Wave Propagation in Linear Media | re-examined

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4 One-dimensional quantum tunnelling 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
4.6 Examples of tunnelling events 80 60 40 20 T -30 -20 -10 0 10 20 30 Figure 4.32: Trajectory of the temporal peak of a wave packet with the parameters k =6,n= 4, and =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 Figure 4.33: Trajectory of the temporal peak of a wave packet with the parameters k =6,n= 4, and =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
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DISSERTATION Wave Propagation in Li
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Kurzfassung Seit der Entdeckung des
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Nullum est iam dictum, quod non sit
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Preface Our popular writers and rep
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the quest for superluminality and t
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Contents Part I Wave propagation ph
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7.3.4 PartitionPoints . . . . . . .
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Part I Wave propagation phenomena S
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1.1 Phase and group velocity 1.1 Ph
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1.1 Phase and group velocity ! !c v
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1.2 A few notes on dispersion He co
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1.2 A few notes on dispersion v/c 6
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1.3 Signal velocity dipoles with a
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1.3 Signal velocity The arbitrarine
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1.4 Energy velocity For electromagn
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1.5 Other velocity de nitions For n
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1.5 Other velocity de nitions evide
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2.1 Superluminal wave propagation 2
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2.1 Superluminal wave propagation a
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2.1 Superluminal wave propagation t
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2.2 Quantum mechanical tunnelling e
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2.2 Quantum mechanical tunnelling R
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Chapter 3 Wave propagation in elect
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3.1 Model of a transmission line th
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3.2 Excursion: a delay line section
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3.3 Re ection due to termination mi
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3.4 A simple thought experiment I0
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3.5 A dispersive system: the lossle
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3.6 Inhomogeneous transmission line
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3.7 Turn-on e ects in a lossless pl
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Page 77 and 78: 3.8 Turn-on e ects in a wave guide
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Page 81 and 82: 3.9 A Gaussian pulse in plasma Like
Page 83 and 84: 3.9 A Gaussian pulse in plasma 250
Page 85 and 86: 3.9 A Gaussian pulse in plasma 250
Page 87 and 88: 3.9 A Gaussian pulse in plasma Note
Page 89 and 90: 4.1 The potential step 4.1 The pote
Page 91 and 92: 4.1 The potential step Inside the b
Page 93 and 94: 4.2 Initial wave forms -60 -50 -40
Page 95 and 96: 4.2 Initial wave forms -60 -50 -40
Page 97 and 98: 4.3 Examples of scattering processe
Page 109 and 110: 4.4 The square barrier they have va
Page 111 and 112: 4.4 The square barrier 1 0.8 0.6 0.
Page 113 and 114: 4.5 Tunnelling time de nitions for
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Page 117 and 118: 4.6 Examples of tunnelling events P
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Page 123 and 124: 4.6 Examples of tunnelling events t
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Page 135 and 136: Interlude Wave functions in graphic
Page 137 and 138: Wave functions in graphical represe
Page 139 and 140: Part II Numerical aspects of wave e
Page 141 and 142: 5.1 Univariate numerical quadrature
Page 143 and 144: 5.1 Univariate numerical quadrature
Page 145 and 146: 5.2 Convergence acceleration one, t
Page 147 and 148: 5.2 Convergence acceleration (1) ,1
Page 149 and 150: 6.1 Partitioning the integration in
Page 155 and 156: 6.2 Choosing the rst partition poin
Page 157 and 158: 6.3 How to compute the rst integral
Page 159 and 160: 6.3 How to compute the rst integral
Page 161 and 162: 6.4 Asymptotic partition consuming
Page 163 and 164: 6.4 Asymptotic partition of the int
Page 165 and 166: 6.5 Considerations for a Mathematic
Page 171 and 172: 6.6 Controlling the accuracy of the
Page 177 and 178: Chapter 7 Mathematica implementatio
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7.1 User interface of the function
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7.3 Implementation of OscInt and re
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7.4 Auxiliary functions While[itera
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7.4 Auxiliary functions Linear appr
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7.4 Auxiliary functions For the sak
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7.4 Auxiliary functions Out[8]= 3 f
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7.5 Test of the quadrature routine
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8.1 Preparation of the wave integra
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8.2 Outline of the program structur
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8.2 Outline of the program structur
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8.3 Implementation of the quadratur
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8.3 Implementation of the quadratur
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8.4 Test of the package 8.4 Test of
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8.4 Test of the package -200 -150 -
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8.4 Test of the package 0.4 0.2 -0.
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8.4 Test of the package 8.4.2 Forma
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8.4 Test of the package Out[24]= 0
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A.1 Numerical quadrature Asymptotic
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A.1 Numerical quadrature WynnDegree
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A.1 Numerical quadrature Which[ft =
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A.2 Solutions for the step potentia
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A.3 Solutions for the square barrie
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A.3 Solutions for the square barrie
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A.4 Electromagnetic waves function
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A.5 Utilities for displaying points
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Bibliography Bibliography [1] James
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Bibliography [29] Kurt Edmund Oughs
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Bibliography [59] Ch. Spielmann, R.
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Bibliography [91] C. R. Leavens and
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Bibliography [123] T. O. Espelid an
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Index Index absorption, 7, 9 accura
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Index saddle point integration, 11,
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Curriculum vitae Dipl.-Ing. Thilo S
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