 

of the sequence, n is the desired number of points. The variable offs is the o set o obtained 

from PartitionOffs. This function could have been merged into PartitionPoints (and in 

fact had been in a former version of PartitionTable), but as for instance OscIntControlled 

makes multiple calls to PartitionPoints to gradually build up the table of subdivision points, 

is was deemed more e cient to split these two functional blocks. The parameter irel denotes 

the index r 0 of the rst point and can be used to compute subdivision tables in two or 

more portions. 

Like PartitionOffs, this function also behaves di erently for di erent values of the option 

FunctionType. The simplest case is when zerof returns the partition points themselves 

(FunctionType->ZeroList). Then we just need to build up the table fx (r);x (r+ 

1);::: ;x (r+n,1)g. 

Things are more complicated when (x) is given directly (FunctionType->SinArgument or 

CosArgument). Then the evaluation depends on the behaviour of (x) at x ! 1. If the 

limit is de nite, then the integrand is not in nitely oscillating, and a partition extrapolation 

quadrature is not applicable at all. In this case, we return Infinity so that the rst partial 

integral already covers the entire range. For an inde nite limit, we must distinguish between 

increasing and decreasing functions and nally obtain the partition point from the following 

equations: 

zi : 

8 

>< 

>: 

xi = 1 if ,1Infinity]]], 

Table[Infinity,fk,0,n-1g], 

N[zerof'[offs+1]] > 0, 

koffs = N[Ceiling[zerof[offs]/Pi],2 wp]; 

Re[Table[x/.FindRoot[zerof[x] == (k+koffs+cfoffs) Pi, 

fx,offs,offs+1g], 

fk,irel,irel+n-1g]], 

True, 

170 

(7.2)


], 

koffs = N[Floor[zerof[offs]/Pi],2 wp]; 

Re[Table[x/.FindRoot[zerof[x] == (-k+koffs-cfoffs) Pi, 

fx,offs,offs+1g], 

fk,irel,irel+n-1g]] 

Remark (Finding the limit) The test for a de nite limit of (x) was implemented 

directly with the built-in function Limit. This works well enough for any kind of polynomial 

and rational functions, but usually gives problems with trancendental functions, 

which, however, are not supposed to be part of (x) anyhow. The outcome of Limit is 

then converted to a machine number and examined whether it is a number or not to lter 

out the value Infinity. 

Whether (x) is increasing or decreasing can be identi ed with the sign of 0 (x) for x>o. 

Accordingly, the table of subdivision points is computed with FindRoot, which is again 

straightforward because the respective equations have only one solution above the starting 

points o and o +1. Although these solutions must be real-valued, there are often spurious 

imaginary parts left, which are eliminated by taking only the real part of the resulting 

table. 

7.3.5 PartInt 

This function actually computes the integral R 1 

f(x) dx based on a set of n subdivision points 

a 

fx0;x1;::: ;xn,1g; xi a by means of extrapolation. Its syntax is 

PartInt[f, zerotab, a, opts], 

where f is the integrand function (de ned either as pure function or like f[x]), zerotab 

is the list of partition points, and a is the lower integration limit. The essential part of the 

function body is listed below without most of the option assignments. The rst statement isa 

test that should always fail under normal conditions, because the functions that generate the 

subdivision points already make sure that the rst point x0 a. If this condition, however, is 

not satis ed, then all members greater than a are ltered out for further use, and a warning 

message is issued to inform the user that only fewer sequence members than intended will be 

computed. 

Remark (Selecting members of lists) The ltering of the partition points is accomplished 

with the function Select[list, crit], which tests each element of a list with a 

given criterion and picks out those for which the test yields True. 

The value of the integral consists of two parts. The rst is the de nite integral R x0 

f(x) dx 

a 

(if, of course, x0 6= a) that is computed with the double exponential rule according to the 

results of chapter 6.3. The second contribution is the extrapolation result that is obtained 

in the well-known manner by computing a sequence of partial integrals R xi+1 

f(x) dx and 

xi 

nding the numerical limit of this series. If the rst partition point equals Infinity, then 

this extrapolation is skipped. This feature is imperative in order to include all special cases 

of parameter-dependent integrands that lose their in nite oscillations under certain circumstances. 

171


If[N[zeros[[1]]] < N[a], 

zeros = Select[zeros,(N[#] >= N[a])&]; 

Message[OscInt::lowfirst,Length[zeros]]]; 

If[N[zeros[[1]]] == N[a], 0, 

NIntegrate[f[x],fx,a,zeros[[1]]g, 

Method->DoubleExponential]] + 

If[N[zeros[[1]]] == Infinity, 0, 

seqtab = Table[NIntegrate[f[x],fx,zeros[[i]],zeros[[i+1]]g, 

Method->GaussKronrod], 

fi,1,Length[zeros]-1g]; 

SequenceLimit[Take[FoldList[Plus,0,seqtab],-ne],WynnDegree->wd]] 

7.3.6 OscIntControlled 

This function is probably the most intricate of the package. It is basically a copy ofPartInt 

extended by a control loop. The user interface is exactly the same as that of OscInt, it is 

invoked by 

OscIntControlled[f, fzero, a, opts]. 

The only di erence in the behaviour towards the user is that the setting of WorkingPrecision 

is not touched in any way, which means that the user himself must choose a reasonable value 

(in particular one greater than the machine precision). The meta algorithm that is used is 

shown in g. 7.3. This algorithm is of course applied only if error control is enabled, otherwise 

the function works much like PartInt. 

compute initial set of partition points fx0;x1;::: ;xng 

compute rst partial integral R x0 

a f(x) dx 

compute partial integrals R xi+1 

f(x) dx 

xi 

compute sequence limit 

iterations := 1 

while not reached precision or accuracy or maximum iterations do 

compute il additional partition points 

compute additional partial integrals 

iterations := iterations + 1 

compute sequence limit 

check error conditions 

Figure 7.3: The meta algorithm for OscIntControlled 

The portion of the function body that implements the control loop is given below. Note 

that the table of partition points parttab is not augmented in each pass, but only the last 

element is kept before the new points are added. For the sake of simplicity, the variable ne 

that initially holds the value of NSumExtraTerms is increased by the number il of additional 

sequence members directly. 

172

7.4 Auxiliary functions 

While[iterations < it && Accuracy[intlim] < ac && Precision[intlim] < pc, 

parttab = Join[Last[parttab], 

PartitionPoints[fzero,a,il,offs,ne+nt+1,opts]]; 

seqtab = Join[seqtab, 

Table[NIntegrate[f[x],x,parttab[[i]],parttab[[i+1]], 

Method->GaussKronrod], 

i,1,Length[parttab]-1]]; 

ne = ne + il; 

iterations++; 

intlim = SequenceLimit[Take[FoldList[Plus,firstval,seqtab], 

-ne],WynnDegree->wd] 

]; 

Example 7.3.2 We want to compute R 1 


x 

dx to the precision of 10 signi cant digits. Note that 

0 

the integrand as well as the argument are given as pure functions. If we choose the working precision 

too low, we hit the iteration limit. 

In[5]:= OscIntControlled[Sin[#]/#&,#&,0, 

WorkingPrecision->17,PrecisionControl->10] 

OscInt::accfail: 

OscInt failed to achieve the desired accuracy or precision after 6 

iterations. Increase MaxIterations or WorkingPrecision (currently 17). 

Out[5]= 1.570798 

If we increase the working precision, we obtain the desired result. 

In[6]:= OscIntControlled[Sin[#]/#&,#&,0, 

WorkingPrecision->20,PrecisionControl->10] 

Out[6]= 1.570796327 


In addition to the functions that are needed for pure quadrature, a number of supporting 

routines have been added to the package. These functions facilitate the use of the quadrature 

programs in that they provide information required by the quadrature algorithm. 

A set of such functions returns the zeros fx0;x1;x2;:::g of sin f(x) for quadratic and hyperbolic 

arguments f(x). In order to be suitable as partition points for the quadrature, they 

are de ned such that the rst zero x0 always lies to the right of the extremum of f(x). It is 

thus necessary to nd solutions for the equation f(x) =(k+o) , where k 0 and the o set 

o = f(xe)= is chosen so that f(x) is monotonic for x>xe. 

173

7.4.1 Zero computation for quadratic arguments 


Let f(x) be a quadratic polynomial. We thus need to solve the equation 

ax 2 + bx + c =(k+o) ; k 0 (7.3) 

for arbitrary coe cients and for the right branch of the parabola. Depending on the coe - 

cients, we nd the solutions 

x = 

8 

>< 

>: 

p 

,b+ b2 ,4a(c,(k+o) ) 

2a 

,b, p b 2 ,4a(c,(,k+o) ) 

2a 

(k+o) ,c 

b 

(,k+o) ,c 

b 

if a>0 

if a0 

if a =0, b< 

>: 

d 4ac,b2 

4a e if a>0 

b 4ac,b2 

4a c if a


Linear approximation 

For x 0, we can approximate (7.6) by 

with the solutions 

(a + b)x + d =(k+o) ; k 0 (7.7) 

x = 

8 

>< 

>: 

(k+o) ,d 

a+b 

(,k+o) ,d 

a+b 

if a + b>0 

if a + b< 

>: 

f 0 (x) =a+ 

q 

+ a2c (7.8) 

x 

p x 2 +c =0; (7.9) 

b2 ,a2 if jbj > jaj; a b jaj; a b>0: 

With these values we can now solve (7.7) for o and get 

o = 

8 

>< 

>: 

d (a+b)xe+d e if jbj > jaj; b>0 

b (a+b)xe+d c if jbj > jaj; b0 

b d c if jbj jaj; a+b

Exact solution 


For the exact solution of (7.6) we must distinguish several cases: when f(x) is monotonically 

decreasing (a + b0) or below (b< 

>: 

a((k+o) ,d), p b2 ((a2 ,b2 )c+((k+o) ,d) 2 ) 

a2 ,b2 p 

a((k+o) ,d)+ b2 ((a2 ,b2 )c+((k+o) ,d) 2 ) 

a2 ,b2 p 

a((,k+o) ,d)+ b2 ((a2 ,b2 )c+((,k+o) ,d) 2 ) 

a2 ,b2 a((,k+o) ,d), p b 2 ((a 2 ,b 2 )c+((,k+o) ,d) 2 ) 

a 2 ,b 2 

((k+o) ,d) 2 ,a 2 c 

2a((k+o) ,d) 

if a + b>0; b>0 

if a + b>0; b< 

>: 

b d,p (b 2 ,a 2 )c c if jbj > jaj; b0 

b d+bp c c if jbj jaj; a+b


For the sake of computing e ciency, two distinct functions have been de ned for each case. 

The one that computes the index o set requires nothing but the coe cients as an input, 

whereas the other also needs the pre-computed o set value and the index of the subdivision 

point. 

HypOffsetExact[a_,b_,c_,d_] := Which[ 

(Abs[b] > Abs[a]) && (c >= 0) && b > 0, 

Ceiling[(Sqrt[c (b^2-a^2)] + d)/Pi], 

(Abs[b] > Abs[a]) && (c >= 0) && b < 0, 

Floor[(-Sqrt[c (b^2-a^2)] + d)/Pi], 

(a+b) > 0, Ceiling[(b Sqrt[c] + d)/Pi], 

(a+b) < 0, Floor[(b Sqrt[c] + d)/Pi], 

True, 0]; 

HypZeroExact[a_,b_,c_,d_,o_][k_] := Which[ 

a == b && a != 0, 

(((k+o) Pi - d)^2 - a^2 c)/(2 a ((k+o) Pi - d)), 

(a+b) > 0, 

(a (( k+o) Pi-d) - 

Sign[b] Sqrt[b^2 ((a^2-b^2) c + (( k+o) Pi-d)^2)])/(a^2-b^2), 

(a+b) < 0, 

(a ((-k+o) Pi-d) + 

Sign[b] Sqrt[b^2 ((a^2-b^2) c + ((-k+o) Pi-d)^2)])/(a^2-b^2), 

True, Infinity]; 

In the example, the use of the signum function allowed to contract the cases in (7.12) that 

di er only in the sign of b. Although this brings no advantage in terms of execution time, it 

renders the code more compact. 

Remark (Implementation) The Which[test1, exp1, test2, exp2,...] statement 

that is used to implement the functions consists of a sequence of conditional statements 

and expressions. The tests are evaluated in turn until one of them yields True. In this 

case, the following expression is evaluated, and the result is returned. The tests therefore 

need not be mutually exclusive, but care must be taken to arrange them in the correct 

order. It is also advisable to include a default value with a preceeding True statement as 

last element to ensure that the function always returns a result. 

It is important that the coe cients used in the function call be numbers and not symbolic 

expressions. The reason is that the conditional statements could not be evaluated when used 

with symbolic expressions unless their values are computed explicitly. 2 Pi > Pi is therefore 

left unevaluated, whereas N[2 Pi] > N[Pi] calculates the numerical values of both sides of 

the expression prior to comparing them and yields the correct answer. To make the functions 

versatile with respect to the numerical precision, such a conversion has not been included in 

the functions. 

177

7.4.3 Approximation error control 


Another auxilliary function implements the criterion of equation (6.13), 

0 (x) 

0 (x) , 0 (x) = k; 

which has been given as a quality measure for asymptotic partition. It relates the argument 

function and its polynomial approximation by means of their respective oscillation frequencies. 

The di erence between these two cause the actual partition points to deviate from the ideal 

phase invariant positions, which in turn produces wavepacket-like pictures in the sequence of 

partial sums. Hence the above formula hence gives the length of such awave packet depending 

on the abscissa value. We would like, however, to prescribe k and get the corresponding x as 

a result in order to know where to start the extrapolation. The functions needed to this end 

exploit the symbolical calculation capabilities of Mathematica. 

To nd the polynomial approximation of a given function f(x) for x ! 1, we de ne two 

small routines that determine the polynomial degree of f(x) and then compute the asymptotic 

expansion to the correct order. 

PolynomialDegree[f_,x_Symbol] := 

Length[CoefficientList[Series[f,fx,Infinity,1g],x]]-1; 

AsymptoticExpand[f_,x_Symbol] := 

Normal[Series[f,fx,Infinity,PolynomialDegree[f,x]g]]; 

Remark (Series expansion) The function Series[f, x, x0, n] generates a power series 

expansion for a given function to the order n about the point x0. In PolynomialDegree 

we use it just to determine the highest order terms, that is why we set n = 1. With 

CoefficientList we then obtain a list of the coe cients of the resulting polynomial 

and, by regarding the length of this list, the degree of the polynomial. 

In AsymptoticExpand, we again calculate the power series, but now to the correct order. 

The command Normal nally converts the series expansion to a normal expression by 

cutting o the error term that is also returned by Series. 

Example 7.4.1 Consider the function f(x) =x+ p x 4 +x 3 ,x+1. The series expansion calculated 

to nd the degree of the polynomial includes only the highest order terms and an error term. 

In[7]:= Series[x+Sqrt[x^4+x^3-x+1],fx,Infinity,1g] 

Out[7]= 2 3 x 1 0 

x + --- + O[-] 

2 x 

The length of the list of coe cients (starting with x 0 ) gives the desired value. 

In[8]:= CoefficientList[%,x] 

178


Out[8]= 3 

f0, -, 1g 

2 

The correct asymptotic expansion comprises all terms in x down to x 0 , but of course no error term. 

In[9]:= AsymptoticExpand[x+Sqrt[x^4+x^3-x+1],x] 

Out[9]= 1 3 x 2 

-(-) + --- + x 

8 2 

These supporting functions are used inside a more intricate module that rst determines the 

asymptotic expansion, then solves the equation given above and returns the largest real-valued 

solution. There are several cases that lead to error or warning messages: 

As the asymptotic expansion must be of the form (x) =xpP1ai i=0 xi , p being a nonnegative 

integer, all expansions with other than integer exponents are rejected (for instance 

(x) = p x3 ). 

If no solution exists, an error is issued. 

The existence of more than one solution is reported to the user, and the largest solution 

is returned. 

The function ApproxLimGeneric takes as inputs the argument function that is to be approximated 

and the number of subdivision points that are sought within one wavepacket. 

ApproxLimGeneric[f_,goal_] := 

Module[fx,fapprox,g,coeffs,respos,resneg,resg, 

fapprox = AsymptoticExpand[f[x],x]; 

coeffs = CoefficientList[fapprox,x]; 

If[D[coeffs,x] =!= Table[0,fLength[coeffs]g], 

Message[ApproxLimGeneric::nopoly], 

g = D[fapprox,x] / D[fapprox - f[x],x]; 

respos = Select[NSolve[g == goal,x],(Im[x/.#] == 0)&]; 

resneg = Select[NSolve[g == -goal,x],(Im[x/.#] == 0)&]; 

res = Select[Join[respos,resneg],Positive[x/.#]&]; 

If[Length[res] == 0, 

Message[ApproxLimGeneric::nosolution], 

If[Length[res] > 1, 

Message[ApproxLimGeneric::nounique,Length[res], 

Map[x/.#&,res]]]; 

x/.Last[Sort[res]]]] 

] 

179


Remark (Implementation) To check whether the exponential expansion comprises 

only terms with integer exponents, we simply form the derivative of the list of coe cients. 

Since terms with non-integer exponents are listed together with the constant terms, the 

derivative insuch cases is di erent from zero. Note that to check this, we must test the 

two expressions for symbolic identity with the relational operator =!=, because a simple 

test for equality with != will not be evaluated if the variables in the symbolic expression 

D[f, x] are not assigned any values. 

Since it is not possible to numerically solve equations that involve functions like Abs, 

NSolve must be invoked twice with both the positive and negative possibility for the 

variable goal. The results are ltered such that only the real and positive solutions are 

retained. As NSolve returns a list of replacement rules fx->x1, x->x2 ...g, wemust 

apply the replacement operator /. to the elements to enable testing for a vanishing 

imaginary part or the like. The operation of the Select command has been described in 

section 7.3.5 . 

Example 7.4.2 The next two examples show the error checking mechanisms. The function in the 

rst one cannot be expressed as apower series with integer exponents, whereas the second one has 

multiple solutions. Note that the functions are speci ed as pure functions that need no separate 

declaration. 

In[10]:= ApproxLimGeneric[Sqrt[#^3]&,10] 

ApproxLimGeneric::nopoly: 

Function has no polynomial expansion with integer exponents. 

In[11]:= ApproxLimGeneric[(#^2-20 Sqrt[#^2+1])&,20] 

ApproxLimGeneric::nounique: 

There are 3 real solutions f3.98475, 8.68827, 10.8449g, 

here is the largest. 

Out[11]= 10.8449 

There are also two special cases of this module included in the package: ApproxLimQuad 

for parabola-like functions ax 2 + b p x 2 + c + d and ApproxLimHyp for functions of the form 

ax + b p x 2 + c + d. They can be found together with the code listing of the package in the 

appendix. 

180

7.5 Test of the quadrature routine 


We now want to apply our quadrature routine to some of the test functions that are commonly 

used in the literature. The objective of such benchmarks usually is to examine how 

many evaluations of the integrand function a given algorithm needs to return a result of the 

required accuracy. The number of function evaluations is in turn a measure for the e ectiveness 

of competing algorithms. In the case of our quadrature routine, however, we obtain no 

information how often Mathematica needs to evaluate the integrand for its internal computations. 

All we can test is the accuracy of the result depending on the settings of the algorithm 

parameters. So the main goal of this section is to verify that the algorithm works correctly 

rather than to compare it with other routines. 

As the execution time of the routine is not our main concern here, we do not use explicit 

enumeration functions that de ne the partition points for the quadrature. We just specify 

the argument function of the oscillating factors of the integrand and leave the determination 

of the appropriate subdivision of the integration range to Mathematica. 

Remark (Pure functions) To avoid unnecessary declarations, we use pure functions 

[140] to specify the functions that are needed to compute the partition. In such a function 

the places where the independent variables are to be inserted upon evaluation are marked 

with slots (#, ##, #1, ::: ), the function itself is de ned as pure by a trailing ampersand (&). 

So while we would de ne a quadratic polynomial in full notation as p[x_] := x^2 + 3 x, 

we write the pure function as (#^2 + 3 #)&. This concept enables us to write very 

compact code, but it is useful only if the function is not required to have a name. 

Prior to testing we must load the package. 

In[12]:=

Out[14]= 0.537450389063711 


The result is correct to 13 digits, although Mathematica initially considers it less reliable. The di erence 

in the last two digits is due to roundo errors. We can still improve the result by using a higher 

working precision and including more terms in the extrapolation. 

In[15]:= OscInt[f,#&,0,WorkingPrecision->30,NSumExtraTerms->25] 

Out[15]= 0.53745038906373283 

It is interesting to notice that the last digit of the èxact' value given by Ehrenmark seems to be wrong, 

for if we let Mathematica compute the integral analytically, we obtain a slightly di erent result. 

In[16]:= Integrate[f[x],fx,0,Infinityg] 

Out[16]= Pi BesselI[0, 2] 3 3 

---------------- - 2 HypergeometricPFQ[f1g, f-, -g, 1] 

2 2 2 

In[17]:= N[%,20] 

Out[17]= 0.5374503890637328029 

Example 7.5.2 

Z 1 

In[18]:= f[x_] := Sin[x]/Sqrt[x]; 

OscInt[f,#&,0] - N[Sqrt[Pi/2],20] 

Out[18]= -12 

5.05929 10 

0 

r 


p dx = 

x 2 

Again this result could be improved with a higher working precision, albeit at the expense of an 

increased computing time. 

In[19]:= OscInt[f,#&,0,NSumTerms->30,NSumExtraTerms->20, 

WorkingPrecision->34] - N[Sqrt[Pi/2],30] 

Out[19]= -19 

0. 10 

The next integrals have been used by Espelid and Overholt as benchmarks [123]. Since their 

algorithm was tailored to regularly oscillating integrands, the arguments of the sin-functions 

are at least asymptotically linear. 

182


Example 7.5.3 

Z 1 

In[20]:= f[x_] := Sin[x]/Sqrt[1+x]; 

SetPrecision[OscInt[f,#&,0],16] 

Out[20]= 0.8095254817472858 

Example 7.5.4 

0 

Z 1 

1 


p 1+x dx =0:809525481747 

sin(x + 1 

x ) 

p dx =0:232948197094 

x 

In[21]:= f[x_] := Sin[x+1/x]/Sqrt[x]; 

SetPrecision[OscInt[f,(#+1/#)&,1],16] 

Out[21]= 0.23294819709403225 

Example 7.5.5 

Z 1 

1 

sin(x + 1 p ) x 

p dx =0:0416328516893 

x 

In[22]:= f[x_] := Sin[x+1/Sqrt[x]]/Sqrt[x]; 

SetPrecision[OscInt[f,(#+1/Sqrt[#])&,1],16] 

Out[22]= 0.04163285168937769 

In the last two examples it would have also been possible to use only the linear part of the 

argument function for the determination of the partition points by substituting the pure 

function #& for the respective arguments in the function call. The results would in fact have 

been the same. 

