Wave Propagation in Linear Media | re-examined

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]; AccuracyGoal->ag, MaxIterations->ma], {k,irel,irel+n-1}]] ], True, Message[PartitionTable::badparam,ft]] OscIntControlled[f_, fzero_, a_, opts___Rule] := Module[{seqtab,parttab,iterations=1,offs,firstval,intlim, wp = WorkingPrecision/.{opts}/.Options[OscInt], ag = AccuracyGoal/. {opts}/.Options[OscInt], pg = PrecisionGoal/. {opts}/.Options[OscInt], mi = MinRecursion/. {opts}/.Options[OscInt], ma = MaxRecursion/. {opts}/.Options[OscInt], nt = NSumTerms/. {opts}/.Options[OscInt], ne = NSumExtraTerms/. {opts}/.Options[OscInt], wd = WynnDegree/. {opts}/.Options[OscInt], it = MaxIterations/. {opts}/.Options[OscInt], ac = AccuracyControl/. {opts}/.Options[OscInt], il = IterationLength/. {opts}/.Options[OscInt]}, offs = PartitionOffs[zerof,a,opts]; parttab = PartitionPoints[fzero,a,ne+nt+1,offs,0,opts]; If[N[parttab[[1]]] < a, parttab = Select[parttab,(N[#] >= a)&]; Message[OscInt::lowfirst,Length[parttab]]]; firstval = If[N[parttab[[1]]] == N[a], 0, NIntegrate[f[x],{x,a,parttab[[1]]}, Method->DoubleExponential, WorkingPrecision->wp, AccuracyGoal->ag, PrecisionGoal->pg, MinRecursion->mi, MaxRecursion->ma]]; If[N[parttab[[1]]] =!= Infinity, seqtab = Table[NIntegrate[f[x],{x,parttab[[i]],parttab[[i+1]]}, Method->GaussKronrod, WorkingPrecision->wp, AccuracyGoal->ag, PrecisionGoal->pg, MinRecursion->mi, MaxRecursion->ma], {i,1,Length[parttab]-1}]]; Which[N[parttab[[1]]] == Infinity, firstval, ac === False, SequenceLimit[Take[FoldList[Plus,firstval,seqtab], {nt+1,nt+ne}],WynnDegree->wd], IntegerQ[ac], intlim = SequenceLimit[Take[FoldList[Plus,firstval,seqtab], {nt+1,nt+ne}],WynnDegree->wd]; While[iterations < it && Accuracy[intlim] < ac, parttab = Join[{Last[parttab]}, PartitionPoints[fzero,a,il,offs,ne+nt+1,opts]]; seqtab = Join[seqtab, Table[NIntegrate[f[x],{x,parttab[[i]],parttab[[i+1]]}, Method->GaussKronrod, WorkingPrecision->wp, AccuracyGoal->ag, PrecisionGoal->pg, MinRecursion->mi, MaxRecursion->ma], {i,1,Length[parttab]-1}]]; ne = ne + il; iterations++; intlim = SequenceLimit[Take[FoldList[Plus,firstval,seqtab], {nt+1,nt+ne}],WynnDegree->wd] ]; If[Accuracy[intlim] < ac, Message[OscInt::accfail,iterations]]; intlim, True, Message[OscInt::badparam,ac]] ]; OscInt[fint_, fzero_, {a_, a0_}, opts___Rule] := Module[{parttab,lowlim, wp = WorkingPrecision/.{opts}/.Options[OscInt], nt = NSumTerms/. {opts}/.Options[OscInt], ne = NSumExtraTerms/. {opts}/.Options[OscInt]}, lowlim = If[a == a0, a, N[Max[a,a0],2 wp]]; parttab = PartitionTable[fzero,lowlim,ne+nt+1,opts]; PartInt[fint,parttab,a,opts] ]; 220 A Mathematica packages
A.2 Solutions for the step potential OscInt[fint_, fzero_, a_, opts___Rule] := OscInt[fint,fzero,{a,a},opts] End[] SetAttributes[PolynomialDegree, ReadProtected] SetAttributes[AsymptoticExpand, ReadProtected] SetAttributes[ApproxLimGeneric, ReadProtected] SetAttributes[ApproxLimQuad, ReadProtected] SetAttributes[ApproxLimLinear, ReadProtected] SetAttributes[QuadZero, ReadProtected] SetAttributes[QuadOffset, ReadProtected] SetAttributes[HypZero, ReadProtected] SetAttributes[HypOffset, ReadProtected] SetAttributes[HypZeroExact, ReadProtected] SetAttributes[HypOffsetExact, ReadProtected] SetAttributes[ZerosInBetween, ReadProtected] SetAttributes[HypApproxError, ReadProtected] SetAttributes[PartInt, ReadProtected] SetAttributes[PartitionTable, ReadProtected] SetAttributes[PartitionOffs, ReadProtected] SetAttributes[PartitionPoints, ReadProtected] SetAttributes[OscInt, ReadProtected] SetAttributes[OscIntControlled, ReadProtected] (* Protect[PolynomialDegree, AsymptoticExpand, ApproxLimGeneric] Protect[ApproxLimQuad, ApproxLimLinear] Protect[QuadZero, QuadOffset] Protect[HypZero, HypOffset] Protect[HypZeroExact, HypOffsetExact] Protect[ZerosInBetween, HypApproxError] Protect[PartInt, PartitionTable, PartitionOffs, PartitionPoints] Protect[OscInt, OscIntControlled] *) EndPackage[] A.2 