Wave Propagation in Linear Media | re-examined

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P 0 2 1.5 1 0.5 0 0 1 50 2 X T 100 3 150 4 4 One-dimensional quantum tunnelling 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
4.3 Examples of scattering processes 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
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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
Page 53 and 54: 3.3 Re ection due to termination mi
Page 55 and 56: 3.4 A simple thought experiment I0
Page 57 and 58: 3.5 A dispersive system: the lossle
Page 63 and 64: 3.6 Inhomogeneous transmission line
Page 65 and 66: 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
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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
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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
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
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6.2 Choosing the rst partition poin
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6.3 How to compute the rst integral
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6.3 How to compute the rst integral
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6.4 Asymptotic partition consuming
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6.4 Asymptotic partition of the int
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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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