Some other functions with linearly growing numbers of oscillations have been examined by 

Hasegawa and Torii [127]. 

Example 7.5.6 

Z 1 

0 

e ,x cos xdx = 1 

2 

183

In[23]:= f[x_] := Exp[-x] Cos[x]; 

OscInt[f,#&,0] 

Out[23]= 0.5 


In this case we do not even have the slightest approximation error, as can be shown by the attempt 

to reveal further digits of the result. 

In[24]:= SetPrecision[%,40] 

Out[24]= 0.5 

Example 7.5.7 

Z 1 

0 

x 

x 2 +1 sin xdx = 2e 

In[25]:= f[x_] := x/(x^2+1) Sin[x]; 

SetPrecision[OscInt[f,#&,0],16] - N[Pi/(2 E),20] 

Out[25]= -14 

-1.93 10 

Example 7.5.8 

Z 1 

0 

cos x 

p x 2 +1 dx = K0(1) 

In[26]:= f[x_] := Cos[x]/Sqrt[x^2+1]; 

SetPrecision[OscInt[f,#&,0],16] - N[BesselK[0,1]] 

Out[26]= -13 

-1.12188 10 

Example 7.5.9 

Z 1 

0 

ln x2 +4 

x2 +1 cos !x dx = , ,! ,2! 

e , e 

! 

In[27]:= f[x_] := Log[(x^2+4)/(x^2+1)] Cos[omega x]; 

omega = 5; 

SetPrecision[OscInt[f,omega #&,0],16] - 

N[(Exp[-omega]-Exp[-2 omega]) Pi/omega,20] 

184


Out[27]= -15 

-8.471 10 

The next test functions were taken from Sidi [128]. 

Example 7.5.10 

Z 1 

0 


x 

sinh x 

10 

sinh 2x dx = arctan tan 

10 

1 

4 

tanh 10 

4 

In[28]:= f[x_] := Sin[x]/x Sinh[1/10 x]/Sinh[2/10 x]; 

OscInt[f,#&,0] - N[ArcTan[Tan[1/4 Pi] Tanh[10 Pi/4]]] 

Out[28]= -13 

-2.72449 10 

Increasing both WorkingPrecision and NSumExtraTerms,we obtain a result with more correct digits. 

Although Sidi gives the exact value of the integral as 0.785398012695720765, the last two digits di er 

in the result found by Mathematica as well as in the numerical evaluation of the analytical solution. 

In[29]:= OscInt[f,#&,0,WorkingPrecision->34,NSumExtraTerms->30] 

Out[29]= 0.78539801269572077061037 

In[30]:= N[ArcTan[Tan[1/4 Pi] Tanh[10 Pi/4]],20] 

Out[30]= 0.78539801269572077061 


Z 1 

In[31]:= f[x_] := Sin[Pi/2 x^2]; 

OscInt[f,Pi/2 #^2&,0] - 0.5 

Out[31]= -14 

-2.08167 10 

0 

sin x2 

2 

185 

dx = 1 

2


Z 1 

0 

sin x 2 


cos x2 

4 

dx 

x2 = e, , 1 

4 p 2 

This integral, taken from an earlier work of Sidi [126], is di erent from the others examined so far 

in that the integrand has an in nite number of oscillations both for x ! 1 and x ! 0. Thus it 

cannot be treated in a straightforward manner like the integrals before and we split the integration 

interval at the point x =1. The right part I1 = R 1 

sin 0 x2 cos x2 dx 

4 x2 remains unchanged, whereas we 

transform the left part by achange of variable x ! 1=x to obtain a second integral in our standard 

form I2 = R 1 

sin( x 1 2 ) cos 4x2 dx. 

In[32]:= f1[x_] := Sin[Pi/x^2] Cos[Pi x^2/4]/x^2; 

f2[x_] := Sin[Pi x^2] Cos[Pi/(x^2 4)]; 

OscInt[f1,Pi/4 #^2&,1] + OscInt[f2,Pi #^2&,1] - 

N[(Exp[-Pi] - 1)/(4 Sqrt[2])] 

Out[32]= -14 

4.54636 10 

The next integrals are taken from the tables of Abramowitz and Stegun [148] and involve 

Bessel functions that are known to behave like cos x for large values of x. Thus the partition 

points may be computed by a linear function. 


In[33]:= f[x_] := BesselJ[0,x]; 

OscInt[f,#&,0] - 1 

Out[33]= -14 

-4.74065 10 


Z 1 

0 

Z 1 

0 

J0(x) dx =1 

cos xK0(x)dx =1 

In[34]:= f[x_] := Cos[x] BesselK[0,x]; 

OscInt[f,#&,0] - N[Pi/(2 Sqrt[2])] 

Out[34]= -14 

-2.84217 10 

186


The following integrals can be found in the tables of Grobner and Hofreiter [149]. Some of 

them have complex-valued integrands to show that the quadrature routine is able to treat 

complex functions as well. 


Z 1 

0 

x , 

e 10 sin p xdx = 10 

2 

p 10 e , 10 4 

In[35]:= f[x_] := Exp[-x/10] Sin[Sqrt[x]]; 

OscInt[f,Sqrt[#]&,0] - N[5 Sqrt[10 Pi] Exp[-10/4]] 

Out[35]= -11 

2.87295 10 


Z 1 

,1 

e j(x2 +x) dx = p e j( 4 , 1 4 ) 

In order to be able to compute this integral, we divide it at the origin and transform the left half 

onto the positive real axis. We thereby obtain two integrals in our standard notation R 1 

0 ej(x2 +x) dx + 

R 1 

0 ej(x2 ,x) dx. The result is accurate only to nine digits if we leave the options at their default values. 

A larger number of terms, however, signi cantly improves the result. 

In[36]:= f1[x_] := Exp[I (x^2 + x)]; 

f2[x_] := Exp[I (x^2 - x)]; 

OscInt[f1,(#^2+#)&,0,NSumExtraTerms->30] + 

OscInt[f2,(#^2-#)&,0,NSumExtraTerms->30] - 

N[Sqrt[Pi] Exp[I (Pi/4 - 1/4)],20] 

Out[36]= -13 -13 

1.42331 10 - 1.4494 10 I 


Z 1 

0 

1 j(x, 

e x ) dx 

r 

p =(1+j) 

x 2 e,2 

Since the integrand in this case, too, has an in nite number of zeros as it approaches the lower 

integration limit, we again apply the successful strategy of dividing the integration interval at the 

point x = 1. We map the interval [0; 1] to the semi-in nite range [1; 1[ by the transformation 

x ! 1=x and obtain the two integrals R 1 

0 ej(x, 1 x ) dx 

px and R 1 

0 ej( 1 x ,x) p x 

x 2 dx that are amenable to our 

quadrature routine. 

187

In[37]:= f1[x_] := Exp[I (x - 1/x)]/Sqrt[x]; 

f2[x_] := Exp[I (1/x - x)] Sqrt[x]/x^2; 

OscInt[f1,(#-1/#)&,1] + OscInt[f2,(1/#-#)&,1] - 

N[(1+I) Sqrt[Pi/2] Exp[-2]] 

Out[37]= -13 -13 

9.02889 10 + 2.14218 10 I 


Instead of using the exact argument functions x , 1=x and 1=x , x to determine the partition points 

we could as well take their polynomial parts x and ,x, respectively, which yields even slightly better 

results. 

In[38]:= OscInt[f1,#&,1] + OscInt[f2,-#&,1] - 

N[(1+I) Sqrt[Pi/2] Exp[-2]] 

Out[38]= -15 -14 

2.52576 10 + 3.71925 10 I 


Z 1 

0 

sin 3 x 

x 2 

In[39]:= f[x_] := Sin[x]^3/x^2; 

OscInt[f,#&,0] - N[3/4 Log[3]] 

Out[39]= -14 

-7.39409 10 


Z 1 

0 

sin x , x cos x 

x 3 

In[40]:= f[x_] := (Sin[x] - x Cos[x])/x^3; 

OscInt[f,#&,0] - N[Pi/4] 

Out[40]= -13 

-2.498 10 


Z 1 

0 

sin x 2 , 1 

x 2 

dx = 3 

ln 3 

4 

dx = 4 

dx = 1 

r 

2 2 e,2 

Dividing the range of integration and with the change of variable x2 ! 1=x2 , we obtain the two 

2 dx 

integrals R 1 

1 sin , x 2 , 1 

x 2 

dx and R 1 

1 sin , 1 

x 2 , x 

188 

x 2 , respectively.


In[41]:= f1[x_] := Sin[x^2 - 1/x^2]; 

f2[x_] := Sin[1/x^2 - x^2]/x^2; 

OscInt[f1,(#^2-1/#^2)&,1] + OscInt[f2,(1/#^2-#^2)&,1] - 

N[Sqrt[Pi/2] Exp[-2]/2] 

Out[41]= -13 

-1.51323 10 

With only the polynomial parts of the argument functions and an increased number of sequence 

members for the extrapolation, we obtain a comparable result. 

In[42]:= OscInt[f1,(#^2)&,1,NSumExtraTerms->20] + 

OscInt[f2,(-#^2)&,1,NSumExtraTerms->20] - 

N[Sqrt[Pi/2] Exp[-2]/2] 

Out[42]= -13 

-1.78635 10 

The last integral is again taken from Sidi [128] and shows a limitation of our routine. 


Z 1 

0 

J0(x) J1(x) dx 

x 

Since the Bessel functions for large x behave like cos x, wechoose the linear function to compute the 

partition points. Still, the result of the quadrature is rather poor and any improvement needs a large 

number of terms to be included in the summation. But even for several thousand terms the accuracy 

is limited to nine digits. 

In[43]:= f[x_] := BesselJ[0,x] BesselJ[1,x]/x; 

OscInt[f,#&,0] - N[2/Pi] 

Out[43]= -7 

-2.31225 10 

We have seen in this section that our quadrature routine is versatile and can be applied 

successfully to a wide variety of integrals with irregularly oscillating behaviour. In some 

cases such asintegration intervals that are in nite at both ends, the integral must be divided 

and transformed before it can be treated with the algorithm. We now can nd a proper 

formulation for the wave integrals to render them amenable to our routine. 

189 

= 2

Chapter 8 

8 Application of the quadrature routine 

Application of the quadrature 

routine 

Newton Verstehen Sie etwas von Elektrizitat, Richard? 

Inspektor Ich bin kein Physiker. 

Newton Ich verstehe auch wenig davon. Ich stelle nur aufgrund von 

Naturbeobachtungen eine Theorie daruber auf. Diese Theorie schreibe ich 

in der Sprache der Mathematik nieder und erhalte mehrere Formeln. Dann 

kommen die Techniker. Sie kummern sich nur noch um die Formeln. Sie 

gehen mit der Elektrizitat um wie der Zuhalter mit der Dirne. Sie nutzen 

sie aus. Sie stellen Maschinen her, und brauchbar ist eine Maschine erst 

dann, wenn sie von der Erkenntnis unabhangig geworden ist, die zu ihrer 

Er ndung fuhrte. Friedrich Durrenmatt, Die Physiker 

Having a quadrature routine at hand that is capable of dealing with irregularly oscillating 

functions, we can now apply it to the wave integrals of the rst part of this work | as an 

example, we choose the tunneling particle. To this end, a little bit of rewriting is necessary 

to permit an e cient computation. In this chapter, we shall develop a Mathematica package 

comprising all functions necessary to compute the wave function inside and outside the tunnel 

as well as providing a simple user interface. Finally, we shall test the implementation both 

numerically and analytically. 

There is a good reason why weselect the tunnel e ect to demonstrate what the quadrature 

routine is capable of: from all physical models we discussed in the rst part, this is by far the 

most intricate one. While we mighthave succeeded with standard algorithms in all other cases, 

this one really calls for our non-standard approach. But not only are the integrals di cult to 

compute, the variety of functions also results in a complex structure of the package. Having 

understood the implementation of this example, one can easily deduce the others. 

190

8.1 Preparation of the wave integrals for quadrature 


We recall from section 4.1 the integrals describing the wave function in the di erent regions 

of a step potential barrier: 

inc = !p 

c 

ref = !p 

c 

+ !p 

c 

+ !p 

c 

tun = !p 

c 

+ !p 

c 

+ !p 

c 

Z 1 

A( ) e 

,1 

jX ,j 2T d ; (8.1) 

Z ,1 

,1 

Z 1 

,1 

Z 1 

1 

Z ,1 

,1 

Z 1 

,1 

Z 1 

1 

A( ) + p 2 , 1 

, p ,jX ,jT 2 

e d + 

2 , 1 

A( ) , jp1 , 2 

+ j p 2 ,jT 

e,jX d + 

1 , 2 

A( ) , p 2 , 1 

+ p ,jX ,jT 2 

e d ; 

2 , 1 

A( ) 

A( ) 

A( ) 

2 

, p e 

2 , 1 ,jX 

p 

2,1,jT 2 

d + 

2 

+ j p 1 , 

2 e,X 

p 1, 2 ,jT 2 

d + 

2 

+ p e 

2 , 1 jX 

p 

2,1,jT 2 

d : 

In section 4.2, we found the spectra of the initial wave forms, 

A rect ( ) 

p l = j 

4 2 k 

A tria ( ) 

p l 

A gauss( ) 

p l 

p 

3 

= 

4 3k2 = 

1 

1 , 1 

p 

1 

1 , 1 

p 

1 

p 2 n p 2 e 

, 

0 

,j2 k 1, p1 

e 

,j k 1, p 

1 

2e 2 

@ 2 k p 1 

,1 

2n 

191 

1 

A 2 

(8.2) 

(8.3) 

, 1 ; (8.4) 

,j k 1, 1 

p 

,j2 k 1, p 

1 

, e 

, 1 ; (8.5) 

: (8.6)


If we take a closer look at the integrands, we recognise that they all have a similar structure. 

The integrands of the wave functions have the general form 

f( )=CrA( )cr( )e jpr( ) ; (8.7) 

with a constant Cr, a coe cient function cr( ) that is de ned by the transmission and reection 

coe cients and a function pr( ) describing the phase of the integrand. All these 

functions depend on the region where the wave function is evaluated and on the direction 

of propagation, respectively. In addition, they might be slightly di erent for positive and 

negative wavenumbers because of the dispersion relations. 

For the step potential barrier, these functions are given below. Note that for the evanescent 

part of the spectrum in the interval ,1 < 1ifwe take the principal value of the complex root. The 

constant Cr = !c=c is independent of both the region and the direction of propagation, so we 

disregard it in the sequel. 

cinc( )=1 (8.8) 

pinc( )=X , T 2 

c ref( )= 

8 

>< 

>: 

p 

+ 2,1 

, p 2 ,1 

, p 2 ,1 

p 

+ 2,1 

p ref( )=,X , T 2 

ctun( )= 

ptun( )= 

8 

>< 

>: 

8 

< 

: 

2 

, p 2 ,1 

2 p 

+ 2,1 

if ,1 

if ,1 

,X p 2 ,1,T 2 if ,1 

(8.9) 

(8.10) 

(8.11) 

(8.12) 

(8.13) 

The spectral functions, too, can be written in a uniform manner as the sum of exponential 

functions, 

A( )=Csss( ) 

mX 

i=1 

as;i e jgs;i( ) ; (8.14) 

where Cs is some waveform-dependent constant and ss is a function that de nes the shape of 

the spectrum. Taking equations (8.7) and (8.14) together, we obtain the complete structure 

of the wave integrands, 

f( )=Csss( )cr( ) 

mX 

i=1 

192 

as;i e j(gs;i( )+pr( )) : (8.15)


Now we are ready to formulate the integrals in an appropriate way for the implementation 

with Mathematica. 

The equations (8.1) to (8.6) still permit no straightforward application of our standard quadrature 

routine. Remembering that we can extrapolate only to +1, we must make a variable 

transformation ! , rst for the integrals over the negative real axis. Having done so, 

we nd that the coe cient functions for the incident, re ected, and transmitted part of the 

wave now become unique for both integrals, whereas the exponential arguments as well as the 

coe cient functions of the wave spectra now are di erent in the two integrals. The coe cient 

functions (8.8), (8.10), and (8.12) thus reduce to their positive branches 

cinc( )=1 

cref( )= ,p 2 ,1 

+ p 2 ,1 

ctun( )= 

2 

+ p : 

2 ,1 

For the triangular wave packet, we can combine (8.5) and (8.1 { 8.3) and nd 

p 

3 

Ctria = 

2k 2p 

s + 

1 

tria ( )= 

1, 1 p 

s , tria ( )= 

1 

1+ 1 

p 

2 

2 

(8.16) 

(8.17) 

(8.18) 

With these de nitions we can put the parts together to eventually obtain the nal form of 

the integrals that is suitable for quadrature: 

inc,tria 

p l = Ctria 

+ Ctria 

+ Ctria 

Z 1 

1 

Z 1 

1 

Z 1 

,1 

s + 

tria ( ) 2ej(,T 2 + k 1 p +X , k) 

, 

, e j(,T 2 +2 k 1 

p +X ,2 k) , e j(,T 2 +X ) d + 

s , tria ( ) 2ej(,T 2 , k 1 

p ,X , k) , 

, e j(,T 2 ,2 k 1 

p ,X ,2 k) , e j(,T 2 ,X ) d + 

s + 

tria ( ) 2ej(,T 2 + k 1 

p +X , k) 

, 

, e j(,T 2 +2 k 1 

p +X ,2 k) , e j(,T 2 +X ) d 

193 

(8.19)

ef,tria 

trans,tria 

evan,tria 

p l = Ctria 

+ Ctria 

+ Ctria 

p l = Ctria 

+ Ctria 

p l = Ctria 

Z 1 

1 

Z 1 

1 

Z 1 

,1 

Z 1 

1 

Z 1 

1 

Z 1 

,1 

s + 

tria ( ) cref( ) 2e j(,T 2 + k 1 

p ,X , k) 

, 


, e j(,T 2 +2 k 1 

p ,X ,2 k) , e j(,T 2 +X ) d + 

s , tria ( ) cref( ) 2e j(,T 2 , k 1 

p +X , k) , 

, e j(,T 2 ,2 k 1 

p +X ,2 k) , e j(,T 2 +X ) d + 

s + 

tria ( ) cref( ) 2e j(,T 2 + k 1 

p ,X , k) 

, 

, e j(,T 2 +2 k 1 

p ,X ,2 k) , e j(,T 2 ,X ) d 

s + 

tria ( ) ctun( ) 2e j(,T 2 + k 1 

p 

p +X 2,1, k) 

, 

, e j(,T 2 +2 k 1 

p +X 

s , tria ( ) ctun( ) 2e j(,T 2 , k 1 

p ,X 

p 2,1,2 k) , e j(,T 2 +X ) d + 

p 2,1, k) , 

, e j(,T 2 ,2 k 1 

p 

p ,X 2,1,2 k) j(,T 

, e 2 ,X ) 

d 

s + 

tria ( ) ctun( ) 2e j(,T 2 + k 1 

p 

p +X 2,1, k) 

, 

, e j(,T 2 +2 k 1 

p +X 

p 2,1,2 k) , e j(,T 2 +X ) d : 

(8.20) 

(8.21) 

(8.22) 

We distinguish four di erent parts of the wave function. Inside the tunnel we have the 

evanescent and the transmitted portion, respectively. Outside the tunnel, we separate the 

incident from the re ected part. For these two parts a division of the integration range into 

three parts | by analogy with the tunnel | has no physical reason. Still we stick to this 

structure because we then need only two modules for all parts of the wave function. 

As the integrands consist of three distinct terms, the inde nite integrals are split in order 

to be amenable to the quadrature function. Therefore, the computation of the transmitted 

portion of the wave needs six individual quadrature calls. Obviously the oscillatory parts of 

the integrands are all very similar. Inside the tunnel, they have the structure 

otun( )=e j(a2 p 

+b +c 2,1+d) 

; (8.23) 

on the outside they look like 

oout( )=e j(a 2 +b +c +d) ; (8.24) 

with only the signs of the parameters changing. This notice is important for the implementation 

since it allows us to compute the coe cients inside the module for all function calls and 

to change only their signs as they are actually passed to the quadrature function. 

194


For rectangular initial waves, we nd from (8.4) and (8.1 { 8.3) 

and 

inc,rect 

ref,rect 

trans,rect 

evan,rect 

p l = Crect 

+ Crect 








Z 1 

1 Z 1 

1 Z 1 

,1 

Z 1 

1 Z 1 

1 Z 1 

,1 

Z 1 

1 

Z 1 

1 

Z 1 

,1 

Crect = 

j 

p 

2 

(8.25) 

s + 

1 

rect ( )= 

1, 1 p 

(8.26) 

s , rect ( )= 

1 

1+ 1 

p 

s + 

h j(,T 

rect ( ) e 2 +2 k 1 p +X ,2 k) j(,T 

, e 2 i 

+X ) 

d + 

s , rect ( ) 

s + 

rect ( ) 

h j(,T 

e 2 ,2 k 1 

p ,X ,2 k) j(,T 

, e 2 i 

,X ) 

d + 

h j(,T 

e 2 +2 k 1 

p +X ,2 k) j(,T 

, e 2 i 

+X ) 

d 

s + 

rect ( ) c h j(,T 

ref( ) e 2 +2 k 1 p ,X ,2 k) j(,T 

, e 2 i 

,X ) 

d + 

h j(,T 

e 2 ,2 k 1 p +X ,2 k) j(,T 

, e 2 i 

+X ) 

d + 

s , rect ( ) c ref( ) 

s + 

rect ( ) c ref( ) 

h j(,T 

e 2 +2 k 1 

p ,X ,2 k) j(,T 

, e 2 i 

,X ) 

d 

s + 

rect ( ) ctun( ) e j(,T 2 +2 k 1 

p 

p +X 2,1,2 k) 

, e j(,T 2 +X ) d + 

s , rect ( ) ctun( ) e j(,T 2 ,2 k 1 

p 

p ,X 2,1,2 k) 

, e j(,T 2 ,X ) d 

s + 

rect ( ) ctun( ) e j(,T 2 +2 k 1 

p 

p +X 2,1,2 k) 

, e j(,T 2 +X ) d : 

(8.27) 

(8.28) 

(8.29) 

(8.30) 

(8.31) 

The basic structure of the integrals is the same as before, however in this case the computation 

of the transmitted wave needs only four calls of the quadrature routine. Like with the 

triangular wave, it is su cient towrite one driver module for all parts of the wave function 

and alter only the parameters in the quadrature call. 

