Wave Propagation in Linear Media | re-examined

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P 0 3 2 1 10 0 -40 T 40 30 20 10 20 30 -20 40 0 X 20 4 One-dimensional quantum tunnelling 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
4.6 Examples of tunnelling events 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
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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 71 and 72: 3.7 Turn-on e ects in a lossless pl
Page 77 and 78: 3.8 Turn-on e ects in a wave guide
Page 79 and 80: 3.8 Turn-on e ects in a wave guide
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
Page 115 and 116: 4.5 Tunnelling time de nitions for
Page 117 and 118: 4.6 Examples of tunnelling events P
Page 119 and 120: 4.6 Examples of tunnelling events 2
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Page 125 and 126: 4.6 Examples of tunnelling events P
Page 133 and 134: 4.6 Examples of tunnelling events T
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
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6.6 Controlling the accuracy of the
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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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