Wave Propagation in Linear Media | re-examined

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Chapter 4 One-dimensional quantum tunnelling 4 One-dimensional quantum tunnelling Das Bestreben, der Dispersion Herr zu werden, leuchtet mir fur langsame Wellen ein, das Ubrige ist mir aus den Andeutungen nicht klar geworden. Das Au allende ist, wie viel man mit klassischer Mechanik befriedigend machen kann. Wenn es nur einmal gelange, das Prinzipielle an den Quanten einigermassen zu erklaren! Aber meine Ho nung, das zu erleben, wird immer kleiner. Albert Einstein in a letter to Arnold Sommerfeld (1. 2. 1918 [103]) Having exhausted the question of electromagnetic wave propagation we now turn to the second area where tunnelling occurs and where, after all, the term `tunnelling' was coined. To start with, however, we shall restrict our investigation to the comparatively simple step potential barrier, where we have to cope with only one interface. In a second step, we regard tunnelling in its original meaning as the motion of a particle through a classically impenetrable barrier. In the simplest form of a rectangular potential wall, such an investigation requires the inclusion of two interfaces, which renders the treatment more complicated albeit by no means impossible. In both cases, we study how the particle evolves in time and space upon impinging on the barrier. The chapter begins with the well-known formulation of the scattering process consisting of an incident free particle being partly re ected and partly penetrating the obstacle. The solutions will be given as Fourier integrals and primarily be independent of the actual initial condition. We shall then explore several possibilities of initial wave forms or particle shapes that nally are used to obtain numerical results of the wave propagation both inside and outside the barrier. The square potential barrier will be considered next, and we shall brie y examine the variety of tunnelling time de nitions known from the literature for monochromatic waves. The concluding numerical evaluation of tunnelling events is restricted to Gaussian wave packets. 72
4.1 The potential step 4.1 The potential step Being interested chie y in the dynamical evolution of a wave inside an evanescent barrier, we assume the barrier to be in nitely extended into the right half space. The potential function is therefore ( 0 if 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
Page 37 and 38: 2.1 Superluminal wave propagation 2
Page 39 and 40: 2.1 Superluminal wave propagation a
Page 41 and 42: 2.1 Superluminal wave propagation t
Page 43 and 44: 2.2 Quantum mechanical tunnelling e
Page 45 and 46: 2.2 Quantum mechanical tunnelling R
Page 47 and 48: Chapter 3 Wave propagation in elect
Page 49 and 50: 3.1 Model of a transmission line th
Page 51 and 52: 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: 3.9 A Gaussian pulse in plasma Note
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
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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
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Part II Numerical aspects of wave e
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5.1 Univariate numerical quadrature
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5.1 Univariate numerical quadrature
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5.2 Convergence acceleration one, t
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5.2 Convergence acceleration (1) ,1
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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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