195

Likewise, we collect the de nitions for a Gaussian wave, 

p 

2 k 

Cgauss = p p 

n 2 

s + 

, k 

n 

gauss( )=e 

s , k 

gauss ( )=e, n 

1 

p ,1 

1 , p ,1 

to write the nal form of the integrals 

p Z 1 

l = Cgauss s + 

gauss( ) e j(,T 2 + k 1 p +X , k) 

d + 

inc,gauss 

ref,gauss 

trans,gauss 

evan,gauss 

+ Cgauss 

+ Cgauss 

p l = Cgauss 

+ Cgauss 

+ Cgauss 

p l = Cgauss 

+ Cgauss 

p l = Cgauss 

1 Z 1 

1 Z 1 

,1 

Z 1 

1 Z 1 

1 Z 1 

,1 

Z 1 

1 Z 1 

1 

Z 1 

,1 

s , gauss ( ) ej(,T 2 , k 1 

p ,X , k) d + 

s + 

gauss ( ) ej(,T 2 + k 1 

p +X , k) 

d 


s + gauss ( ) c ref( ) e j(,T 2 + k 1 

p ,X , k) d + 

s , gauss( ) c ref( ) e j(,T 2 , k 1 

p +X , k) d + 

s + gauss( ) c ref( ) e j(,T 2 + k 1 

p ,X , k) d 

2 

2 

(8.32) 

(8.33) 

; (8.34) 

s + gauss ( ) ctun( ) e j(,T 2 + k 1 p 

p 

+X 2,1, k) 

d + 

s , gauss ( ) ctun( ) e j(,T 2 , k 1 p 

p 

,X 2,1, k) 

d 

(8.35) 

(8.36) 

(8.37) 

s + gauss ( ) ctun( ) e j(,T 2 + k 1 

p 

p +X 2,1, k) 

d : (8.38) 

The computation of the inde nite integrals for a Gaussian wave requires only two calls of the 

quadrature routine. Furthermore, in this special case, there is an alternative to the extrapolating 

quadrature strategy: as the integrands decay exponentially, we can safely truncate 

them at appropriate values of the integration variable and integrate the remaining de nite 

integral in a conventional way. A safe value for the truncation is where the shaping functions 

Cgauss sgauss( ) become smaller than the working precision p, for example 

such that the truncation points are 

trunc = p 

Cgauss sgauss 10 ,(p+1) ; (8.39) 

1 

r 

196 

ln Cgauss 

10 ,(p+1) 

n 

k 

! 

(8.40)

8.2 Outline of the program structure 


Since we have to cope with a large variety of similar functions, we must adopt a modular 

approach for the individual subroutines. This means that the integrands are combined from 

several function modules following the aforementioned structure, which is the only possibility 

to ensure consistency within the package and to provide comparatively easy extendability. A 

reasonable implementation strategy involves three distinct hierarchical levels. 

Driver modules provide the basic interface to the user. The most user-friendly of 

them expects nothing more than the parameters required to compute the wave function 

and an option de ning the shape of the initial waveform. Depending on the spatial 

coordinate it then returns either the wave function inside the tunnel or the sum of the 

incident and re ected parts on the outside. These partial solutions are computed in 

turn by other driver modules that can also be accessed directly. All modules must take 

into account that di erent waveforms may need di erent numbers of parameters (such 

as the Gaussian wave packet, which requires an additional geometry parameter for the 

variance of the distribution). 

Quadrature modules are distinct for all waveforms. The reason for this is that the 

number of exponential functions in the sum of equation (8.15) as well as the respective 

coe cients as;i are di erent for each possible waveform. Furthermore, some of 

the in nite integrals can be computed in di erent ways. So these modules are typically 

implemented as waveform-dependent templates that need the more general part 

ss( ) cr( ) of the integrand as input parameter. The they add the speci c functions 

P m 

i=1 as;i e j(gs;i( )+pr( )) and call the actual quadrature routine with the appropriate 

options. 

The only exception to this scheme is the computation of the evanescent part of the 

integrals. As this involves only a single de nite integral, there is no point in dividing the 

integrands like we do for the in nite integrals in order to meet the formal requirements 

of our quadrature strategy. Hence there is only one template for this task, and the 

whole integrand function is passed to this routine from the calling module. Because all 

these routines are highly specialised, they are not made available to the user. 

Function modules are the actual building blocks of the integrands. There are atomic 

functions describing the shape- and region-dependent coe cients ss( ), cr( )aswell as 

the constants Cs. These elements are then combined to form the integrands for the two 

inde nite transmissive parts and the evanescent section of the integration range. These 

functions, too, are internal to the package and cannot be accessed by the user. 

For the triangular and Gaussian wave packet, the gradient of the wave function inside the 

tunnel is also of interest for the computation of the energy density and nally the energy 

velocity. These functions have not been included in the general framework, in that they 

cannot be accessed from the same driver routine as the wave functions themselves. Although 

the gradient functions share the quadrature modules of the plain wave computation, they use 

separate driver and, of course, function modules. 

197


Knowing that we will end up with a plethora of very similar functions, we establish a consistent 

naming convention to avoid confusion and to reduce the risk of programming errors. All 

function names are thus composed of distinct elements to distinguish them from each other. 

A severe obstacle to the legibility of the names is the fact that Mathematica permits no 

underscores within a function's name because the underscore is reserved to designate patterns. 

So we have no possibility toseparate the tokens that make up a function name as in other 

programming languages. 

We start with the de nition of a number of tokens, each of which describes a part of the 

purpose the function is intended for. 

Form 2 fRect, Tria, Gaussg denotes the shape of the initial wavepacket and applies 

to all waveform-dependent functions. 

Region 2fInc, Ref, Out, Trans, Evan, Tung distinguishes the regions a function may 

be applicable to. Some modules depend on the direction of the wave propagation or 

whether the evanescent or the transmitted part of the integral is regarded. Others 

simply are di erent inside and outside the barrier. 

FunctionType 2 fConst, Shape, Oscg denotes constants, shaping functions for the 

spectra of the waves, and the oscillatory factors of the integrands. 

Polarity 2 fNULL, Pos, Negg is used to distinguish integrands de ned for either the 

positive or negative real axis. The empty element NULL applies to all functions that 

need no distinction between the two cases. 

Gradient 2fNULL, Gradg signi es whether the function is used for the computation of 

the wave function or its gradient. In the latter case, the name includes the additional 

token Grad. 

These tokens are used to compose the various function modules. Which tokens are actually 

combined and what values are meaningful depends on the class of modules under consideration. 

Region-dependent factors have the form kRegionjFunctionTypek. For example, 

RefCoeff is the coe cient function for the re ected wave, TunOsc means the oscillatory 

factor of the integrands in the tunnel. 

Integrands and waveform-dependent factors have the more complicated structure 

kFormjGradientjfFunctionType, RegiongjPolarityk. Examples are GaussConst for a 

constant factor, RectShapeNeg for a factor of an integrand and nally TriaGradEvan 

or GaussRefNeg for the entire integrands. 

Quadrature templates are rather heterogenous and have the naming structure kfNULL, 

FormgjfEvan, TransgjfNULL, Exact, TruncgjTempk, as for instance TriaTransExactTemp 

or GaussEvanTruncTemp. The elements Exact and Trunc refer to alternative computing 

strategies for the non-Gaussian and Gaussian wave packets, respectively. 

198


Driver modules are named according to the scheme kPhijfNULL, Inc, Ref, Trans, 

Evangk, depending on whether the function depends on the region and the propagation 

direction or not. 

Two options, Shape and PPoints, are provided to control the execution of the user accessible 

modules. The allowed values of the latter depend on the waveform. 

Shape 2fRect, Tria, Gaussg de nes the shape of the wave packet under consideration. 

The default value is Tria. 

PPoints 2 fApproximate, Zeros, Truncateg selects the way the integrals are evaluated. 

Which one is applicable depends on the waveform. For triangular and rectangular 

waves inside the tunnel, where the exponential functions have no polynomial arguments, 

the partition points can be determined either using a polynomial approximation or by 

computing the zeros of the integrand numerically. Outside the tunnel, the partition 

points are exactly the zeros, anyhow, and this option is irrelevant. 

For the Gaussian wave packet, on the contrary, it is possible not to use extrapolation 

but to compute the integrals directly by truncating them at reasonably large abscissa 

values. This is possible both inside and outside the tunnel and also for the evanescent 

part of the wave. For the other shapes, the option is ignored for the evaluation of the 

de nite integrals. The default value of this option is Approximate. 

Beside these two options all other options of the quadrature function OscInt can be used. 

They take, however, no in uence on the evaluation of the modules in the package but are 

simply passed to the quadrature routine. 

The list of arguments that have to be passed to the driver module comprises the spatial and 

time coordinate X and T , the ratio between cut-o frequency and the signal frequency 

and the number of wavelengths k that the initial wave packet spans. For Gaussian waves, 

the geometry parameter n is needed, otherwise an error message will be generated. For all 

other shapes, this additional parameter is not necessary and will be ignored. Following these 

arguments, options may be given, like in the examples below. 

Phi[x,t,w,k,Shape->Rect] 

PhiInc[x,t,w,k,n,Shape->Gauss,PPoints->Truncate] 

PhiRef[x,t,w,k,Shape->Tria,NSumTerms->30] 

PhiEvan[x,t,w,k,Shape->Rect,WorkingPrecision->25] 

PhiTrans[x,t,w,k,Shape->Tria,PPoints->Zeros] 

PhiGrad[x,t,w,k,n,Shape->Gauss] 

PhiGradTrans[x,t,w,k,Shape->Tria] 

PhiGradEvan[x,t,w,k,Shape->Tria] 

These examples are a comprehensive list of all available functions of the package. 

199

8.3 Implementation of the quadrature modules 


Now that we have formulated all integrals in a way that suits our needs, we can write the 

quadrature modules. Since they are all based on the same structure, we shall discuss here 

only the example of the rectangular wave packet and a few functions that are related to it. All 

other modules are very similar in their structure and can be understood without di culties 

from the listing of the package in the appendix. 

The atomic function modules we use to de ne the integrands are the transmission and reection 

coe cients and the oscillatory factors of the integrands within (TunOsc) and outside 

(OutOsc) the tunnel. In addition, we need the argument function of the oscillation inside 

the tunnel for the optional exact determination of the integrand's zeros. These functions are 

independent of the waveform. 

TunCoeff[xi_] := 2 xi/(xi + Sqrt[xi^2-1]); 

RefCoeff[xi_] := (xi - Sqrt[xi^2-1])/(xi + Sqrt[xi^2-1]); 

TunOsc[a_,b_,c_,d_,xi_] := Exp[I (a xi^2 + b xi + c Sqrt[xi^2 - 1] + d)]; 

OutOsc[a_,b_,c_,d_,xi_] := Exp[I (a xi^2 + (b+c) xi + d)]; 

PhiArg[a_,b_,c_,d_][xi_] := a xi^2 + b xi + c Sqrt[xi^2 - 1] + d; 

Other basic functions like those describing the shape of the spectrum are di erent for each 

waveform. Note that in the implementation we use the variable w for the parameter . 

RectConst[w_,k_] := I / (Sqrt[w] 2 Pi); 

RectShapePos[w_,xi_] := 1/(1-xi/Sqrt[w]); 

RectShapeNeg[w_,xi_] := 1/(1+xi/Sqrt[w]); 

With these building blocks we de ne the actual integrands that are to be used inside the 

quadrature routines. As we saw while formulating the respective equations, we need two 

distinct functions for the positive and negative inde nite integrals. The integrand for the 

evanescent region is augmented by the coe cients and oscillatory factors that stem from the 

shape of the spectrum. Therefore we can compute the integral with only one single quadrature 

call. Owing to the particular structure of the oscillatory factors, we need to pass only values 

for the parameters a and b to the quadrature routine and must set b and d to zero in this 

case. 

RectTransPos[a_,b_,c_,d_,w_,k_][xi_] := 

TunCoeff[xi] * RectShapePos[w,xi] * TunOsc[a,b,c,d,xi]; 

RectTransNeg[a_,b_,c_,d_,w_,k_][xi_] := 

TunCoeff[xi] * RectShapeNeg[w,xi] * TunOsc[a,b,c,d,xi]; 

RectEvan[a_,b_,c_,d_,w_,k_][xi_] := 

RectTransPos[a,b,c,d,w,k][xi] * 

(-1 + Exp[I 2 (Pi k/Sqrt[w] xi - Pi k)]); 

The constant RectConst is not part of the integrand because it is useless to compute it 

with each function evaluation. Instead, the constant is multiplied to the result of the entire 

200


quadrature. There are of course other function declarations necessary for the other regions 

(incident and re ected waves). However, they look very much the same as the mentionend 

integrands and thus are not given here. 

The quadrature modules consist of two parts, the computation of the parameters required for 

the function calls and the quadrature itself. Note that the parameters b and d are evaluated 

numerically. In the case of b this is necessary to ensure the correct operation of the functions 

QuadOffset and QuadZero that comprise conditional statements and are not prepared to 

take symbolical arguments. On the other hand, it improves the performance of the module 

because it reduces the computation time by up to ten percent (depending on the actual 

parameter values). For the same reason, the o set parameter of the function enumerating the 

zeros is calculated before the quadrature starts. To prevent inaccuracies due to the numerical 

evaluation of the parameters, they are calculated to a precision two digits higher than the 

WorkingPrecision. 

Remark (Version peculiarities) In the original version of the package the o set values 

were not computed numerically as shown in the example given below. The function 

QuadOffset thus returned results like Ceiling[const/Pi] that were left unevaluated 

because Mathematica treats Pi as a symbol rather than as a number. This worked 

well most of the time, however, for particular sets of input parameters the quadrature 

function would never terminate. This peculiar behaviour seemingly depended on the 

software version (2.2) but not on the platform, which was veri ed on PCs as well as on 

SUN and HP workstations. Neither version 2.1 nor the new version 3.0 caused similar 

troubles. The problem could be overcome only by changing the results of QuadOffset 

into numbers, which in turn provides another slight performance improvement. 

The quadrature function OscInt is called four times with the arguments of the integrands set 

to the appropriate values taken from (8.30). In this connection it is important to notice that 

owing to the uniform structure of the integrand functions we need to pass only the name of the 

function to the template, the arguments are then added inside the module. Therefore we can 

use the template for all parts of the wave. The results of the individual function calls are then 

summed up with the coe cients given in (8.30) and multiplied with the waveform-dependent 

constant. 

RectTransTemp[fpos_, fneg_, x_, t_, w_, k_, opts___Rule] := 

Module[fa,b,c,d,o11,o12,o21,o22, 

wp = WorkingPrecision/.foptsg/.Options[OscInt]g, 

a = -t; 

b = N[2 Pi k/Sqrt[w],wp+2]; 

c = x; 

d = N[-2 Pi k,wp+2]; 

o11 = N[QuadOffset[a, c,0],wp+2]; 

o12 = N[QuadOffset[a,-c,0],wp+2]; 

o21 = N[QuadOffset[a, b+c,d],wp+2]; 

o22 = N[QuadOffset[a,-b-c,d],wp+2]; 

(-(OscInt[fpos[a, 0, c,0,w,k], 

QuadZero[a, c,0,o11],1,FunctionType->ZeroList,opts] + 

201


OscInt[fneg[a, 0,-c,0,w,k], 

QuadZero[a, -c,0,o12],1,FunctionType->ZeroList,opts]) + 

(OscInt[fpos[a, b, c,d,w,k], 

QuadZero[a, b +c,d,o21],1,FunctionType->ZeroList,opts] + 

OscInt[fneg[a,-b,-c,d,w,k], 

QuadZero[a,-b -c,d,o22],1,FunctionType->ZeroList,opts]) 

) RectConst[w,k] 

]; 

The second possibility to compute this integral is not to use the approximate partitioning 

strategy but employ the exact zeros for this purpose. Since we do not need this module for 

any other function than the one inside the tunnel, we need not write it as a template. So we 

pass only the parameters of the wave to the module but keep the names of the integrands 

xed. This time we call the quadrature routine without a value for the option FunctionType 

because it defaults to Argument which is exactly what we want in order to compute the zeros 

numerically. The rest of the module is the same as the previously discussed version. 

RectTransExact[x_, t_, w_, k_, opts___Rule] := 

Module[fa,b,c,d, 

wp = WorkingPrecision/.foptsg/.Options[OscInt]g, 

a = -t; 


c = x; 

d = N[-2 Pi k,wp+2]; 

(-(OscInt[RectTransPos[a, 0, c,0,w,k], 

PhiArg[a, 0, c,0],1,opts] + 

OscInt[RectTransNeg[a, 0,-c,0,w,k], 

PhiArg[a, 0,-c,0],1,opts]) + 

(OscInt[RectTransPos[a, b, c,d,w,k], 

PhiArg[a, b, c,d],1,opts] + 

OscInt[RectTransNeg[a,-b,-c,d,w,k], 

PhiArg[a,-b,-c,d],1,opts]) 


]; 

The template for the calculation of the evanescent part of the wave is di erent. First, we 

do not split the integrand and thus have only one call of the integration routine. Second, 

as we use the built-in Mathematica function NIntegrate, we must set a number of options 

according to the values given by the user. As method for the evaluation of this de nite integral 

we choose again DoubleExponential. A third di erence is that this module can be used for 

all waveforms and therefore the waveform-dependent constant must be passed to the module 

as an argument. 

EvanTemp[f_, const_, opts___Rule] := 

Module[fwp = WorkingPrecision/.foptsg/.Options[OscInt], 

ag = AccuracyGoal/. foptsg/.Options[OscInt], 

202


pg = PrecisionGoal/. foptsg/.Options[OscInt], 

mi = MinRecursion/. foptsg/.Options[OscInt], 

ma = MaxRecursion/. foptsg/.Options[OscInt]g, 

NIntegrate[f[xi],fxi,-1,1g, 

Method->DoubleExponential, 


AccuracyGoal->ag, 

PrecisionGoal->pg, 

MinRecursion->mi, 

MaxRecursion->ma] const 

]; 

The functions discussed so far are internal only and not visible to the user | in contrast 

to the driver functions. These are used to provide a consistent user interface that is easy to 

control. For the transmitted part of the wave function in the tunnel, the module PhiTrans, 

for example, looks at the user speci ed values of the options Shape and PPoints or their 

default values, respectively, to call the appropriate quadrature module. Furthermore, warning 

and error messages are generated if an option has an unkown or inappropriate value. The 

argument n is a geometrical parameter that applies only to the Gaussian waveform. Thus 

we must provide an alternative way to call the module where n is not speci ed. This is the 

second rule given for the function name, which is applicable for rectangular and triangular 

waves, but issues an error when called with Shape->Gauss. In the other cases, it calls the 

main function with n set to the dummy value one. The message bodies are declared in the 

package header. 

PhiTrans[x_, t_, w_, k_, n_?Positive, opts___Rule] := 

Module[fsh = Shape/. foptsg/.Options[Phi], 

pt = PPoints/.foptsg/.Options[Phi]g, 

Switch[sh, 

Rect, 

Switch[pt, 

Approximate, RectTransTemp[RectTransPos, 

RectTransNeg,x,t,w,k,opts], 

Zeros, RectTransExact[x,t,w,k,opts], 

_, Message[Phi::invalidpart,pt]], 

Tria, 

Switch[pt, 

Approximate, TriaTransTemp[TriaTransPos, 

TriaTransNeg,x,t,w,k,opts], 

Zeros, TriaTransExactTemp[TriaTransPos, 

TriaTransNeg,x,t,w,k,opts], 


Gauss, 

Switch[pt, 

Approximate, GaussTransTemp[GaussTransPos, 

GaussTransNeg,x,t,w,k,n,opts], 

Truncate, GaussTransTruncTemp[GaussTransPos, 

203

] 

]; 


GaussTransNeg,x,t,w,k,n,opts], 


_, 

Message[Phi::invalidshape,sh] 

PhiTrans[x_, t_, w_, k_, opts___Rule] := 

If[(Shape/.foptsg/.Options[Phi]) === Gauss, 

Message[Phi::missingval], 

PhiTrans[x,t,w,k,1,opts]]; 

Remark (Input checking) Apart from checking the option values, the function provides 

other tests of the user's input, too. The parameter n, for example must be greater than 

zero, which is expressed with the conditional pattern matching statement n ?Positive. 

Therefore the function evaluates only if n satis es this requirement. Otherwise no error 

message is generated, but the function is left unevaluated. The same is done with the 

additional arguments that match the pattern opts Rule only if they have the syntax of 

a rule, that is if they look like lhs->rhs. However, this does not encompass a validity 

check for the option names or their values. 

The modules PhiTrans and PhiEvan compute the wave function inside the barrier, PhiInc and 

PhiRef return the incident and re ected wave outside the barrier. A still more comfortable 

driver is the function Phi that decides (based upon the spatial coordinate given by the user) 

which module to call. For the boundary case X =0it takes the functions that are de ned 

for the interior of the barrier (which must, of course, yield the same results as those for the 

outside). This function also traps the somewhat inconvenient case X = T = 0 for rectangular 

waves and sets the return value to the exactly known result 1=2. 

Phi[x_, t_, w_, k_, n_?Positive, opts___Rule] := 

Which[t == 0 && x == 0 && Shape/.foptsg/.Options[Phi] === Rect, 0.5, 

t < 0, 0, 

x >= 0, PhiEvan[x,t,w,k,n,opts] + PhiTrans[x,t,w,k,n,opts], 

x < 0, PhiInc[x,t,w,k,n,opts] + PhiRef[x,t,w,k,n]]; 

Phi[x_, t_, w_, k_, opts___Rule] := 

If[(Shape/.foptsg/.Options[Phi]) === Gauss, 


Phi[x,t,w,k,1,opts]]; 

All the driver functions dealing with the wave function inside the barrier also exist in a version 

that computes the gradient of the wave function, namely PhiGradTrans, PhiGradEvan, and 

PhiGrad. They are implemented only for triangular and Gaussian waves. For a rectangular 

shape, the integral would diverge for the boundary and for T =0. 

204

8.4 Test of the package 


We now have to verify that the functions in our package are fault free and yield the desired 

results. There are three possibilities to do so. First we can check that the initial values are 

correct, namely that the inital wave impinging on the barrier is indeed rectangular, triangular, 

or of Gaussian shape outside the barrier and zero inside the tunnel at time zero: 

(X; T ) T =0 = 

( 0 

if X 0 

0 if X > 0 : 

(8.41) 

Second, the boundary conditions must be satis ed and the wave function as well as its space 

derivatives must be continuous at the edge of the tunnel, 

lim 

X!,0 

(X; T )= lim 

X!+0 

(X; T) = (X; T)jX=0 ; 8T (8.42) 

@ (X; T ) @ (X; T) 

lim 

= lim 

= 

X!,0 @X X!+0 @X 

@ (X; T) 

@X jX=0 ; 8T (8.43) 

The third test is merely to verify the di erent implementations of the inde nite integration, 

because the results for all possible values of the option PPoints must be identical within the 

numerical precision. 

