Wave Propagation in Linear Media | re-examined

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4 One-dimensional quantum tunnelling 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
4.6 Examples of tunnelling events 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
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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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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
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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 119 and 120: 4.6 Examples of tunnelling events 2
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Page 123 and 124: 4.6 Examples of tunnelling events t
Page 125 and 126: 4.6 Examples of tunnelling events P
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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
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6.5 Considerations for a Mathematic
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6.5 Considerations for a Mathematic
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6.6 Controlling the accuracy of the
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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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