Wave Propagation in Linear Media | re-examined

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30 25 20 15 10 5 T 4 One-dimensional quantum tunnelling -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
4.6 Examples of tunnelling events -40 40 30 20 10 -20 0 X 40 30 20 T 20 0 -40 -20 0 20 40 10 0 1 0 40 Figure 4.24: Probability density P = j j ofawave packet with the parameters k =6,n= 4, and =0:8interacting with a medium-sized barrier (D = 5). 105 2 P
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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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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
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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 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 127 and 128: 4.6 Examples of tunnelling events 8
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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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6.6 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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