8.4.1 Numerical tests 

To verify the boundary conditions, we chose a wave packet spanning ten wavelengths with 

=0:1, set the time coordinate to zero and compute a list of points along the spatial axis 

to achieve a reasonable resolution. We begin with a rectangular and triangular wave. Note 

the use of the option Shape to set the appropriate waveform. 

Remark (Extracting real parts) Since the wave function is complex-valued, we plot 

only its real part. This is done with the rule fx ,y g:>fx,Re[y]g that replaces the 

second element of each point (which is essentially the wave function for the respective 

x-coordinate) with its real part. 

Example 8.4.1 

In[1]:= t = 0; 

w = 0.1; 

k = 10; 

initialrect = Table[fx,Phi[x,t,w,k,Shape->Rect]g,fx,-230,10,1.25g]; 

ListPlot[initialrect/.fx_,y_g:>fx,Re[y]g,PlotJoined->True]; 

205

-200 -150 -100 -50 

Example 8.4.2 

1 

0.5 

-0.5 

-1 


In[2]:= initialtria = Table[fx,Phi[x,t,w,k,Shape->Tria]g, 

fx,-230,10,1.25g]; 

ListPlot[initialtria/.fx_,y_g:>fx,Re[y]g,PlotJoined->True]; 

-200 -150 -100 -50 

1.5 

1 

0.5 

-0.5 

-1 

-1.5 

For the Gaussian wave, we need the additional parameter describing the variance of the 

probability distribution. We set it to one third of the distance between the maximum and 

the edge of the barrier. 

Example 8.4.3 

In[3]:= n = 6; 

initialgauss = Table[fx,Phi[x,t,w,k,n,Shape->Gauss, 

PPoints->Truncate]g,fx,-230,10,1.25g]; 

ListPlot[initialgauss/.fx_,y_g:>fx,Re[y]g,PlotJoined->True, 

PlotRange->All]; 

206


-200 -150 -100 -50 

2 

1 

-1 

-2 

The examples show that the initial conditions are met. In order to test if the boundary 

conditions are satis ed, we sum up the parts of the wave function on either side of the barrier 

edge and compute their di erence for a number of points along the time axis. We then plot 

the absolute value j (,0;T), (+0;T)j of this di erence. Let us start with the triangular 

wave. 

Example 8.4.4 

In[4]:= x = 0; 

w = 0.2; 

k = 5; 

boundary = Table[ft,PhiInc[x,t,w,k,Shape->Tria]+ 

PhiRef[x,t,w,k,Shape->Tria]- 

(PhiTrans[x,t,w,k,Shape->Tria]+ 

PhiEvan[x,t,w,k,Shape->Tria])g,ft,0,20g]; 

ListPlot[boundary/.fx_,y_g:>fx,Abs[y]g,PlotJoined->True, 


-7 

6. 10 

-7 

5. 10 

-7 

4. 10 

-7 

3. 10 

-7 

2. 10 

-7 

1. 10 

5 10 15 20 

207


Things are di erent with the rectangular wave packet, which is to be treated next. Examining 

this case, we see that the di erence at the origin is extremely high. This is no surprise because 

at this point wemust not split the integrand like we did in (4.22). The reason is that one of 

the resulting integrals does not converge, and this is why we ought to exclude the origin from 

the computation (which in fact has been done for the more general driver function Phi). 

Example 8.4.5 

In[5]:= boundary = Table[ft,PhiInc[x,t,w,k,Shape->Rect]+ 

PhiRef[x,t,w,k,Shape->Rect]- 

(PhiTrans[x,t,w,k,Shape->Rect]+ 

PhiEvan[x,t,w,k,Shape->Rect])g,ft,0,20g]; 



0.02 

0.015 

0.01 

0.005 

5 10 15 20 

The best congruence is found for the Gaussian wave, where we have virtually no di erence at 

all (this is why Mathematica chooses a wide plot range on the y-axis). 

Example 8.4.6 

In[6]:= n = 6; 

boundary = Table[ft,PhiInc[x,t,w,k,n,Shape->Gauss]+ 

PhiRef[x,t,w,k,n,Shape->Gauss]- 

(PhiTrans[x,t,w,k,n,Shape->Gauss]+ 

PhiEvan[x,t,w,k,n,Shape->Gauss])g,ft,0,20g]; 



208


0.4 

0.2 

-0.2 

-0.4 

5 10 15 20 

As for the di erent implementations of the evaluation of the inde nite integrals, we compare 

the results they yield as well as their computing time. Not surprisingly, we nd that it takes 

longer to use the zeros as partition points and that for Gaussian waves the truncation of the 

integrals is much faster. 

Example 8.4.7 

In[7]:= x = 1; 

t = 4; 

w = 0.1; 

k = 2; 

Timing[N[Phi[x,t,w,k,Shape->Tria]]] 

Out[7]= f21.696 Second, 0.0717104 - 0.0469158 Ig 

In[8]:= Timing[N[Phi[x,t,w,k,Shape->Tria,PPoints->Zeros]]] 

Out[8]= f29.056 Second, 0.0717104 - 0.0469158 Ig 

In[9]:= Timing[N[Phi[x,t,w,k,Shape->Rect]]] 

Out[9]= f18.785 Second, 0.0188638 - 0.264976 Ig 

In[10]:= Timing[N[Phi[x,t,w,k,Shape->Rect,PPoints->Zeros]]] 

Out[10]= f22.903 Second, 0.0188638 - 0.264976 Ig 

In[11]:= n = 6; 

Timing[N[Phi[x,t,w,k,n,Shape->Gauss]]] 

209

Out[11]= f19.773 Second, -0.00034339 + 0.00140152 Ig 

In[12]:= Timing[N[Phi[x,t,w,k,n,Shape->Gauss,PPoints->Truncate]]] 

Out[12]= f4.998 Second, -0.00034339 + 0.00140152 Ig 

Example 8.4.8 

In[13]:= x = 1; 

t = 4; 

w = 0.1; 

k = 20; 

Timing[N[Phi[x,t,w,k,Shape->Tria]]] 

Out[13]= f284.017 Second, 0.00688579 - 0.00506005 Ig 

In[14]:= Timing[N[Phi[x,t,w,k,Shape->Tria,PPoints->Zeros]]] 

Out[14]= f289.894 Second, 0.00688579 - 0.00506005 Ig 

In[15]:= Timing[N[Phi[x,t,w,k,Shape->Rect]]] 

Out[15]= f218.381 Second, -0.00471446 - 0.282477 Ig 

In[16]:= Timing[N[Phi[x,t,w,k,Shape->Rect,PPoints->Zeros]]] 

Out[16]= f222.116 Second, -0.00471446 - 0.282477 Ig 

In[17]:= n = 6; 

Timing[N[Phi[x,t,w,k,n,Shape->Gauss]]] 

Out[17]= f138.467 Second, 0.0000199533 - 0.0000757562 Ig 

In[18]:= Timing[N[Phi[x,t,w,k,n,Shape->Gauss,PPoints->Truncate]]] 

Out[18]= f1.373 Second, 0.0000199533 - 0.0000757562 Ig 

210 

8 Application of the quadrature routine


8.4.2 Formal veri cation 

Apart from these numerical examples, there is also a rather formal way to check the implementation. 

Since the wave function is in principle a Fourier integral over the space of wave 

numbers , each of the partial waves that make up the whole solution is itself a solution of 

Schrodinger's equation. In addition, the partial waves on both sides of the tunnel for a given 

wave number must satisfy the boundary conditions. Thus we can use the integrands instead 

of the entire integrals for veri cation. Although the conditions must be met irrespective of the 

wave number, we must, of course, keep in mind that we haveintroduced di erent de nitions 

for di erent ranges of . 

To perform the checks we must rst declare the function modules from which the integrands 

are composed because they are not visible from outside the Mathematica package. As this is 

quite a lengthy input, we only quote a few de nitions. 

In[19]:= (*-- Functions independent of the waveform --*) 


TunOsc[a_,b_,c_,d_,xi_] := 

Exp[I (a xi^2 + b xi + c Sqrt[xi^2 - 1] + d)]; 



(*-- Triangular Shape inside the tunnel --*) 

TriaConst[w_,k_] := Sqrt[3/w] / (2 k Pi^2); 

TriaShapePos[w_,xi_] := 1/(1-xi/Sqrt[w])^2; 

In[20]:= TriaShapeNeg[w_,xi_] := 1/(1+xi/Sqrt[w])^2; 

TriaTransPos[a_,b_,c_,d_,w_,k_][xi_] := 

TunCoeff[xi] * TriaShapePos[w,xi] * TunOsc[a,b,c,d,xi]; 

TriaTransNeg[a_,b_,c_,d_,w_,k_][xi_] := 

TunCoeff[xi] * TriaShapeNeg[w,xi] * TunOsc[a,b,c,d,xi]; 

TriaEvan[a_,b_,c_,d_,w_,k_][xi_] := 

TriaTransPos[a,b,c,d,w,k][xi] * 

(-1 + 2 Exp[I (Pi k/Sqrt[w] xi - Pi k)] - 

Exp[2 I (Pi k/Sqrt[w] xi - Pi k)]); 

(*-- and so on for the rest of the shapes --*) 

To start with the veri cation of the boundary conditions, we rst check that the waves 

inside and outside the tunnel have the same value and the same derivative at the edge. The 

arguments a to d are the coe cients of the oscillatory factor. According to, for instance, (8.30) 

we know that a = ,t and c = x for waves propagating in positive or negative directions, 

211


respectively. The parameters b and d depend on the shape of the initial waveform and are 

irrelevant as long as they are constant for all terms. 

Example 8.4.9 Considering the evanescent parts of a rectangular wave packet, we form the di erence 

of the waves outside and inside the barrier, replace the spatial coordinate by its boundary value 

and simplify the result. As expected, we obtain zero independent of the parameters. 

Remark (Clearing de nitions) To avoid interference with the de nitions made for 

the numerical examples, we must rst clear the values we have assigned to x and t, for 

example by using the command Remove[x,t], prior to executing the symbolic operations 

in the sequel. 

In[21]:= (RectIncEvan[-t,b,x,d,w,k][xi] + RectRefEvan[-t,b,-x,d,w,k][xi] - 

RectEvan[-t,b,x,d,w,k][xi]) /.x->0 //Together 

Out[21]= 0 

We do the same with the derivatives at the edge of the barrier. 

In[22]:= (D[RectIncEvan[-t,b,x,d,w,k][xi] + RectRefEvan[-t,b,-x,d,w,k][xi] - 

RectEvan[-t,b,x,d,w,k][xi], x]) /.x->0 //Together 

Out[22]= 0 

We can carry out these tests for all parts of the integrands and for all di erent wavefoms to 

nd that the de nitions of the integrands meet the boundary conditions. 

To see whether the integrands really are solutions of Schrodinger's equation, we rst write 

the di erential equation in our normalised variables, which is 

, @2 @ 

, j 

@X2 @T 

+ V (X) =0 V = 

( 0 if X < 0 

1 if X 0 

: (8.44) 

For convenience, we de ne a di erential operator that we can apply to the integrands. The 

potential V must be passed as an argument and depends on the region where the di erential 

equation is evaluated. 

In[23]:= Schroedinger[V_] := (-D[#,fx,2g] + V # - I D[#,t])& 

Example 8.4.10 Wecheck if the transmitted part of the rectangular wave for positivewavenumbers 

satis es the di erential equation inside the tunnel. Note that the potential is set to one. Collecting 

all the terms of the result together we nd the expected answer. 

In[24]:= Schroedinger[1][RectTransPos[-t,b,x,d,w,k][xi]] //Together 

212


Out[24]= 0 

Example 8.4.11 Outside the barrier we must combine the incident and re ected waves to obtain 

the desired result. Here the potential vanishes of course. 

In[25]:= Schroedinger[0][TriaIncNeg[-t,b,x,d,w,k][xi] + 

TriaRefNeg[-t,b,-x,d,w,k][xi]] //Together 

Out[25]= 0 

Performing these tests on all di erent functions, we see that they satisfy the requirements. 

However, with this formal veri cation we can only ensure the correct implementation of the 

integrand functions. We cannot check if those parts of the integrals that comprise several 

distinct terms (i. e. all transmitted portions of the waves) are put together without error. Nor 

can we prove the correctness of the quadrature modules directly. This must be done with 

numerical experiments like the ones above. But taking together the results of both formal 

and numerical veri cation, we conclude that the implementation is right. 

213

Appendix A 

Mathematica packages 

A Mathematica packages 

The appendix contains source code listings of the most important Mathematica packages that 

were needed to carry out the numerical evaluation of the wave integrals in the rst part of 

this work. The last section describes a number of utilities that made life easier with the 

three-dimensional pictures of propagating waves. 

A.1 Numerical quadrature 

The package OscInt was written to compute semi-in nite integrals of strongly oscillating 

integrands. The user interface and implementation are described in detail in chapter 7. 

(* Copyright: Copyright 1997, Institute of Computertechnology, *) 

(* Vienna University of Technology *) 

(*:Version: Mathematica 2.2.3 *) 

(*:Title: OscInt *) 

(*:Revision: 1.2 *) 

(*:Author: Thilo Sauter *) 

(*:Keywords: Semi-infinite Quadrature, Very Oscillating Integrands, 

Series Acceleration *) 

(*:Requirements: None. *) 

(*:Warnings: None so far *) 

(*:Summary: This package provides a set of functions for the numerical 

quadrature of univariate oscillating integrands over the 

semi-infinte range {a,Infinity}. The oscillations of the 

integrand may be irregular and need not show an ultimately 

constant period. The integrand must be of the form 

f(x) u(g(x)) with u(x) being Sin[x], Cos[x] or - more 

generally - Exp[I x]. The argument function g(x) must have 

an asymptotic expansion at infinity. 

*) 

(*:History: 27-12-1996 ZerosInBetween extended to cope with special cases 

25-01-1997 PartInt, PartitionTable added, OscInt rewritten 

24-03-1997 OscIntControlled, PartitionOffs, PartitionPoints 

added to allow accuracy control 

25-03-1997 Check of the limit of zerof[x] added to functions 

PartitionTable and PartitionPoints to detect 

argument functions with finite limit (and thus 

finite set of zeros) 

*) 

BeginPackage["OscInt`"] 

PolynomialDegree::usage = 

"PolynomialDegree[f,x] returns the degree of the polynomial 

asymptotic expansion of f regarding the variable x at Infinity. If f 

is a pure function or is defined as f[...][x] then x need not be 

specified." 

214


AsymptoticExpand::usage = 

"AsymptoticExpand[f,x] returns the polynomial part of the 

asymptotic expansion of f regarding the variable x at Infinity. If f 

is a pure function or is defined as f[...][x] then x need not be 

specified." 

ApproxLimGeneric::usage = 

"ApproxLimGeneric[f,goal] is used to give a figure of merit for the quality 

of a polynomial approximation to an argument f of a circular function. It returns 

the abscissa value where the ratio of the circular velocity of the approximation 

polynomial and the approximation error, respectively, reaches a given goal (the 

larger the better). This is equivalent to the number of zeros the approximating 

function has within one half period of the frequency defined by the difference 

between f and its approximation." 

ApproxLimQuad::usage = 

"ApproxLimHyp[a,b,c,goal] is a special case of ApproxLimGeneric for the 

function a x^2 + b Sqrt[x^2 + c] + d and its approximation a x^2 + b x + d." 

ApproxLimLinear::usage = 

"ApproxLimHyp[a,b,c,goal] is a special case of ApproxLimGeneric for the 

function a x + b Sqrt[x^2 + c] + d and its approximation (a + b) x + d." 

QuadZero::usage = 

"QuadZero[a,b,c,offs][k] returns the k-th root of the quadratic equation 

a x^2 + b x + c == (k+offs)*Pi for ascending x (the branch to the right of the 

extremum). The parameter offs has to be determined such that the 0-th solution is 

already larger than the abscissa value of the extremum (see QuadOffset)." 

QuadOffset::usage = 

"QuadOffset[a,b,c] returns the offset value used in QuadZero, which is 

essentially the ordinate value of the extremum divided by Pi." 

HypZero::usage = 

"HypZero[a,b,c,d,offs][k] returns the k-th root of the quadratic equation 

(a+b) x + d == (k+offs)*Pi, which is the polynomial approximation of 

a x + b Sqrt[x^2 + c] + d at Infinity for ascending x (the branch to the right of the 


already larger than the abscissa value of the extremum (see HypOffset)." 

HypOffset::usage = 

"HypOffset[a,b,c,d] returns the offset value used in HypZero, which is 

essentially the ordinate value of the extremum or bending point divided by Pi." 

HypZeroExact::usage = 

"HypZeroExact[a,b,c,d,offs][k] returns the k-th root of the equation 

a x + b Sqrt[x^2 + c] + d == (k+offs)*Pi for ascending x (the branch to the right of the 


already larger than the abscissa value of the extremum (see HypOffsetExact)." 

HypOffsetExact::usage = 

"HypOffsetExact[a,b,c,d] returns the offset value used in HypZeroExact, which is in 

principle the ordinate value of the extremum or bending point divided by Pi." 

ZerosInBetween::usage = 

"ZerosInBetween[f,a,b] returns the number of solutions of the function 

g[x] == k*Pi for x in (a,b). The function f[k] is required to give the k-th 

solution of g[x] == k*Pi. If f[0] lies in the interval (a,b) only the part 

to the right of f[0] is regarded. With this function, the parameter NSumTerms 

of NSum can be determined." 

HypApproxError::usage = 

"HypApproxError[b,c,goal] returns the abscissa value where the error between 

the function b*Sqrt[x^2 + c] and its approximation b*x reaches a given value." 

PartInt::usage = 

"PartInt[f,partpoints,a,(opts)] computes the indefinite integral of a function 

f over the range (a,Infinity) using a partition-extrapolation method when the 

partition points partpoints are explicitly given. The first interval (a,p1) is 

obtained using the double-exponential method, the others by a Gauss-Kronrod formula. 

The partial integrals between the partition points are summed up and passed to 

SequenceLimit to determine the value of the integral. If the first member of the 

list of partition points in Infinity, the no extrapolation is performed, and the 

integral is entirely calculated with the double-exponential rule. Note that the 

partition points must be of ascending order and chosen such that extrapolation 

is possible (in case of an oscillating integrand, they must be to the right of 

the rightmost extremum or saddle point of the argument function)." 

PartitionTable::usage = 

"PartitionTable[f,a,n,(opts)] returns a list of partition points suitable for 

PartInt. Depending on the option FunctionType the function f is interpreted either 

as argument function of Sin[f[x]] or as a function giving the k-th zero of Sin[arg[x]]. 

In the first case, the first n solutions of f[x] == K*Pi to the right of the lower 

215

integration limit a or the rightmost extremum of f, whichever is greater, are 

calculated. In the second case, the first n function values of f to the right of a 

are sought. Note that in this case, f is supposed to return values only to the 

right of the rightmost extremum. In any case, f must be of the form f[...][x]." 

OscInt::usage = 

"OscInt[f,fpartition,a,(opts)] returns the univariate integral of an oscillating 

function f over the range (a,Infinity) by means of a partition extrapolation strategy. 

The function f is required to of the form f[...][x], x being the 

integration variable. Quadrature is carried out between successive partition points 

defined by the function fpartition[...][x] that either returns the zeros of the 

integrand or the argument of the oscillating part. The sequence of partition 

points is computed by the function PartitionTable, the actual integration is carried 

out with the function PartInt. The first partition point can be given explicitly 

with OscInt[f,fpartition,{a,a0},(opts)]. The options are passed to PartitionTable and 

PartInt." 

OscIntControlled::usage = 

"OscIntControlled[f,fpartition,a,(opts)] works like OscInt with the additional 

feature that the accuracy of the result may be specified in advance." 

FunctionType::usage = 

"FunctionType is an option for PartitionTable (and OscInt) that specifies how 

the function passed to PartitionTable is to be interpreted. 

FunctionType -> ZeroList means that the function returns consecutive zeros of the 

integrand. FunctionType -> SinArgument or CosArgument means that it is the argument 

function of the respective cyclic function." 

ZeroList::usage = 

"ZeroList is a value to FunctionType." 

SinArgument::usage = CosArgument::usage = 

"Argument is a value to FunctionType." 

AccuracyControl::usage = 

"AccuracyControl specifies whether the accuracy of the result of OscInt should 

be controlled. With AccuracyControl -> n, at most MaxIterations are performed 

to achieve the number of n correct digits. With AccuracyControl -> False, no 

checking is done. The control loop uses the built-in function Accuracy." 

IterationLength::usage = 

"IterationLength specifies the number of terms that are added to the sequence 

of partial sums with each iteration." 

ApproxLimGeneric::nounique = 

"There are `1` real solutions `2`, here is the largest."; 

ApproxLimGeneric::nosolution = 

"There is no positive real valued solution."; 

QuadZero::noquad = "Function is not a quadratic polynomial"; 

ApproxLimQuad::noapprox = "No approximation needed"; 

ApproxLimLinear::noosc = "Function not infinitely oscillating"; 

ZerosInBetween::outside = "Function has no zeros inside the interval"; 

OscInt::lowfirst = 

"First partition point is smaller than lower limit, only 

`1` points are greater. This may cause problems."; 

OscInt::badparam = 

"\"`1`\" is no valid parameter for AccuracyControl."; 

OscInt::accfail = 

"OscInt failed to achieve the desired accuracy after `1` iterations. Increase 

MaxIterations or lower AccuracyControl."; 

PartitionTable::badparam = 

"\"`1`\" is no valid parameter for FunctionType."; 

Begin["`Private`"] 

Options[OscInt] = {WorkingPrecision->$MachinePrecision, 

AccuracyGoal->Infinity, 

PrecisionGoal->Automatic, 

MinRecursion->0, 

MaxRecursion->20, 

NSumTerms->10, 

NSumExtraTerms->12, 

216 

A Mathematica packages


WynnDegree->Infinity, 

MaxIterations->5, 

AccuracyControl->False, 

IterationLength->2}; 

Options[PartInt] = Options[OscInt]; 

Options[PartitionTable] = 

{WorkingPrecision->$MachinePrecision, 

AccuracyGoal->Automatic, 

MaxRecursion->20, 

FunctionType->SinArgument}; 

PolynomialDegree[f_] := 

Length[CoefficientList[Series[f[x],{x,Infinity,1}],x]]-1; 

PolynomialDegree[f_,x_Symbol] := 

Length[CoefficientList[Series[f,{x,Infinity,1}],x]]-1; 

AsymptoticExpand[f_] := 

Normal[Series[f[x],{x,Infinity,PolynomialDegree[f]}]]; 

AsymptoticExpand[f_,x_Symbol] := 

Normal[Series[f,{x,Infinity,PolynomialDegree[f,x]}]]; 

ApproxLimGeneric[f_,goal_] := 

Module[{fapprox,respos,resneg,res}, 

fapprox = AsymptoticExpand[f]; 

respos = Select[ 

NSolve[Evaluate[ 

D[fapprox,x] / 

D[fapprox - f[x],x]] == goal,x],(Im[x/.#] == 0)&]; 

resneg = Select[ 

NSolve[Evaluate[ 

D[fapprox,x] / 

D[fapprox - f[x],x]] == -goal,x],(Im[x/.#] == 0)&]; 

res = Select[Join[respos,resneg],Positive[x/.#]&]; 

If[Length[res] == 0, 

Message[ApproxLimGeneric::nosolution], 

If[Length[res] > 1, 

Message[ApproxLimGeneric::nounique,Length[res],res]]; 

x/.Last[res]] 

] 

ApproxLimQuad[a_,b_,c_,d_,goal_] := Which[ 

a != 0 && c != 0 && d != 0, 

x/.Flatten[Last[ 

NSolve[(2 a x + b + c) / 

(c (1-x/Sqrt[x^2+d])) == Sign[a c d] goal,x]]], 

a == 0 && c != 0 && d != 0, 

ApproxLimLinear[b,c,d,goal], 

True, 

Message[ApproxLimQuad::noapprox];0] 

ApproxLimLinear[a_,b_,c_,goal_] := Which[ 

a + b == 0, 

Message[ApproxLimLinear::noosc];0, 

Abs[(a + b) / b] > goal, 0, 

True, 

x/.Flatten[ 

NSolve[(a + b) / 

(b (1-x/Sqrt[x^2+c])) == Sign[(a+b) b c] goal,x]]]; 

QuadOffset[a_,b_,c_] := Which[ 

a == 0, Ceiling[c/Pi], 

a > 0,-Floor[(b^2 - 4 a c)/(4 a Pi)], 

True, -Ceiling[(b^2 - 4 a c)/(4 a Pi)]]; 

QuadZero[a_,b_,c_,o_][k_] := Which[ 

a > 0, (-b+Sqrt[b^2-4 a (c-( k+o) Pi)])/(2 a), 

a < 0, (-b-Sqrt[b^2-4 a (c-(-k+o) Pi)])/(2 a), 

a == 0 && b > 0, (( k+o) Pi - c)/b, 

a == 0 && b < 0, ((-k+o) Pi - c)/b, 

True, k]; 

HypApproxError[b_,c_,goal_] := 

x/.Last[Solve[b (Sqrt[x^2 + c]-x) == Sign[c b] goal,x]]; 

ZerosInBetween[zerof_,a_,b_] := 

Module[{min=Min[a,b], max=Max[a,b]}, 

Which[N[zerof[0]] < min, 

Ceiling[i/.FindRoot[zerof[i] == max,{i,0,1}]] - 

Ceiling[i/.FindRoot[zerof[i] == min,{i,0,1}]], 

217

]; 

N[zerof[0]] < max, 

Ceiling[i/.FindRoot[zerof[i] == max,{i,0,1}]], 

True, 

Message[ZerosInBetween::outside];0] 

HypOffset[a_,b_,c_,d_] := Which[ 

(Abs[b] > Abs[a]) && (c >= 0) && (a+b) > 0, 

Ceiling[((a+b) (-Sign[a b] Sqrt[a^2 c/(b^2-a^2)]) + d)/Pi], 

(Abs[b] > Abs[a]) && (c >= 0) && (a+b) < 0, 

Floor[((a+b) (-Sign[a b] Sqrt[a^2 c/(b^2-a^2)]) + d)/Pi], 

(a+b) > 0, Ceiling[d/Pi], 

(a+b) < 0, Floor[d/Pi], 

True, 0]; 

HypZero[a_,b_,c_,d_,o_][k_] := Which[ 

(a+b) > 0, (( k + o) Pi - d)/(a + b), 

(a+b) < 0, ((-k + o) Pi - d)/(a + b), 


HypOffsetExact[a_,b_,c_,d_] := Which[ 

(Abs[b] > Abs[a]) && (c >= 0) && (a+b) > 0, 

Ceiling[((a x + b Sqrt[x^2+c] + d)/Pi)/. 

x->(-Sign[a b] Sqrt[a^2 c/(b^2-a^2)])], 

(Abs[b] > Abs[a]) && (c >= 0) && (a+b) < 0, 

Floor[((a x + b Sqrt[x^2+c] + d)/Pi)/. 

x->(-Sign[a b] Sqrt[a^2 c/(b^2-a^2)])], 

(a+b) > 0, Ceiling[(b Sqrt[c] + d)/Pi], 

(a+b) < 0, Floor[(b Sqrt[c] + d)/Pi], 

True, 0]; 

HypZeroExact[a_,b_,c_,d_,o_][k_] := Which[ 

(a+b) > 0 && (Abs[b] > Abs[a]), 

(a (( k+o) Pi-d) - Sqrt[b^2 ((a^2-b^2) c + (( k+o) Pi-d)^2)])/(a^2-b^2), 

(a+b) < 0 && (Abs[b] > Abs[a]), 

(a ((-k+o) Pi-d) - Sqrt[b^2 ((a^2-b^2) c + ((-k+o) Pi-d)^2)])/(a^2-b^2), 

(a+b) < 0 && (Abs[b] < Abs[a]), 

(a ((-k+o) Pi-d) - Sign[a b] Sqrt[b^2 ((a^2-b^2) c + ((-k+o) Pi-d)^2)])/(a^2-b^2), 

(a+b) > 0 && (Abs[b] < Abs[a]), 

(a (( k+o) Pi-d) - Sign[a b] Sqrt[b^2 ((a^2-b^2) c + (( k+o) Pi-d)^2)])/(a^2-b^2), 

a == b && N[(2 a ((k+o) Pi - d))] != 0, 

(((k+o) Pi - d)^2 - a^2 c)/(2 a ((k+o) Pi - d)), 


PartInt[f_, zerotab_List, a_, opts___Rule] := 

Module[{zeros = zerotab, seqtab, 

wp = WorkingPrecision/.{opts}/.Options[OscInt], 

ag = AccuracyGoal/. {opts}/.Options[OscInt], 

pg = PrecisionGoal/. {opts}/.Options[OscInt], 

mi = MinRecursion/. {opts}/.Options[OscInt], 

ma = MaxRecursion/. {opts}/.Options[OscInt], 

nt = NSumTerms/. {opts}/.Options[OscInt], 

ne = NSumExtraTerms/. {opts}/.Options[OscInt], 

wd = WynnDegree/. {opts}/.Options[OscInt]}, 

If[N[zeros[[1]]] < N[a], 

zeros = Select[zeros,(N[#] >= a)&]; 

Message[OscInt::lowfirst,Length[zeros]]]; 

If[N[zeros[[1]]] == N[a], 0, 

NIntegrate[f[x],{x,a,zeros[[1]]}, 






MaxRecursion->ma]] + 

If[N[zeros[[1]]] == Infinity, 0, 

seqtab = Table[NIntegrate[f[x],{x,zeros[[i]],zeros[[i+1]]}, 

Method->GaussKronrod, 





MaxRecursion->ma], 

{i,1,Length[zeros]-1}]; 

SequenceLimit[Take[FoldList[Plus,0,seqtab],{nt+1,nt+ne}], 

WynnDegree->wd]] 

]; 

PartitionTable[zerof_, a_, n_, opts___Rule] := 

Module[{ofs=0,x,xoffs,koffs,extrema,cfoffs=0, 

wp = WorkingPrecision/.{opts}/.Options[PartitionTable], 

ag = AccuracyGoal/. {opts}/.Options[PartitionTable], 

ma = MaxRecursion/. {opts}/.Options[PartitionTable], 

ft = FunctionType/. {opts}/.Options[PartitionTable]}, 

218 



Which[ft === ZeroList, 


ofs = Ceiling[x/.FindRoot[zerof[x] == a,{x,0,1}, 



MaxIterations->ma]]]; 

Table[zerof[x],{x,ofs,ofs+n-1}], 

ft === SinArgument || ft === CosArgument, 

If[ft === CosArgument, cfoffs=1/2]; 

extrema = Select[NSolve[zerof'[x] == 0,x,WorkingPrecision->wp], 

(Im[x/.#] == 0)&]; 

xoffs = If[Length[extrema] == 0, a, 

Max[x/.Last[extrema],a]]; 

Which[NumberQ[Limit[zerof[x],x->Infinity]], 

Table[Infinity,{k,0,n-1}], 

N[zerof'[xoffs+1]] > 0, 

koffs = N[Ceiling[zerof[xoffs]/Pi],2 wp]; 


{x,xoffs,xoffs+1}, 



MaxIterations->ma], 

{k,0,n-1}]], 

True, 

koffs = N[Floor[zerof[xoffs]/Pi],2 wp]; 


{x,xoffs,xoffs+1}, 




{k,0,n-1}]] 

], 

True, 

Message[PartitionTable::badparam,ft]] 

]; 

PartitionOffs[zerof_, a_, opts___Rule] := 

Module[{x,extrema,cfoffs=0, 







Ceiling[x/.FindRoot[zerof[x] == a,{x,0,1}, 



MaxIterations->ma]], 

0], 



extrema = Select[NSolve[zerof'[x] == 0,x,WorkingPrecision->wp], 

(Im[x/.#] == 0)&]; 

If[Length[extrema] == 0, a, 

Max[x/.Last[extrema],a]], 

True, 


]; 

PartitionPoints[zerof_, a_, n_, offs_, irel_, opts___Rule] := 

Module[{x,koffs,cfoffs=0, 






Table[zerof[x],{x,offs+irel,offs+irel+n-1}], 



Which[NumberQ[Limit[zerof[x],x->Infinity]], 

Table[Infinity,{k,0,n-1}], 

N[zerof'[offs+1]] > 0, 

koffs = N[Ceiling[zerof[offs]/Pi],2 wp]; 


{x,offs,offs+1}, 




{k,irel,irel+n-1}]], 

True, 

koffs = N[Floor[zerof[offs]/Pi],2 wp]; 


{x,offs,offs+1}, 


219

]; 



{k,irel,irel+n-1}]] 

], 

True, 


OscIntControlled[f_, fzero_, a_, opts___Rule] := 

Module[{seqtab,parttab,iterations=1,offs,firstval,intlim, 







ne = NSumExtraTerms/. {opts}/.Options[OscInt], 

wd = WynnDegree/. {opts}/.Options[OscInt], 

it = MaxIterations/. {opts}/.Options[OscInt], 

ac = AccuracyControl/. {opts}/.Options[OscInt], 

il = IterationLength/. {opts}/.Options[OscInt]}, 

offs = PartitionOffs[zerof,a,opts]; 

parttab = PartitionPoints[fzero,a,ne+nt+1,offs,0,opts]; 

If[N[parttab[[1]]] < a, 

parttab = Select[parttab,(N[#] >= a)&]; 

Message[OscInt::lowfirst,Length[parttab]]]; 

firstval = If[N[parttab[[1]]] == N[a], 0, 

NIntegrate[f[x],{x,a,parttab[[1]]}, 






MaxRecursion->ma]]; 

If[N[parttab[[1]]] =!= Infinity, 

seqtab = Table[NIntegrate[f[x],{x,parttab[[i]],parttab[[i+1]]}, 







{i,1,Length[parttab]-1}]]; 

Which[N[parttab[[1]]] == Infinity, 

firstval, 

ac === False, 

SequenceLimit[Take[FoldList[Plus,firstval,seqtab], 

{nt+1,nt+ne}],WynnDegree->wd], 

IntegerQ[ac], 


{nt+1,nt+ne}],WynnDegree->wd]; 

While[iterations < it && Accuracy[intlim] < ac, 

parttab = Join[{Last[parttab]}, 

PartitionPoints[fzero,a,il,offs,ne+nt+1,opts]]; 

seqtab = Join[seqtab, 

Table[NIntegrate[f[x],{x,parttab[[i]],parttab[[i+1]]}, 







{i,1,Length[parttab]-1}]]; 

ne = ne + il; 

iterations++; 


{nt+1,nt+ne}],WynnDegree->wd] 

]; 

If[Accuracy[intlim] < ac, Message[OscInt::accfail,iterations]]; 

intlim, 

True, 

Message[OscInt::badparam,ac]] 

]; 

OscInt[fint_, fzero_, {a_, a0_}, opts___Rule] := 

Module[{parttab,lowlim, 



ne = NSumExtraTerms/. {opts}/.Options[OscInt]}, 

lowlim = If[a == a0, a, N[Max[a,a0],2 wp]]; 

parttab = PartitionTable[fzero,lowlim,ne+nt+1,opts]; 

PartInt[fint,parttab,a,opts] 

]; 

220 


A.2 Solutions for the step potential 

OscInt[fint_, fzero_, a_, opts___Rule] := 

OscInt[fint,fzero,{a,a},opts] 

End[] 

SetAttributes[PolynomialDegree, ReadProtected] 

SetAttributes[AsymptoticExpand, ReadProtected] 

SetAttributes[ApproxLimGeneric, ReadProtected] 

SetAttributes[ApproxLimQuad, ReadProtected] 

SetAttributes[ApproxLimLinear, ReadProtected] 

SetAttributes[QuadZero, ReadProtected] 

SetAttributes[QuadOffset, ReadProtected] 

SetAttributes[HypZero, ReadProtected] 

SetAttributes[HypOffset, ReadProtected] 

SetAttributes[HypZeroExact, ReadProtected] 

SetAttributes[HypOffsetExact, ReadProtected] 

SetAttributes[ZerosInBetween, ReadProtected] 

SetAttributes[HypApproxError, ReadProtected] 

SetAttributes[PartInt, ReadProtected] 

SetAttributes[PartitionTable, ReadProtected] 

SetAttributes[PartitionOffs, ReadProtected] 

SetAttributes[PartitionPoints, ReadProtected] 

SetAttributes[OscInt, ReadProtected] 

SetAttributes[OscIntControlled, ReadProtected] 

(* 

Protect[PolynomialDegree, AsymptoticExpand, ApproxLimGeneric] 

Protect[ApproxLimQuad, ApproxLimLinear] 

Protect[QuadZero, QuadOffset] 

Protect[HypZero, HypOffset] 

Protect[HypZeroExact, HypOffsetExact] 

Protect[ZerosInBetween, HypApproxError] 

Protect[PartInt, PartitionTable, PartitionOffs, PartitionPoints] 

Protect[OscInt, OscIntControlled] 

*) 

EndPackage[] 


The package Tunnel computes the solutions of the Schrodinger equation for the step potential 

barrier discussed in section 4.3. Its structure is explained in chapter 8. For the calculation of 

the integrals, the package OscInt is used. 




(*:Title: Tunnel *) 


(*:Keywords: Tunnel Effect *) 



(*:Packages: OscInt *) 

(*:Summary: This package contains functions for the time-dependent solutions 

of Schroedinger's equation. The exact solution has been evaluated 

for wave packet impinging on a given step potential barrier. The 

initial shape of the wave packet can be either rectangular, 

triangular, or Gaussian. The solution of the differential 

equation consists of Fourier integrals and must be evaluated 

numerically. A severe obstacle to a straightforward integration 

are the heavily oscillating integrands. Thus integration is 

carried out by using the functions in the package OscInt 

*) 

(*:History: 15-11-1996 written 

25-01-1997 templates added 

02-02-1997 new naming convention for functions 

17-02-1997 truncation quadrature of Gaussian waves added 

*) 

BeginPackage["Tunnel`", "OscInt`"] 

Phi::usage = 

"Phi[x,t,w,k,(n),(opts)] returns the wave function at a given coordinate 

in space and time. The shape of the incident wave is selected with the option 

Shape. The step barrier is supposed to begin at x=0. Depending on the spatial 

221

coordinate the functions either returns the sum of incident and reflected wave 

or the portion of the wave inside the tunnel. The parameter w is the ratio of 

the carrier frequency of the incident wave and the characteristic frequency of 

the barrier. For tunnelling, 0


(*-- Functions independent of the waveform --*) 





PhiArg[a_,b_,c_,d_][xi_] := a xi^2 + b xi + c Sqrt[xi^2 - 1] + d; 

GaussTruncLims[w_,k_,n_,wp_] := 

{N[Sqrt[w] (1-n/(Pi k) * 

Sqrt[Log[Sqrt[2 Pi] k/Sqrt[n w Sqrt[2 Pi]]/10^-(wp+1)]]), wp+2], 

N[Sqrt[w] (1+n/(Pi k) * 

Sqrt[Log[Sqrt[2 Pi] k/Sqrt[n w Sqrt[2 Pi]]/10^-(wp+1)]]), wp+2]}; 

(*-- Triangular Shape inside the tunnel --*) 

TriaConst[w_,k_] := Sqrt[3/w] / (2 k Pi^2); 

TriaShapePos[w_,xi_] := 1/(1-xi/Sqrt[w])^2; 

TriaShapeNeg[w_,xi_] := 1/(1+xi/Sqrt[w])^2; 

TriaTransPos[a_,b_,c_,d_,w_,k_][xi_] := 

TunCoeff[xi] * TriaShapePos[w,xi] * TunOsc[a,b,c,d,xi]; 

TriaTransNeg[a_,b_,c_,d_,w_,k_][xi_] := 

TunCoeff[xi] * TriaShapeNeg[w,xi] * TunOsc[a,b,c,d,xi]; 

TriaGradTransPos[a_,b_,c_,d_,w_,k_][xi_] := 

TunCoeff[xi] * TriaShapePos[w,xi] * TunOsc[a,b,c,d,xi] * 

I Sqrt[xi^2 - 1]; 

TriaGradTransNeg[a_,b_,c_,d_,w_,k_][xi_] := 

TunCoeff[xi] * TriaShapeNeg[w,xi] * TunOsc[a,b,c,d,xi] * 

(-I Sqrt[xi^2 - 1]); 

TriaEvan[a_,b_,c_,d_,w_,k_][xi_] := 

TriaTransPos[a,b,c,d,w,k][xi] * 



TriaGradEvan[a_,b_,c_,d_,w_,k_][xi_] := 

TriaGradTransPos[a,b,c,d,w,k][xi] * 



(*-- Triangular Shape outside the tunnel --*) 

TriaRefPos[a_,b_,c_,d_,w_,k_][xi_] := 

RefCoeff[xi] * TriaShapePos[w,xi] * OutOsc[a,b,c,d,xi]; 

TriaRefNeg[a_,b_,c_,d_,w_,k_][xi_] := 

RefCoeff[xi] * TriaShapeNeg[w,xi] * OutOsc[a,b,c,d,xi]; 

TriaRefEvan[a_,b_,c_,d_,w_,k_][xi_] := 

TriaRefPos[a,b,c,d,w,k][xi] * 



TriaIncPos[a_,b_,c_,d_,w_,k_][xi_] := 

TriaShapePos[w,xi] * OutOsc[a,b,c,d,xi]; 

TriaIncNeg[a_,b_,c_,d_,w_,k_][xi_] := 

TriaShapeNeg[w,xi] * OutOsc[a,b,c,d,xi]; 

TriaIncEvan[a_,b_,c_,d_,w_,k_][xi_] := 

TriaIncPos[a,b,c,d,w,k][xi] * 



(*-- Rectangular Shape inside the tunnel --*) 

RectConst[w_,k_] := I / (Sqrt[w] 2 Pi); 

RectShapePos[w_,xi_] := 1/(1-xi/Sqrt[w]); 

RectShapeNeg[w_,xi_] := 1/(1+xi/Sqrt[w]); 

RectTransPos[a_,b_,c_,d_,w_,k_][xi_] := 

TunCoeff[xi] * RectShapePos[w,xi] * TunOsc[a,b,c,d,xi]; 

RectTransNeg[a_,b_,c_,d_,w_,k_][xi_] := 

TunCoeff[xi] * RectShapeNeg[w,xi] * TunOsc[a,b,c,d,xi]; 

RectEvan[a_,b_,c_,d_,w_,k_][xi_] := 

RectTransPos[a,b,c,d,w,k][xi] * 


(*-- Rectangular Shape outside the tunnel --*) 

223

RectRefPos[a_,b_,c_,d_,w_,k_][xi_] := 

RefCoeff[xi] * RectShapePos[w,xi] * OutOsc[a,b,c,d,xi]; 

RectRefNeg[a_,b_,c_,d_,w_,k_][xi_] := 

RefCoeff[xi] * RectShapeNeg[w,xi] * OutOsc[a,b,c,d,xi]; 

RectRefEvan[a_,b_,c_,d_,w_,k_][xi_] := 

RectRefPos[a,b,c,d,w,k][xi] * 


RectIncPos[a_,b_,c_,d_,w_,k_][xi_] := 

RectShapePos[w,xi] * OutOsc[a,b,c,d,xi]; 

RectIncNeg[a_,b_,c_,d_,w_,k_][xi_] := 

RectShapeNeg[w,xi] * OutOsc[a,b,c,d,xi]; 

RectIncEvan[a_,b_,c_,d_,w_,k_][xi_] := 

RectIncPos[a,b,c,d,w,k][xi] * 


(*-- Exponential Shape inside the tunnel --*) 

GaussConst[w_,k_,n_] := k Sqrt[2 Pi / (w n Sqrt[2 Pi])]; 

GaussShapePos[w_,k_,n_,xi_] := Exp[-(Pi k (xi/Sqrt[w] - 1)/n)^2]; 

GaussShapeNeg[w_,k_,n_,xi_] := Exp[-(Pi k (-xi/Sqrt[w] - 1)/n)^2]; 

GaussTransPos[a_,b_,c_,d_,w_,k_,n_][xi_] := 

TunCoeff[xi] * GaussShapePos[w,k,n,xi] * TunOsc[a,b,c,d,xi]; 

GaussTransNeg[a_,b_,c_,d_,w_,k_,n_][xi_] := 

TunCoeff[xi] * GaussShapeNeg[w,k,n,xi] * TunOsc[a,b,c,d,xi]; 

GaussGradTransPos[a_,b_,c_,d_,w_,k_,n_][xi_] := 

TunCoeff[xi] * GaussShapePos[w,k,n,xi] * TunOsc[a,b,c,d,xi] * 

I Sqrt[xi^2 - 1]; 

GaussGradTransNeg[a_,b_,c_,d_,w_,k_,n_][xi_] := 

TunCoeff[xi] * GaussShapeNeg[w,k,n,xi] * TunOsc[a,b,c,d,xi] * 

(-I Sqrt[xi^2 - 1]); 

GaussEvan[a_,b_,c_,d_,w_,k_,n_][xi_] := 

GaussTransPos[a,b,c,d,w,k,n][xi]; 

GaussGradEvan[a_,b_,c_,d_,w_,k_,n_][xi_] := 

GaussGradTransPos[a,b,c,d,w,k,n][xi]; 

(*-- Exponential Shape outside the tunnel --*) 

GaussRefPos[a_,b_,c_,d_,w_,k_,n_][xi_] := 

RefCoeff[xi] * GaussShapePos[w,k,n,xi] * OutOsc[a,b,c,d,xi]; 

GaussRefNeg[a_,b_,c_,d_,w_,k_,n_][xi_] := 

RefCoeff[xi] * GaussShapeNeg[w,k,n,xi] * OutOsc[a,b,c,d,xi]; 

GaussRefEvan[a_,b_,c_,d_,w_,k_,n_][xi_] := 

GaussRefPos[a,b,c,d,w,k,n][xi]; 

GaussIncPos[a_,b_,c_,d_,w_,k_,n_][xi_] := 

GaussShapePos[w,k,n,xi] * OutOsc[a,b,c,d,xi]; 

GaussIncNeg[a_,b_,c_,d_,w_,k_,n_][xi_] := 

GaussShapeNeg[w,k,n,xi] * OutOsc[a,b,c,d,xi]; 

GaussIncEvan[a_,b_,c_,d_,w_,k_,n_][xi_] := 

GaussIncPos[a,b,c,d,w,k,n][xi]; 

(*-- Templates for wave computations depending on the shape --*) 

TriaTransTemp[fpos_, fneg_, x_, t_, w_, k_, opts___Rule] := 

Module[{a,b,c,d,o11,o12,o21,o22,o31,o32, 

wp = WorkingPrecision/.{opts}/.Options[OscInt]}, 

a = -t; 

b = N[Pi k/Sqrt[w],wp+2]; 

c = x; 

d = N[-Pi k,wp+2]; 

o11 = QuadOffset[a, c,0]; 

o12 = QuadOffset[a,-c,0]; 

o21 = QuadOffset[a, b+c,d]; 

o22 = QuadOffset[a,-b-c,d]; 

o31 = QuadOffset[a, 2 b + c,2 d]; 

o32 = QuadOffset[a,-2 b - c,2 d]; 



224 








QuadZero[a,-b -c,d,o22],1,FunctionType->ZeroList,opts])*2 - 

(OscInt[fpos[a, 2 b, c,2 d,w,k], 

QuadZero[a, 2 b +c,2 d,o31],1,FunctionType->ZeroList,opts] + 

OscInt[fneg[a,-2 b,-c,2 d,w,k], 

QuadZero[a,-2 b -c,2 d,o32],1,FunctionType->ZeroList,opts]) 

) TriaConst[w,k] 

]; 

TriaTransExactTemp[fpos_, fneg_, x_, t_, w_, k_, opts___Rule] := 

Module[{a,b,c,d, 


a = -t; 


c = x; 

d = N[-Pi k]; 








PhiArg[a,-b,-c,d],1,opts])*2 - 

(OscInt[fpos[a, 2 b, c,2 d,w,k], 

PhiArg[a, 2 b, c,2 d],1,opts] + 

OscInt[fneg[a,-2 b,-c,2 d,w,k], 

PhiArg[a,-2 b,-c,2 d],1,opts]) 

) TriaConst[w,k] 

]; 

EvanTemp[f_, const_, opts___Rule] := 

Module[{wp = WorkingPrecision/.{opts}/.Options[OscInt], 




ma = MaxRecursion/. {opts}/.Options[OscInt]}, 

NIntegrate[f[xi],{xi,-1,1}, 






MaxRecursion->ma] const 

]; 

RectTransTemp[fpos_, fneg_, x_, t_, w_, k_, opts___Rule] := 

Module[{a,b,c,d,o11,o12,o21,o22, 


a = -t; 


c = x; 

d = N[-2 Pi k,wp+2]; 

o11 = N[QuadOffset[a, c,0],wp+2]; 

o12 = N[QuadOffset[a,-c,0],wp+2]; 

o21 = N[QuadOffset[a, b+c,d],wp+2]; 

o22 = N[QuadOffset[a,-b-c,d],wp+2]; 








QuadZero[a,-b -c,d,o22],1,FunctionType->ZeroList,opts]) 


]; 

RectTransExact[x_, t_, w_, k_, opts___Rule] := 

Module[{a,b,c,d, 


a = -t; 


c = x; 

d = N[-2 Pi k,wp+2]; 

(-(OscInt[RectTransPos[a, 0, c,0,w,k], 


OscInt[RectTransNeg[a, 0,-c,0,w,k], 


(OscInt[RectTransPos[a, b, c,d,w,k], 

225


OscInt[RectTransNeg[a,-b,-c,d,w,k], 

PhiArg[a,-b,-c,d],1,opts]) 


]; 

GaussTransTemp[fpos_, fneg_, x_, t_, w_, k_, n_, opts___Rule] := 

Module[{a,b,c,d,o1,o2, 


a = -t; 


c = x; 

d = N[-Pi k,wp+2]; 

o1 = QuadOffset[a, b+c,d]; 

o2 = QuadOffset[a,-b-c,d]; 

(OscInt[fpos[a, b, c,d,w,k,n], 


OscInt[fneg[a,-b,-c,d,w,k,n], 

QuadZero[a,-b -c,d,o2],1,FunctionType->ZeroList,opts] 

) GaussConst[w,k,n] 

]; 

GaussTransTruncTemp[fpos_, fneg_, x_, t_, w_, k_, n_, opts___Rule] := 






limpos, limneg}, 

{limneg,limpos} = GaussTruncLims[w,k,n,wp]; 

(If[limpos DoubleExponential, 






If[limneg >= -1, 0, 

NIntegrate[fneg[-t,-Pi k/Sqrt[w],-x,-Pi k,w,k,n][xi], 

{xi,1,-limneg}, 






MaxRecursion->ma]] 


]; 

GaussEvanTruncTemp[f_, w_, k_, n_, opts___Rule] := 

Module[{limpos,limneg, 







limneg = Max[limneg,-1]; 

limpos = Min[limpos,1]; 

NIntegrate[f[xi],{xi,limneg,limpos}, 






MaxRecursion->ma] GaussConst[w,k,n] 

]; 

(*-- Driver functions for the computation of the wave integrals --*) 

PhiTrans[x_, t_, w_, k_, n_?Positive, opts___Rule] := 

Module[{sh = Shape/. {opts}/.Options[Phi], 

pt = PPoints/.{opts}/.Options[Phi]}, 

Switch[sh, 

Rect, 

Switch[pt, 

Approximate, RectTransTemp[RectTransPos,RectTransNeg,x,t,w,k,opts], 

Zeros, RectTransExact[x,t,w,k,opts], 


Tria, 

Switch[pt, 

226 



] 

]; 

Approximate, TriaTransTemp[TriaTransPos,TriaTransNeg,x,t,w,k,opts], 

Zeros, TriaTransExactTemp[TriaTransPos,TriaTransNeg,x,t,w,k,opts], 


Gauss, 

Switch[pt, 

Approximate, GaussTransTemp[GaussTransPos,GaussTransNeg,x,t,w,k,n,opts], 

Truncate, GaussTransTruncTemp[GaussTransPos,GaussTransNeg,x,t,w,k,n,opts], 


_, 


PhiTrans[x_, t_, w_, k_, opts___Rule] := 

If[(Shape/.{opts}/.Options[Phi]) === Gauss, 


PhiTrans[x,t,w,k,1,opts]]; 

PhiGradTrans[x_, t_, w_, k_, n_?Positive, opts___Rule] := 



Switch[sh, 

Tria, 

Switch[pt, 

Approximate, TriaTransTemp[TriaGradTransPos,TriaGradTransNeg,x,t,w,k,opts], 

Zeros, TriaTransExactTemp[TriaGradTransPos,TriaGradTransNeg,x,t,w,k,opts], 


Gauss, 

Switch[pt, 

Approximate, GaussTransTemp[GaussGradTransPos,GaussGradTransNeg,x,t,w,k,n,opts], 

Truncate, GaussTransTruncTemp[GaussGradTransPos,GaussGradTransNeg,x,t,w,k,n,opts], 


_, 


] 

]; 

PhiGradTrans[x_, t_, w_, k_, opts___Rule] := 



PhiGradTrans[x,t,w,k,1,opts]]; 

PhiEvan[x_, t_, w_, k_, n_?Positive, opts___Rule] := 



Switch[sh, 

Rect, 

EvanTemp[RectEvan[-t,0,x,0,w,k],RectConst[w,k],opts], 

Tria, 

EvanTemp[TriaEvan[-t,0,x,0,w,k],TriaConst[w,k],opts], 

Gauss, 

Switch[pt, 

Approximate, EvanTemp[GaussEvan[-t,Pi k/Sqrt[w],x,-Pi k,w,k,n], 

GaussConst[w,k,n],opts], 

Truncate, GaussEvanTruncTemp[GaussEvan[-t,Pi k/Sqrt[w],x,-Pi k,w,k,n], 

w,k,n,opts], 


_, 


] 

]; 

PhiEvan[x_, t_, w_, k_, opts___Rule] := 


Message[PhiTransmit::missingval], 

PhiEvan[x,t,w,k,1,opts]]; 

PhiGradEvan[x_, t_, w_, k_, n_?Positive, opts___Rule] := 



Switch[sh, 

Tria, 

EvanTemp[TriaGradEvan[-t,0,x,0,w,k],TriaConst[w,k],opts], 

Gauss, 

Switch[pt, 

Approximate, EvanTemp[GaussGradEvan[-t,Pi k/Sqrt[w],x,-Pi k,w,k,n], 


Truncate, GaussEvanTruncTemp[GaussGradEvan[-t,Pi k/Sqrt[w],x,-Pi k,w,k,n], 

w,k,n,opts], 


_, 


] 

227

]; 

PhiGradEvan[x_, t_, w_, k_, opts___Rule] := 



PhiGradEvan[x,t,w,k,1,opts]]; 

(*-- outside the tunnel --*) 

(* Note that in this case the spatial coordinate is inverted in the function calls 

because of the reverse motion of the reflected wave. *) 

PhiRef[x_, t_, w_, k_, n_?Positive, opts___Rule] := 



Switch[sh, 

Rect, 

RectTransTemp[RectRefPos,RectRefNeg,-x,t,w,k,opts] + 

EvanTemp[RectRefEvan[-t,0,-x,0,w,k],RectConst[w,k],opts], 

Tria, 

TriaTransTemp[TriaRefPos,TriaRefNeg,-x,t,w,k,opts] + 

EvanTemp[TriaRefEvan[-t,0,-x,0,w,k],TriaConst[w,k],opts], 

Gauss, 

Switch[pt, 

Approximate, GaussTransTemp[GaussRefPos,GaussRefNeg,-x,t,w,k,n,opts] + 

EvanTemp[GaussRefEvan[-t,Pi k/Sqrt[w],-x,-Pi k,w,k,n], 


Truncate, GaussTransTruncTemp[GaussRefPos,GaussRefNeg,-x,t,w,k,n,opts] + 

GaussEvanTruncTemp[GaussRefEvan[-t,Pi k/Sqrt[w],-x,-Pi k,w,k,n], 

w,k,n,opts], 


_, 


] 

]; 

PhiRef[x_, t_, w_, k_, opts___Rule] := 



PhiRef[x,t,w,k,1,opts]]; 

PhiInc[x_, t_, w_, k_, n_?Positive, opts___Rule] := 



Switch[sh, 

Rect, 

RectTransTemp[RectIncPos,RectIncNeg,x,t,w,k,opts] + 

EvanTemp[RectIncEvan[-t,0,x,0,w,k],RectConst[w,k],opts], 

Tria, 

TriaTransTemp[TriaIncPos,TriaIncNeg,x,t,w,k,opts] + 

EvanTemp[TriaIncEvan[-t,0,x,0,w,k],TriaConst[w,k],opts], 

Gauss, 

Switch[pt, 

Approximate, GaussTransTemp[GaussIncPos,GaussIncNeg,x,t,w,k,n,opts] + 

EvanTemp[GaussIncEvan[-t,Pi k/Sqrt[w],x,-Pi k,w,k,n], 


Truncate, GaussTransTruncTemp[GaussIncPos,GaussIncNeg,x,t,w,k,n,opts] + 

GaussEvanTruncTemp[GaussIncEvan[-t,Pi k/Sqrt[w],x,-Pi k,w,k,n], 

w,k,n,opts], 


_, 


] 

]; 

PhiInc[x_, t_, w_, k_, opts___Rule] := 



PhiInc[x,t,w,k,1,opts]]; 

Phi[x_, t_, w_, k_, n_?Positive, opts___Rule] := 

Which[t == 0 && x == 0 && (Shape/.{opts}/.Options[Phi]) === Rect, 0.5, 

t < 0, 0, 

x >= 0, PhiEvan[x,t,w,k,n,opts] + PhiTrans[x,t,w,k,n,opts], 

x < 0, PhiInc[x,t,w,k,n,opts] + PhiRef[x,t,w,k,n,opts]]; 

Phi[x_, t_, w_, k_, opts___Rule] := 



Phi[x,t,w,k,1,opts]]; 

PhiGrad[x_, t_, w_, k_, n_?Positive,opts___Rule] := 

If[x < 0 || t < 0 || x == 0 && t == 0, 0, 

PhiGradEvan[x,t,w,k,n,opts] + PhiGradTrans[x,t,w,k,n,opts]]; 

228 


A.3 Solutions for the square barrier 

PhiGrad[x_, t_, w_, k_, opts___Rule] := 

If[x < 0 || t < 0 || x == 0 && t == 0, 0, 

PhiGradEvan[x,t,w,k,opts] + PhiGradTrans[x,t,w,k,opts]]; 

End[] 

SetAttributes[PhiTrans, ReadProtected] 

SetAttributes[PhiGradTrans, ReadProtected] 

SetAttributes[PhiEvan, ReadProtected] 

SetAttributes[PhiGradEvan, ReadProtected] 

SetAttributes[PhiRef, ReadProtected] 

SetAttributes[PhiInc, ReadProtected] 

SetAttributes[Phi, ReadProtected] 

SetAttributes[PhiGrad, ReadProtected] 

(* 

Protect[PhiTrans, PhiGradTrans, PhiEvan, PhiGradEvan] 

Protect[PhiRef, PhiInc, Phi, PhiGrad] 

*) 

EndPackage[] 


The package Gauss was written to compute the solutions of the Schrodinger equation for the 

tunnelling of an electron through a square barrier (section 4.6). The initial shape of the wave 

packet is Gaussian. The details of the implementation are not treated in previous chapters, 

however the structure is identical to that of Tunnel. 




(*:Title: Gauss *) 


(*:Keywords: Tunnel Effect *) 




(*:Summary: This package computes the solutions of the time-dependent 

Schroedinger equation for an initially Gaussian wave packet 

tunnelling through a square barrier. It is based on the same 

structure as the package Tunnel. The wave integrals are 

computed by truncation of the infinite integration range. 

*) 

(*:History: 31-10-1997 written 

*) 

BeginPackage["Gauss`", "OscInt`"] 

Phi::usage = 

"Phi[x,t,w,k,n,l,(opts)] returns the wave function of an initially Gaussian 

wave packet at a given coordinate in space and time. The square barrier is 

supposed to span the interval x=0 to x=l. Depending on the spatial 

coordinate the function either returns the sum of incident and reflected wave, 

the portion of the wave inside the tunnel, or the transmitted wave. The parameter 

w is the ratio of the carrier frequency of the incident wave and the 

characteristic frequency of the barrier. For tunnelling, 0

PhiFro::usage = 

"PhiFro[x,t,w,k,n,l,(opts)] returns the left-going part in the barrier. 

For further information see Phi" 

PhiTun::usage = 

"PhiTun[x,t,w,k,n,l,(opts)] returns the wave inside the barrier. For 

further information see Phi" 


(*-- Functions independent of the waveform --*) 

RefCoeffPos[xi_,l_] := -I Sin[l Sqrt[xi^2-1]]/ 

(2 xi Sqrt[xi^2-1] Cos[l Sqrt[xi^2-1]] - I (2 xi^2-1) Sin[l Sqrt[xi^2-1]]); 

RefCoeffNeg[xi_,l_] := I Sin[l Sqrt[xi^2-1]]/ 

(2 xi Sqrt[xi^2-1] Cos[l Sqrt[xi^2-1]] + I (2 xi^2-1) Sin[l Sqrt[xi^2-1]]); 

ToCoeffPos[xi_,l_] := (xi Sqrt[xi^2-1] + xi^2)/ 

(2 xi Sqrt[xi^2-1] Cos[l Sqrt[xi^2-1]] - I (2 xi^2-1) Sin[l Sqrt[xi^2-1]]) 

ToCoeffNeg[xi_,l_] := (xi Sqrt[xi^2-1] + xi^2)/ 

(2 xi Sqrt[xi^2-1] Cos[l Sqrt[xi^2-1]] + I (2 xi^2-1) Sin[l Sqrt[xi^2-1]]) 

FroCoeffPos[xi_,l_] := (xi Sqrt[xi^2-1] - xi^2)/ 


FroCoeffNeg[xi_,l_] := (xi Sqrt[xi^2-1] - xi^2)/ 


TransCoeffPos[xi_,l_] := 2 xi Sqrt[xi^2-1]/ 


TransCoeffNeg[xi_,l_] := 2 xi Sqrt[xi^2-1]/ 




GaussTruncLims[w_,k_,n_,wp_] := 

{N[Sqrt[w] (1-n/(Pi k) * 

Sqrt[Log[Sqrt[2 Pi] k/Sqrt[n w Sqrt[2 Pi]]/10^-(wp+1)]]), wp+2], 

N[Sqrt[w] (1+n/(Pi k) * 

Sqrt[Log[Sqrt[2 Pi] k/Sqrt[n w Sqrt[2 Pi]]/10^-(wp+1)]]), wp+2]}; 

(*-- Wave packet shape --*) 

GaussConst[w_,k_,n_] := k Sqrt[2 Pi / (w n Sqrt[2 Pi])]; 

GaussShapePos[w_,k_,n_,xi_] := Exp[-(Pi k (xi/Sqrt[w] - 1)/n)^2]; 

GaussShapeNeg[w_,k_,n_,xi_] := Exp[-(Pi k (-xi/Sqrt[w] - 1)/n)^2]; 

GaussIncPos[a_,b_,c_,d_,w_,k_,n_,l_][xi_] := 

GaussShapePos[w,k,n,xi] * OutOsc[a,b,c,d,xi]; 

GaussIncNeg[a_,b_,c_,d_,w_,k_,n_,l_][xi_] := 

GaussShapeNeg[w,k,n,xi] * OutOsc[a,b,c,d,xi]; 

GaussRefPos[a_,b_,c_,d_,w_,k_,n_,l_][xi_] := 

RefCoeffPos[xi,l] * GaussShapePos[w,k,n,xi] * OutOsc[a,b,c,d,xi]; 

GaussRefNeg[a_,b_,c_,d_,w_,k_,n_,l_][xi_] := 

RefCoeffNeg[xi,l] * GaussShapeNeg[w,k,n,xi] * OutOsc[a,b,c,d,xi]; 

GaussToPos[a_,b_,c_,d_,w_,k_,n_,l_][xi_] := 

ToCoeffPos[xi,l] * GaussShapePos[w,k,n,xi] * TunOsc[a,b,c,d,xi]; 

GaussToNeg[a_,b_,c_,d_,w_,k_,n_,l_][xi_] := 

ToCoeffNeg[xi,l] * GaussShapeNeg[w,k,n,xi] * TunOsc[a,b,c,d,xi]; 

GaussFroPos[a_,b_,c_,d_,w_,k_,n_,l_][xi_] := 

FroCoeffPos[xi,l] * GaussShapePos[w,k,n,xi] * TunOsc[a,b,c,d,xi]; 

GaussFroNeg[a_,b_,c_,d_,w_,k_,n_,l_][xi_] := 

FroCoeffNeg[xi,l] * GaussShapeNeg[w,k,n,xi] * TunOsc[a,b,c,d,xi]; 

GaussTransPos[a_,b_,c_,d_,w_,k_,n_,l_][xi_] := 

TransCoeffPos[xi,l] * GaussShapePos[w,k,n,xi] * OutOsc[a,b,c,d,xi]; 

GaussTransNeg[a_,b_,c_,d_,w_,k_,n_,l_][xi_] := 

TransCoeffNeg[xi,l] * GaussShapeNeg[w,k,n,xi] * OutOsc[a,b,c,d,xi]; 

(*-- Templates for wave computations --*) 

GaussPassTruncTemp[fpos_, fneg_, x_, t_, w_, k_, n_, l_, opts___Rule] := 





230 




limpos, limneg}, 


(If[limpos DoubleExponential, 






If[limneg >= -1, 0, 

NIntegrate[fneg[-t,-Pi k/Sqrt[w],-x,-Pi k,w,k,n,l][xi], 

{xi,1,-limneg}, 






MaxRecursion->ma]] 


]; 

GaussEvanTruncTemp[f_, x_, t_, w_, k_, n_, l_, opts___Rule] := 

Module[{limpos,limneg, 







limneg = Max[limneg,-1]; 

limpos = Min[limpos,1]; 

NIntegrate[f[-t,Pi k/Sqrt[w],x,-Pi k,w,k,n,l][xi], 

{xi,limneg,limpos}, 






MaxRecursion->ma] GaussConst[w,k,n] 

]; 

(*-- Driver functions for the computation of the wave integrals --*) 

(* Note that for the reflected wave outside and the left-going part 

inside the tunnel spatial coordinate is inverted in the function calls 

because of the reverse motion of the partial waves. *) 

(* Note also that the spatial coordinate for the regions inside and 

behind the barrier is transformed to x-l. *) 

PhiInc[x_, t_, w_, k_, n_?Positive, l_, opts___Rule] := 

GaussPassTruncTemp[GaussIncPos,GaussIncNeg,x,t,w,k,n,l,opts] + 

GaussEvanTruncTemp[GaussIncPos,x,t,w,k,n,l,opts]; 

PhiRef[x_, t_, w_, k_, n_?Positive, l_, opts___Rule] := 

GaussPassTruncTemp[GaussRefPos,GaussRefNeg,-x,t,w,k,n,l,opts] + 

GaussEvanTruncTemp[GaussRefPos,-x,t,w,k,n,l,opts]; 

PhiTo[x_, t_, w_, k_, n_?Positive, l_, opts___Rule] := 

GaussPassTruncTemp[GaussToPos,GaussToNeg,x-l,t,w,k,n,l,opts] + 

GaussEvanTruncTemp[GaussToPos,x-l,t,w,k,n,l,opts]; 

PhiFro[x_, t_, w_, k_, n_?Positive, l_, opts___Rule] := 

GaussPassTruncTemp[GaussFroPos,GaussFroNeg,-x+l,t,w,k,n,l,opts] + 

GaussEvanTruncTemp[GaussFroPos,-x+l,t,w,k,n,l,opts]; 

PhiTrans[x_, t_, w_, k_, n_?Positive, l_, opts___Rule] := 

GaussPassTruncTemp[GaussTransPos,GaussTransNeg,x-l,t,w,k,n,l,opts] + 

GaussEvanTruncTemp[GaussTransPos,x-l,t,w,k,n,l,opts]; 

PhiTun[x_, t_, w_, k_, n_?Positive, l_, opts___Rule] := 

PhiTo[x,t,w,k,n,l,opts] + PhiFro[x,t,w,k,n,l,opts]; 

Phi[x_, t_, w_, k_, n_?Positive, l_, opts___Rule] := 

Which[t < 0, 0, 

x >= l, PhiTrans[x,t,w,k,n,l,opts], 

0 < x && x < l, PhiTun[x,t,w,k,n,l,opts], 

x

SetAttributes[PhiInc, ReadProtected] 

SetAttributes[PhiRef, ReadProtected] 

SetAttributes[PhiTo, ReadProtected] 

SetAttributes[PhiFro, ReadProtected] 

SetAttributes[PhiTun, ReadProtected] 

SetAttributes[PhiTrans, ReadProtected] 

SetAttributes[Phi, ReadProtected] 

(* 

Protect[PhiInc, PhiRef, PhiTo, PhiFro] 

Protect[PhiTun, PhiTrans, Phi] 

*) 

EndPackage[] 

A.4 Electromagnetic waves 


The short package ElMag computes the solutions of the wave equation for a transmission line 

lled with a lossless plasma or, alternatively, the propagation of a wave through an unbounded 

plasma (section 3.7). The boundary condition is given by a cos-like current being switched 

on at t = 0 and switched o at some integral number of periods later. 




(*:Title: ElMag *) 


(*:Keywords: Eletromagnetic Waves, Plasma Waves, Evanescent Waves *) 




(*:Summary: This application specific package contains functions for 

the calculation of wave propagation in one dimensional 

transmission lines filled with a lossless plasma. The model 

is that the wave is excited by a sinusoidal current source 

switched on at t=0. Voltage and current along the line may 

be calculated as a function of time, distance, and the ratio 

of the plasma frequency and the signal frequency. 

*) 

BeginPackage["ElMag`", "OscInt`"] 

CurrStat::usage = 

"CurrStat[x,t,omega] returns the stationary solution of the wave equation 

for the current. See also Curr." 

CurrTrans::usage = 

"CurrTrans[x,t,omega,(opts)] returns the stationary solution of the wave 

equation for the current. The options opts are passed to the quadrature 

function OscInt. For further explanations see also Curr." 

Curr::usage = 

"Curr[x,t,omega,(opts)] gives the complete solution of the wave equation 

for the current in a one dimensional, infinitely long transmission line 

filled with lossless plasma. Excitation of the wave is accomplished by a 

sinusoidal current source at the entrance of the line (x=0) being switched 

on at t=0. Evanescence is presumed, so the signal frequency must be lower 

than the plasma frequency. The excitation frequency is specified by the 

parameter omega which denotes the ratio between signal and plasma frequency, 

hence 0

A.4 Electromagnetic waves 

function OscInt. For further explanations see also Volt." 

Volt::usage = 

"Volt[x,t,omega,(opts)] gives the complete solution of the wave equation 

for the voltage in a one dimensional, infinitely long transmission line 

filled with lossless plasma. Excitation of the wave is accomplished by a 

sinusoidal current source at the entrance of the line (x=0) being switched 

on at t=0. Evanescence is presumed, so the signal frequency must be lower 

than the plasma frequency. The excitation frequency is specified by the 

parameter omega which denotes the ratio between signal and plasma frequency, 

hence 0

(* 

Protect[CurrStat, CurrTrans, Curr, CurrPack] 

Protect[VoltStat, VoltTrans, Volt, VoltPack] 

*) 

EndPackage[] 

A.5 Utilities for displaying points in 3D 


The Mathematica package Tools3D contains a number of functions that are needed for the 

graphical representation of lists of points in R 3 . Such lists are typically the outcome of surface 

computations over a rectangular grid being obtained with commands like Table[fx,y,f(x,y)g, 

fx,xmin,xmax,dxg, fy,ymin,ymax,dyg]. Therefore all functions in this package assume 

that the data is given as two-dimensional lists of points, like 

fffx11,y11,z11g,fx12,y12,z12g, ::: ,fx1n,y1n,z1ngg, 

ffx21,y21,z21g,fx22,y22,z22g, ::: ,fx2n,y2n,z2ngg, 

. 

. 

. 

. 

. 

. 

. 

ffxm1,ym1,zm1g,fxm2,ym2,zm2g, ::: ,fxmn,ymn,zmnggg. 

This matrix must be rectangular, so each sublist must have an equal number of members. 

Some functions check this property with the statement 

Apply[SameQ,Map[Length,t]], 

which rst counts the number of elements in each row of the list t (by mapping the operation 

Length onto each member of the list; note that these members are again lists) and then 

evaluates if all entries in the resulting list are identical. The outcome of this statement is 

boolean | if it is False, error messages are generated. 

The functions of the package have the following goals: 

OrderGrid takes an unsorted, at list of three-dimensional points fxi,yi,zig as input 

and arranges them in the required two-dimensional array of points. To this end, the 

x-values of all points lying on one grid line must be undisturbed such that they can be 

grouped together. 

Link3DList is useful whenever a surface is computed in more than one chunk and the 

results are stored in di erent les. The function requires a list with the lenames, the 

name for the output le, and optionally a path specifying where the les are located. 

The function then checks whether the output le already exists | and if so issues a 

warning | before merging and sorting the les. The function call is Link3DList[fFFileA",FileB", 

::: g,OutFile"]. 

ReSize rearranges a two-dimensional list by taking only every nth element. Depending 

on whether the reduction should be uniform or not, the function can be called with 

ReSize[list,n] or ReSize[list,m,n] for leaving every mth row and nth column. 

234

A.5 Utilities for displaying points in 3D 

Disp3DList displays surfaces consisting of three-dimensional points by calling the function 

ListSurfacePlot3D, which is part of the standard package Graphics`Graphics3D`. 

Disp3DContour generates a contour plot of the list of points. In this case, the regularity 

of the list is checked for the convenience of the user. The reason is that the built-in 

function ListContourPlot would simply not evaluate (without a warning) if the list 

passed to it was not regular. 

Movie takes the surface and returns cuts in parallel to the x-z plane by increasing the 

y-coordinate. If no special value for PlotRange is given, this option is set so as to cover 

the extrema, being invariant for all pictures. The user can override this default with 

another range. Note that in order to allow this, the user's options must be written 

before the default values in the call of the function ListPlot. 

(* Copyright: Copyright 1994, CAD-Div., IGEE, Vienna Univ. of Techn. *) 


(*:Title: 3DTools *) 


(*:Keywords: *) 


(*:Warnings: The Functions defined herein require the 3D-Lists to 

lie on a rectangular x/y grid *) 

(*:Summary: This package contains functions useful for managing and 

displaying lists of 3D points. The reordering function 

OrderGrid is based on the package ListToArray by Jason F. 

Harris. 

*) 

BeginPackage["Tools3D`","Graphics`Graphics3D`"] 

OrderGrid::usage = 

"Takes a list {{x1,y1,z1},{x2,y2,z2}...} and orders it to the 

two-dimensional form {{{x1,y11,z11},{x1,y12,z12},..},{{x2,y21,z21},..},..} 

of 3D-points." 

ReSize::usage = 

"ReSize[list,n] keeps only every nth row and column of a two-dimensional 

list. ReSize[list,{m,n}] takes every mth row and nth column." 

Link3DList::usage = 

"Link3DList[Names,Output,(Path)] needs a list of filenames, a name for 

the output file and an optional path where to look for the files. The files 

must be a two-dimensional list of 3D-points, the output file has the same 

form (generated by OrderGrid)." 

Disp3DList::usage = 

"Display a surface consisting of points {x,y,z} over a rectangular 

x/y grid (two-dimensional list of 3D-points)" 

Disp3DContour::usage = 

"Disp3DContour[list,(opts)] produces a contour plot of a list of points {x,y,z} 

over a rectangular and equidistant x/y grid (two-dimensional list of 3D-points)" 

Movie::usage = 

"Movie[list,(opts)] produces an animation of a rectangular list of points 

{x,y,z} with {x,z}-Frames and incremented y-value" 

Link3DList::fewlists = "At least two files needed"; 

Link3DList::reggrid = "Warning: Grid not regular!"; 

Disp3DContour::reggrid = "Grid not regular!"; 


OrderGrid[a_List] := Module[{xvals,asort}, 

asort = Sort[a]; 

xvals = Union[Flatten[Cases[asort,{x_,y_,z_}->x]]]; 

Map[Cases[asort,{#,y_,z_}]&, xvals]]; 

TakeOther[l_List,n_] := Map[First[#]&,Partition[l,n]]; 

ReSize[l_List,{n_,m_}] := TakeOther[Map[TakeOther[#,m]&,l],n]; 

235

ReSize[l_List,n_] := ReSize[l,{n,n}]; 

Link3DList[names_,outname_,path__:""] := Module[{dir,mergelist={}}, 

dir = FileNames[names,path]; 

If[Length[dir] < 2, Message[Link3DList::fewlists], 

If[FileNames[outname,path] == {} || 

InputString["Output file exists! Overwrite (y/n)?"] == "y", 

Print["Merging ",Length[dir]," lists"]; 

mergelist = OrderGrid[Flatten[Map[Get,dir],2]]; 

If[!Apply[SameQ,Map[Length,mergelist]], 

Message[Link3DList::reggrid]]; 

If[path != "", SetDirectory[path]]; 

Put[mergelist,outname]; 

If[path != "", ResetDirectory[]] 

]]; 

]; 

Disp3DList[t_List, opts___] := ListSurfacePlot3D[t,opts,BoxRatios->{1,1,0.4}, 

Axes->True]; 

Disp3DContour[t_List, opts___] := If[!Apply[SameQ,Map[Length,t]], 

Message[Disp3DContour::reggrid], 

ListContourPlot[Transpose[t]/.{x_,y_,z_}:>z, 

MeshRange->{{t[[1,1,1]],t[[Length[t],1,1]]}, 

{t[[1,1,2]],t[[1,Length[t[[1]]],2]]}},opts]]; 

Movie[list_List, opts___] := Module[{minz,maxz,optlist}, 

{minz,maxz} = Apply[{Min[#],Max[#]}&, 

{Transpose[Flatten[list,1]][[3]]}]; 

Map[ListPlot[#/.{x_,y_,z_}:>{x,z}, 

opts,PlotJoined->True,PlotRange->{All,{minz,maxz}}]&, 

Transpose[list]] 

]; 

End[] 

SetAttributes[OrderGrid, ReadProtected] 

SetAttributes[ReSize, ReadProtected] 

SetAttributes[Link3DList, ReadProtected] 

SetAttributes[Movie, ReadProtected] 

Protect[OrderGrid, ReSize, Link3DList, Movie] 

EndPackage[] 

236 


Bibliography 

Bibliography 

[1] James R. Newman, editor. The world of mathematics. Simon and Schuster, New York, 

1956. 

[2] Leon Brioullin. Wave Propagation and Group Velocity. Academic Press, New York, 

1960. 

[3] Gerald B. Whitham. Linear and nonlinear waves. John Wiley & Sons, New York, 1976. 

[4] Lord Rayleigh. The Theory of Sound. Dover Publications, New York, 1945, rst published 

in 1877. 

[5] Lord Rayleigh. On progressive waves. In The Theory of Sound, pages 475{480. Dover 

Publications, New York, 1945, rst published in 1877. 

[6] Julius Adams Stratton. Electromagnetic Theory. McGraw-Hill, New York, 1941. 

[7] M. J. Lighthill. Group Velocity. J. Inst. Maths Applics, 1:1{28, 1965. 

[8] Richard L. Smith. The Velocities of Light. American Journal of Physics, 38(8):978{984, 

1970. 

[9] W. V. Prestwich. Precise de nition of group velocity. American Journal of Physics, 

43(9):832{833, 1975. 

[10] Robert Vichnevetsky. Group velocity in general waveforms. Mathematics and Computers 

in Simulation, 27:327{331, 1985. 

[11] John David Jackson. Klassische Elektrodynamik. Walter de Gruyter, Berlin, second 

edition, 1982. 

[12] A. Sommerfeld. Uber die Fortp anzung des Lichtes in dispergierenden Medien. Annalen 

der Physik, 44(10):177{202, 1914. 

[13] E. O. Schulz-DuBois. Energy Transport Velocity of Electromagnetic Propagation in 

Dispersive Media. Proceedings of the IEEE, 57(10):1748{1757, 1969. 

[14] G. R. Baldock and T. Bridgeman. The Mathematical Theory of Wave Motion. Ellis 

Horwood, Chichester, 1981. 

237

Bibliography 

[15] Simon Ramo, John R. Whinnery, and Theodore van Duzer. Fields and Waves in Communication 

Electronics. John Wiley & Sons, New York, 1984. 

[16] Gerhard Piefke. Feldtheorie. Bibliographisches Institut, Mannheim, 1971. 

[17] S. Zhu, A. W. Yu, D. Hawley, and R. Roy. Frustrated total internal re ection: A 

demonstration and review. American Journal of Physics, 54:601{607, 1986. 

[18] Aephraim M. Steinberg and Raymond Y. Chiao. Tunneling delay times in one and two 

dimensions. Physical Review A, 49(5):3283{3295, 1994. 

[19] L. Brillouin. Uber die Fortp anzung des Lichtes in dispergierenden Medien. Annalen 

der Physik, 44(10):203{240, 1914. 

[20] James R. Wait. Propagation of Pulses in Dispersive Media. Radio Science J.Res. NBS, 

69D(11):1387{1401, 1965. 

[21] Hans Georg Baerwald. Uber die Fortp anzung von Signalen in dispergierenden Systemen. 

Annalen der Physik, 7:731{760, 1930. 

[22] Hans Georg Baerwald. Uber die Fortp anzung von Signalen in dispergierenden Systemen. 

Annalen der Physik, 6:295{369, 1930. 

[23] Dennis B. Trizna and Thomas A. Weber. Brillouin revisited: Signal velocity de nition 

for pulse propagation in a medium with resonant anomalous dispersion. Radio Science, 

17(5):1169{1180, 1982. 

[24] Kurt Edmund Oughstun and George C. Sherman. Propagation of electromagnetic pulses 

in a linear dispersive medium with absorption (the Lorentz medium). J. Opt. Soc. Am. 

B, 5(4):817{849, 1988. 

[25] Philippe Wyns, Daniel P. Foty, and Kurt E. Oughstun. Numerical analysis of the 

precursor elds in linear dispersive pulse propagation. J. Opt. Soc. Am. A, 6(9):1421{ 

1429, 1989. 

[26] Kurt E. Oughstun, Philippe Wyns, and Daniel P. Foty. Numerical determination of the 

signal velocity in dispersive pulse propagation. J. Opt. Soc. Am. A, 6(9):1430{1440, 

1989. 

[27] Kurt Edmund Oughstun and George C. Sherman. Uniform asymptotic description 

of electromagnetic pulse propagation of electromagnetic pulses in a linear dispersive 

medium with absorption (the Lorentz medium). J. Opt. Soc. Am. A, 6(9):1394{1420, 

1989. 

[28] Kurt Edmund Oughstun and George C. Sherman. Uniform asymptotic description of 

ultrashort rectangular optical pulse propagation in a linear, causally dispersive medium. 

Physical Review A, 41(11):6090{6113, 1990. 

238

Bibliography 

[29] Kurt Edmund Oughstun. Pulse Propagation in a Linear, Causally Dispersive medium. 

Proceedings of the IEEE, 79(10):1379{1390, 1991. 

[30] Constantinos M. Balictsis and Kurt Edmund Oughstun. Uniform asymptotic description 

of ultrashort Gaussian-pulse propagation in a causal, dispersive medium. Physical 

Review E, 47(5):3645{3669, 1990. 

[31] Shioupyn Shen and Kurt Edmund Oughstun. Dispersive pulse propagation in a doubleresonance 

Lorentz medium. J. Opt. Soc. Am. B, 6(5):948{963, 1989. 

[32] Francesco Mainardi. Energy velocity for hyperbolic dispersive waves. Wave Motion, 

9:201{208, 1985. 

[33] R. Loudon. The propagation of electromagnetic energy through an absorbing medium. 

J. Phys. A: Gen. Phys., 3:233{245, 1970. 

[34] T. H. Havelock. The Propagation of Disturbances in Dispersive Media, volume 17 of 

Cambridge Tract in Math. and Phys. Cambridge University Press, London, 1914. 

[35] M. A. Biot. General Theorems on the Equivalence of Group Velocity and Energy 

Transport. Physical Review, 105(4):1129{1137, 1957. 

[36] Donovan V. Geppert. Energy-Transport Velocity in Electromagnetic Waves. Proceedings 

of the IEEE, 53(11):1790, 1965. 

[37] Fritz Borgnis. Zur elektromagnetischen Energiedichte in Medien mit Dispersion. 

Zeitschrift fur Physik, 159:1{6, 1960. 

[38] Fritz Borgnis. Elektrische Wellen auf Verzogerungsleitungen. Elektrotechnische Zeitschrift 

A, 79(11):383{385, 1958. 

[39] Werner Kleen and Klaus Poschl. Gruppengeschwindigkeit und Energiegeschwindigkeit 

elektromagnetischer Wellen in dampfungsfreien Leitungen. Archiv der Elektrischen 

Ubertragung, 12(12):567{569, 1958. 

[40] Werner Kleen. Einfuhrung in die Mikrowellen-Elektronik. S. Hirzel Verlag, Stuttgart, 

1952. 

[41] Werner Kleen, Johannes Labus, and Klaus Poschl. Raumladungswellen. Ergebnisse der 

exakten Naturwissenschaften, 29:208{274, 1956. 

[42] F. Borgnis. Die Fortp anzungsgeschwindigkeit der Energie monochromatischer elektromagnetischer 

Wellen in dielektrischen Medien. Zeitschrift fur Physik, 117:642{650, 

1941. 

[43] L. J. F. Broer. On the propagation of energy in linear conservative waves. Applied 

scienti c Research, A 2:329{344, 1950. 

239

Bibliography 

[44] George C. Sherman and Kurt Edmund Oughstun. Description of Pulse Dynamics in 

Lorentz Media in Terms of the Energy Velocity and Attenuation of Time-Harmonic 

Waves. Physical Review Letters, 47(20):1451{1454, 1981. 

[45] Kurt Edmund Oughstun and Shioupyn Shen. Velocity of energy transport for a timeharmonic 

eld in a multiple-resonance Lorentz medium. J. Opt. Soc. Am. B, 5(11):2395{ 

2398, 1988. 

[46] George C. Sherman and Kurt Edmund Oughstun. Energy-velocity description of pulse 

propagation in absorbing, dispersive dielectrics. J. Opt. Soc. Am. B, 12(2):229{247, 

1995. 

[47] L. C. Baird. Moments of a Wave Packet. American Journal of Physics, 40:327{329, 

1972. 

[48] Dan G. Anderson and Jan I. H. Askne. Wave Packets in Strongly Dispersive Media. 

Proceedings of the IEEE, 62(11):1518{1523, 1974. 

[49] D. Anderson, J. Askne, and M. Lisak. Wave packets in an absorptive and strongly 

dispersive medium. Physical Review A, 12(4):1546{1552, 1975. 

[50] S. C. Bloch. Eighth velocity of light. American Journal of Physics, 45(6):538{549, 1977. 

[51] Not only is this a remarkable anticipation of data transmission at the speed of 

light(ning), it is even more astonishing in that it was written long before the existence 

of electromagnetic waves was proved by Hertz in 1886. Stifter was, however, very 

well aware of electromagnetism, and there are many passages in his oeuvre that underline 

this interest [150]. Note also the subtle description in this citation of what makes 

up our contemporary `global village'. 

[52] Milton A. Rothman. Things that go faster than light. Scienti c American, pages 

142{152, July 1960. 

[53] A. Enders and G. Nimtz. Evanescent-mode propagation and quantum tunneling. Physical 

Review E, 48(1):632{634, 1993. 

[54] A. Enders and G. Nimtz. Photonic-tunneling experiments. Physical Review B, 

47(15):9605{9609, 1993. 

[55] A. Ranfagni, D. Mugnai, P. Fabeni, and G. P. Pazzi. Optical-tunneling time measures: 

a microwave model. Physica B, 175:283{286, 1991. 

[56] G. Nimtz. Instantanes Tunneln. Phys. Bl., 49(12):1119{1120, 1993. 

[57] P. Thoma and T. Weiland. Wie real ist das Instantane Tunneln? Phys. Bl., 50(4):313, 

360{361, 1994. 

[58] A. M. Steinberg, P. G. Kwiat, and R. Y. Chiao. Measurement of the Single-Photon 

Tunneling Time. Physical Review Letters, 71(5):708{711, 1993. 

240

Bibliography 

[59] Ch. Spielmann, R. Szipocs, A. Stingl, and F. Krausz. Tunneling of Optical Pulses 

through Photonic Band Gaps. Physical Review Letters, 73(17):2308{2311, 1994. 

[60] A. Ranfagni, P. Fabeni, G. P. Pazzi, and D. Mugnai. Anomalous pulse delay in microwave 

propagation: A plausible connection to the tunneling time. Physical Review E, 

48(2):1453{1460, 1993. 

[61] Th. Martin and R. Landauer. Time delay ofevanescent electromagnetic waves and the 

analogy to particle tunneling. Physical Review A, 45(4):2611{2617, 1992. 

[62] J. M. Deutch and F. E. Low. Barrier Penetration and Superluminal Velocity. Annals 

of Physics, 228:184{202, 1993. 

[63] Klaus Hass and Paul Busch. Causality of superluminal barrier traversal. Physics Letters 

A, 185:9{13, 1994. 

[64] Mark Ya. Azbel'. Superluminal Velocity, Tunneling Traversal Time and Causality. Solid 

State Communications, 91(6):439{441, 1994. 

[65] Thorsten Emig. Propagation of an electromagnetic pulse through a waveguide with a 

barrier: A time domain solution within classical electrodynamics. Physical Review E, 

54(5):5780{5787, 1993. 

[66] Eric L. Bolda, Raymond Y. Chiao, and John C. Garrison. Two theorems for the group 

velocity in dispersive media. Physical Review A, 48(5):3890{3893, 1993. 

[67] John S. Toll. Causality and the Dispersion Relation: Logical Foundations. Physical 

Review, 104(6):1760{1770, 1956. 

[68] Aephraim M. Steinberg. Comment on \Photonic tunneling times". J. Phys. I France, 

4:1813{1816, 1994. 

[69] Raymond Y. Chiao, Paul G. Kwiat, and Aephraim Steinberg. Faster Than Light? 

Scienti c American, pages 52{60, August 1993. 

[70] Rolf Landauer. Light faster than light? Nature, 365:692{693, 1993. 

[71] Gunter Nimtz and Winfried Heitmann. Superluminal Photonic Tunneling and Quantum 

Electronics. Progress in Quantum Electronics, 21(2):81{108, 1997. 

[72] Gunter Nimtz. Schneller als das Licht? Physik in unserer Zeit, 28(5):214{218, 1997. 

[73] Raymond Y. Chiao. Superluminal (but causal) propagation of wave packets in transparent 

media with inverted atomic populations. Physical Review A, 48(1):R34{R37, 

1993. 

[74] Eric L. Bolda, John C. Garrison, and Raymond Y. Chiao. Optical pulse propagation at 

negative group velocities due to a nearby gain line. Physical Review A, 49(4):2938{2947, 

1994. 

241

Bibliography 

[75] Aephraim M. Steinberg and Raymond Y. Chiao. Dispersionless, highly superluminal 

propagation in a medium with a gain doublet. Physical Review A, 49(3):2071{2075, 

1994. 

[76] Wang Yun-ping and Zhang Dian-lin. Reshaping, path uncertainty, and superluminal 

travelling. Physical Review A, 52(4):2597{2600, 1995. 

[77] Vijay K. Garg and Joseph E. Wilkes, editors. Wireless and personal communication 

systems. Prentice-Hall, Upper Saddle River, NJ, 1996. 

[78] W. D. Rummler, R. P. Coutts, and M. Liniger. Multipath Fading Channel Models for 

Microwave Digital Radio. IEEE Communications Magazine, 24(11):30{42, 1986. 

[79] Rolf Landauer. Barrier traversal time. Nature, 341:567{568, 1989. 

[80] Thomas E. Hartman. Tunneling of a Wave Packet. Journal of Applied Physics, 

33(12):3427{3433, 1962. 

[81] J. R. Fletcher. Time delay in tunnelling through a potential barrier. J. Phys. C: Solid 

State Phys., 18:L55{L59, 1985. 

[82] K. K. Thornber, Thomas C. McGill, and C. A. Mead. The Tunneling Time of an 

Electron. Journal of Applied Physics, 38(5):2384{2385, 1967. 

[83] S. Collins, David Lowe, and J. R. Barker. The quantum mechanical tunnelling time 

problem|revisited. J. Phys. C: Solid State Phys., 20:6213{6232, 1987. 

[84] John R. Barker. Quantum Theory of Hot Electron Tunnelling in Microstructures. Physica, 

134B:22{31, 1985. 

[85] E. H. Hauge and J. A. St vneng. Tunneling times: a critical review. Reviews of Modern 

Physics, 61(4):917{936, 1989. 

[86] R. Landauer. Barrier interaction time in tunneling. Reviews of Modern Physics, 

66(1):217{228, 1994. 

[87] J. R. Barker, S. Brouard, V. Gasparian, G. Iannaccone, A. P. Jauho, C. R. Leavens, 

J. G. Muga, R. Sala, and D. Sokolovski. First European Workshop on Tunnelling Times. 

Phantoms Newsletter, pages 5{11, October 1994. 

[88] M. Buttiker. Larmor precession and the traversal time for tunneling. Physical Review 

B, 27(10):6178{6188, 1983. 

[89] Aephraim M. Steinberg. How Much Time Does a Tunneling Particle Spend in the 

Barrier Region? Physical Review Letters, 74(13):2405{2409, 1995. 

[90] Aephraim M. Steinberg. Conditional probabilities in quantum theory and the tunnelingtime 

controversy. Physical Review A, 52(1):32{42, 1995. 

242

Bibliography 

[91] C. R. Leavens and G. C. Aers. Bohm Trajectories and the Tunneling Time Problem. 

In Roland Wiesendanger and Hans-Joachim Guntherodt, editors, Scanning Tunneling 

Microscopy III, pages 105{140. Springer, Berlin, 1996. 

[92] Fritz Paschke. Modellbildung. lecture notes, 1993. 

[93] Roger F. Harrington. Time-harmonic Electromagnetic Fields. McGraw-Hill, New York, 

1961. 

[94] Paul Lorrain and Dale R. Corson. Electromagnetic Fields and Waves. W. H. Freeman, 

San Francisco, 1970. 

[95] Arnulf Weinzinger. Die Exponentialleitung als Impedanztransformator und ihre Realisierung 

als Microstrip-Leitung. Diploma thesis, Technische Hochschule Wien, 1971. 

[96] Charles M. Knop. Pulsed Electromagnetic Wave Propagation in Dispersive Media. 

IEEE Transactions on Antennas and Propagation, 12:494{496, 1964. 

[97] Richard E. Haskell and Carl T. Case. Transient Signal Propagation in Lossless, Isotropic 

Plasma. IEEE Transactions on Antennas and Propagation, 15(3):458{464, 1967. 

[98] R. E. McIntosh and Alfonso Malaga. Time dispersion of electromagnetic pulses by the 

ionosphere. Radio Science, 15(3):645{654, 1980. 

[99] Junichi Takanaka, Suguru Yamagishi, Hiroaki Taki, Akihiro Morimoto, and Tetsuro 

Kobayashi. Guiding of a one-dimensional optical beam with nanometer diameter. Optics 

Letters, 22(7):475{477, 1997. 

[100] Fritz Paschke. Zum Verhalten der Streifenleitung bei Submikrometerabstanden. lecture 

notes, 1997. 

[101] G. Diener. Superluminal group velocities and information transfer. Physics Letters A, 

223:327{331, 1996. 

[102] W. Heitmann and G. Nimtz. On causality proofs of superluminal barrier traversal of 

frequency band limited wave packets. Physics Letters A, 196:154{158, 1994. 

[103] Armin Hermann, editor. Albert Einstein / Arnold Sommerfeld, Briefwechsel. Schwabe 

& Co, Basel, 1968. 

[104] Marco A. de Moura and Douglas F. de Albuquerque. Remarks on the traversal time in 

a tunneling process. Solid State Communications, 74(5):353{354, 1990. 

[105] Thomas Weiland. Die Diskretisierung der Maxwell-Gleichungen. Phys. Bl., 42(7):191{ 

201, 1986. 

[106] Richard Albanese, John Penn, and Richard Medina. Short-rise-time microwave pulse 

propagation through dispersive biological media. J. Opt. Soc. Am. A, 6(9):1441{1446, 

1989. 

243

Bibliography 

[107] Abraham Goldberg, Harry M. Schey, and Judah L. Schwartz. Computer-Generated 

Motion Pictures of One-Dimensional Quantum-Mechanical Transmission and Re ection 

Phenomena. American Journal of Physics, 35(3):177{186, 1967. 

[108] S. Collins, David Lowe, and J. R. Barker. A dynamic analysis of resonant tunnelling. 

J. Phys. C: Solid State Phys., 20:6233{6243, 1987. 

[109] Terry Robb. Schroedinger's Equation. Mathematica notebook available on MathSource, 

1994. URL:http://www.mathsource.com/cgi-bin/MathSource/Enhancements/ 

Interfacing/InterCall/0204-657. 

[110] John R. Hiller, Ian D. Johnston, and Daniel F. Styer. Quantum Mechanics Simulations, 

The Consortium for Upper-Level Physics Software. John Wiley & Sons, New York, 1995. 

[111] John R. Merrill. Waves in Dispersive Media: Another Use of Computers in Introductory 

Physics. American Journal of Physics, 39:539{544, 1971. 

[112] Milton Abramowitz. On the Practical Evaluation of Integrals. SIAM Journal of Applied 

Mathematics, 2:20{35, 1954. 

[113] Philip J. Davis and Philip Rabinowitz. Methods of numerical integration. Academic 

Press, Inc., Orlando, Florida, second edition, 1984. 

[114] David Levin. Procedures for computing one and two dimensional integrals of functions 

with rapid irregular oscillations. Mathematics of Computation, 38(158):531{538, 1982. 

[115] G. A. Evans. Two robust methods for irregular oscillatory integrals over a nite range. 

Applied Numerical Mathematics, 14(4):383{395, 1994. 

[116] L. N. G. Filon. On a quadrature formula for trigonometric integrals. Proc. Roy. Soc. 

Edinburgh, 49:38{47, 1928. 

[117] Ulf Torsten Ehrenmark. A three-point formula for numerical quadrature of oscillatory 

integrands with variable frequency. Journal of Computational and Applied Mathematics, 

21:87{99, 1988. 

[118] Pei-Cheng Xu and A. K. Mal. An adaptive integration scheme for irregularly oscillatory 

functions. Wave Motion, 7:235{243, 1985. 

[119] K. T. R. Davies. Complex-plane methods for evaluating integrals with highly oscillatory 

integrands. Journal of Computational Physics, 80(2):498{505, 1989. 

[120] Philip Rabinowitz. Extrapolation methods in numerical integration. Numerical Algorithms, 

3:17{28, 1992. 

[121] J. N. Lyness. Integrating some in nite oscillating tails. Journal of Computational and 

Applied Mathematics, 12 & 13:109{117, 1985. 

[122] James Lyness and Gwendolen Hines. To Integrate Some In nite Oscillating Tails. ACM 

Transactions on Mathematical Software, 12(1):24{25, 1986. 

244

Bibliography 

[123] T. O. Espelid and K. J. Overholt. DQAINF: An algorithm for automatic integration of 

in nite oscillating tails. Numerical Algorithms, 8(2):83{101, 1994. 

[124] Q. Haider and L. C. Liu. Fourier or Bessel transformations of highly oscillatory functions. 

J. Phys. A: Math. Gen., 25(24):6755{6760, 1992. 

[125] Avram Sidi. Extrapolation methods for oscillatory in nite integrals. J. Inst. Maths 

Applics, 26:1{20, 1980. 

[126] Avram Sidi. The Numerical Evaluation of Very Oscillatory In nite Integrals by Extrapolation. 

Mathematics of Computation, 38(158):517{529, 1982. 

[127] Takemitsu Hasegawa and Tatsuo Torii. Inde nite integration of oscillatory functions by 

the Chebyshev series expansion. Journal of Computational and Applied Mathematics, 

17:21{29, 1987. 

[128] Avram Sidi. A user-friendly extrapolation method for oscillatory in nite integrals. 

Mathematics of Computation, 51(183):249{266, 1988. 

[129] Ulf Torsten Ehrenmark. A note on an extension of extrapolative techniques for a class 

of in nite oscillatory integrals. BIT, 30(1):152{155, 1990. 

[130] Bui Doan Khanh. A numerical evaluation of very oscillatory in nite integrals by asymptotics. 

Applied Mathematics and Computation, 40(2):117{124, 1990. 

[131] Christoph Uberhuber. Computer-Numerik. Springer, Berlin, 1995. 

[132] R. Piessens, E. de Doncker-Kapenga, C. W. Uberhuber, and D. K. Kahaner. QUAD- 

PACK, A Subroutine Package for Automatic Integration, volume 1 of Series in Computational 

Math. Springer, Berlin, 1983. 

[133] P. Wynn. On a device for computing the em (Sn) transformation. Math. Comp., 10:91{ 

96, 1956. 

[134] Numerical Algorithms Group, NAG O ce, Oxford. NAG Libary, Mark 17, 1996. URL: 

http://www.nag.co.uk. 

[135] International Mathematical and Statistical Libraries, Inc., Houston, TX. IMSL Library, 

V3.0, 1996. URL: http://www.vni.com/adt.dir/adt.html. 

[136] John P. Boyd. A Lag-Averaged Genralization of Euler's Method for Accelerating Series. 

Applied Mathematics and Computation, 72:143{166, 1995. 

[137] Claude Brezinski and Michela Redivo Zaglia. Extrapolation methods: theory and practice. 

North-Holland, Amsterdam, 1991. 

[138] C. Brezinski and M. Redivo-Zaglia. Extrapolation methods. Applied Numerical Mathematics, 

15(2):123{131, 1994. 

245

Bibliography 

[139] R. Riedel. Einfuhrung in die Theorie der Konvergenzbeschleunigung fur Reihen. Wissenschaftliche 

Beitrage, Martin-Luther-Universitat Halle-Wittenburg, 46:43{61, 1988. 

[140] Stephen Wolfram. Mathematica, A System for Doing Mathematics by Computer. 

Addison-Wesley, Reading, MA, second edition, 1991. 

[141] David A. Smith and William F. Ford. Acceleration of linear and logarithmic convergence. 

J. SIAM Numer. Anal., 16(2):223{240, 1979. 

[142] G. E. Bell and G. M. Phillips. Aitken acceleration of some alternating series. BIT, 

24:70{77, 1984. 

[143] P. Wynn. On the convergence and stability of the epsilon algorithm. J. SIAM Numer. 

Anal., 3(1):91{122, 1966. 

[144] Jerry Keiper. Numerical Computation II. In Reprint from The Mathematica Conference, 

Boston, MA, June 1992, URL: http://www.mathsource.com/cgi-bin/MathSource/ 

Enhancements/Numerical/0204-028, 1992. Wolfram Research, Inc. 

[145] Jerry Keiper. The N functions of Mathematica. In Reprint from The Mathematica 

Conference, Boston, MA, June 1992, URL: http://www.wolfram.com/cgi-bin/ 

MathSource/General/Tutorials/Numerical/0203-948, 1992. Wolfram Research, Inc. 

[146] Roman Maeder. Programming in Mathematica. Addison-Wesley, Reading, MA, second 

edition, 1991. 

[147] William T. Shaw and Jason Tigg. Applied Mathematica, Getting Started, Getting It 

Done. Addison-Wesley, Reading, MA, 1994. 

[148] Milton Abramowitz and Irene A. Stegun. Handbook of Mathematical Functions. Dover 

Publications, New York, ninth edition, 1972. 

[149] Wolfgang Grobner and Nikolaus Hofreiter. Integraltafel. Springer, Wien, fourth edition, 

1966. 

[150] Fritz Paschke. Walter Schottkys erste Arbeitsgebiete. Siemens Forsch.- und Entwickl.- 

Ber., 15(6):287{290, 1986. 

246

Index 

Index 

absorption, 7, 9 

accuracy 

within Mathematica, 141, 156 

approximation 

of integrands, 125 

approximation error, 137, 147, 150 

estimation, 156, 178 

asymptotic expansion, 127, 128, 145, 178 

Buttiker time, 97 

band-limited signal, 25, 71, 100 

Bohm trajectory, 30, 100 

causality, 21, 71, 107 

and dispersion, 24 

and Gaussian pulses, 24 

centroid, 6, 23, 28 

centrovelocity, 18 

Clenshaw-Curtis rule, 125, 128 

continuity equation, 15 

convergence 

linear, 130 

convergence acceleration, 126, 129 

convolution integral, 70 

correlation velocity, 18 

Crank-Nicholson algorithm, 121 

cuto frequency 

optical, 63 

plasma, 50, 66 

quantum mechanical, 73, 95 

wave guide, 22, 61 

2 algorithm, 130, 149 

reliability, 135 

delay line, 15, 21, 35 

dispersion, 4, 7 

and causality, 24 

247 

anomalous, 7, 9, 16 

dispersion relation, 8, 24 

at negative frequencies, 65 

at negative wavenumbers, 74, 192 

delay line, 36 

exponential line, 46 

linear approximation, 5, 68 

Lorentz medium, 11 

plasma, 41, 51, 64 


transmission line, 33 

wave guide, 21, 61 

distortion, 5, 68 

double exponential rule, 141, 145, 171 

dwell time, 29, 96, 97 

e ective, 98 

-algorithm, 128, 130, 149 

reliability, 135 

energy 

propagated, 33, 38, 46 

stored, 15, 33, 36, 38, 46 

energy velocity, 14, 22, 29, 98 

connection to group velocity, 16 

e ective, 42 

electromagnetic wave, 15 

in the pass band, 41, 44, 47 

in the stop band, 42, 44, 47 

negative, 25 

of a current pulse, 39 

terminated transmission line, 38 

envelope, 5, 18, 70 

error 

absolute vs. relative, 142, 158 

Euler transformation, 128, 130, 135 

evanescence, 10

in a plasma, 41 

in wave guides, 22, 61 


exponential line, 46 

extrapolation 

adaptive accuracy control, 155, 172 

increasing the accuracy, 147, 152 

Mathematica implementation, 149 

of sequences, 125 

performance comparison, 150 

Fast Fourier Transform, 18, 120 

Feynman path, 30 

ux, 29, 98 

Fourier integral, 4, 11, 50, 64, 73, 211 

approximate solutions, 119 

numerical evaluation, 120 

free particle, 81, 96, 114 

Gauss rule, 128 

Gauss-Kronrod rule, 141, 145 

Gaussian pulse, 18, 64 

Green's function, 71 

group delay, 28 

group velocity, 4, 8, 22, 28, 68, 96, 102, 112 

abnormal, 24 

connection to phase velocity, 21 

in a plasma, 42, 54 

kinematic derivation, 6 

transmission line, 33 

high-pass lter e ect, 57, 102, 117 

index of refraction, 8, 24 

inductive wall, 36 

information transmission, 24, 70 

ionosphere 

dispersive e ects, 50 

Laplace transform, 120 

Larmor time, 97 

light cone, 71 

Lorentz medium, 10, 24 

mature dispersion, 12 

248 

modulation, 4, 48 

monochromatic wave, 3, 27, 96 

multipath propagation, 25 

on-o -keying, 49 

oscillating integrand, 125, 141 

Index 

partial sums 

sequence of, 133, 138, 146 

partition 

asymptotic, 145, 175 

of integrals, 126, 133 

starting point, 138, 168 

zeros vs. extrema, 135, 146 

phase time, 28, 97, 102, 115 

phase velocity, 3, 17, 34, 60, 68 

connection to group velocity, 21 

plasma 

lossless, 41, 48, 64 

plasma frequency, 41, 54 

precision 

arbitrary, 156 

within Mathematica, 141, 156 

precursor, 11, 17 

probability density, 27, 79 

pulse peak, 24 

temporal vs. spatial, 18, 102, 104 

trajectory, 28, 88, 100, 107, 112 

pulse reshaping, 25, 66, 71, 90, 112, 115 

quadrature, 125 

computer routines, 127 

performance comparison, 141 

quadrature module, 214 

application example, 197, 221 

Mathematica implementation, 161 

options, 162, 167 

partition strategy, 162, 170 

program structure, 164 

test examples, 181 

test of application example, 205 

quantum clock, 30, 97 

re ection coe cient, 29, 73, 192 

resonance, 7

Index 

saddle point integration, 11, 119 

scattering 

examples, 81, 221 

vs. escaping, 28 

Schrodinger equation, 27, 212 

time-independent, 27, 73 

semi-classical time, 96 

sequence 

alternating, 126, 129, 133 

speed of convergence, 137 

transformation, 129 

Shanks transformation, 130 

signal 

small-band, 6, 28, 96 

signal velocity, 10 

for Gaussian pulses, 14 

for rectangular pulses, 13 

measurement of, 13 

Simpson rule, 125 

spectrum 

broad, 49, 85 

energy and momentum, 76 

Gaussian pulse, 64 

Gaussian wave packet, 78 

narrow, 6, 16, 22, 92, 112, 116 

rectangular wave packet, 76 

triangular wave packet, 77 

wave number vs. frequency, 51 

square barrier, 27, 93, 121 

stationary phase 

method of, 11, 16, 28, 140 

step barrier, 73, 81, 122, 191 

physical dimensions, 84 

subdivision 

adaptive, 128, 141 

e ect on convergence acceleration, 133 

superluminal velocity, 21, 70 

experiments, 22 

TE mode, 60 

TEM wave, 35, 41, 49, 60, 63 

termination, 37 

total internal re ection, 10 

transfer function, 65, 70 

249 

transmission coe cient, 29, 73, 95, 104, 192 

transmission line, 17, 32 

characteristic impedance, 33, 42, 44, 46 

dispersion-free, 34 

equivalence to a wave guide, 44 

equivalent circuit, 32, 49 

frequency dependence of losses, 34 

inhomogeneous, 46 

trapezoidal rule, 121, 125, 128, 141 

travelling-wave tube, 16 

truncation 

of integrals, 125, 196 

tunnel e ect, 21, 26 

analogy to wave guides, 26, 117 

examples, 100, 229 

tunnelling time, 28, 96, 112 

negative, 107, 116 

uncertainty principle, 49 

W -transformation, 126, 147 

wave equation, 3 

wave front velocity, 11, 12, 24, 34, 41, 52, 

57, 63, 71 

wave function 

normalisation, 27 

wave guide, 7, 10, 21, 44, 60 

wave number, 3, 8, 50, 54 

quantum mechanical, 27 

wave packet, 4, 16, 76, 129, 149 

centre of gravity, 6, 17, 23 

Gaussian, 88, 96, 100, 121, 196 

moments of a, 18 

rectangular, 81, 195 

triangular, 85, 122, 193

250

Curriculum vitae 

Dipl.-Ing. Thilo Sauter 

3. 9. 1967 geboren in Biberach/Ri , Deutschland 

1973 { 1977 Besuch der Volksschule in Biberach/Ri 

September 1977 Ubersiedlung nach Osterreich 

1977 { 1979 Besuch der Hauptschule Mieming, Tirol 

1979 { 1985 Besuch des realistischen Bundesymnasiums St. Johann/Pongau 

3. 6. 1985 Matura mit Auszeichnung 

Oktober 1985 Beginn des Studiums der Elektrotechnik, Studienzweig Industrielle Elektronik 

und Regelungstechnik an der Technischen Universitat Wien 

19. 3. 1992 Abschlu des Studiums mit Auszeichnung, Diplomarbeit XR-III-Bus 

und XR-III-Bus-Controller uber die Entwicklung eines fehlertoleranten 

Sensor-Aktor-Feldbusses 

1992 { 1996 Tatigkeit als Vertragsassistent am Institut fur Allgemeine Elektrotechnik 

und Elektronik der TU Wien 

Aufbau und Betreuung von Laborubungen zu den Themen Aufbau von 

Operationsverstarkern sowie Parameterextraktion zur Modellierung von 

MOS-Transistoren 

1992 { 1993 Realisierung eines fehlertoleranten Feldbus-Prototyps auf der Basis von 

programmierbaren Logikbausteinen 

ab 1993 Durchfuhrung von Industrie- und Forschungsprojekten auf den Gebieten 

Entwurf gedruckter Schaltungen 

Entwurf programmierbarer integrierter Schaltungen 

Codierung digitaler Signale 

ab 1994 Entwurf analoger integrierter Schaltungen 

Beginn der Dissertation 

ab 1996 Universitatsassistent am Institut fur Computertechnik der TU Wien 

Leiter des Kompetenzzentrums fur Feldbussysteme 

Lehrauftrag zum Thema Entwurf integrierter Schaltungen 

251

Wave Propagation in Linear Media | re-examined

Create successful ePaper yourself

Delete template?

Save as template?