Solutions for the step potential The package Tunnel computes the solutions of the Schrodinger equation for the step potential barrier discussed in section 4.3. Its structure is explained in chapter 8. For the calculation of the integrals, the package OscInt is used. (* Copyright: Copyright 1996, Institute of Computertechnology, *) (* Vienna University of Technology *) (*:Version: Mathematica 2.2.3 *) (*:Title: Tunnel *) (*:Author: Thilo Sauter *) (*:Keywords: Tunnel Effect *) (*:Requirements: None. *) (*:Warnings: None so far *) (*:Packages: OscInt *) (*:Summary: This package contains functions for the time-dependent solutions of Schroedinger's equation. The exact solution has been evaluated for wave packet impinging on a given step potential barrier. The initial shape of the wave packet can be either rectangular, triangular, or Gaussian. The solution of the differential equation consists of Fourier integrals and must be evaluated numerically. A severe obstacle to a straightforward integration are the heavily oscillating integrands. Thus integration is carried out by using the functions in the package OscInt *) (*:History: 15-11-1996 written 25-01-1997 templates added 02-02-1997 new naming convention for functions 17-02-1997 truncation quadrature of Gaussian waves added *) BeginPackage["Tunnel`", "OscInt`"] Phi::usage = "Phi[x,t,w,k,(n),(opts)] returns the wave function at a given coordinate in space and time. The shape of the incident wave is selected with the option Shape. The step barrier is supposed to begin at x=0. Depending on the spatial 221
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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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3.8 Turn-on e ects in a wave guide
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3.8 Turn-on e ects in a wave guide
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3.9 A Gaussian pulse in plasma Like
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3.9 A Gaussian pulse in plasma 250
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3.9 A Gaussian pulse in plasma 250
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3.9 A Gaussian pulse in plasma Note
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4.1 The potential step 4.1 The pote
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4.1 The potential step Inside the b
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4.2 Initial wave forms -60 -50 -40
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4.2 Initial wave forms -60 -50 -40
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4.3 Examples of scattering processe
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4.4 The square barrier they have va
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4.4 The square barrier 1 0.8 0.6 0.
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4.5 Tunnelling time de nitions for
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4.5 Tunnelling time de nitions for
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4.6 Examples of tunnelling events P
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4.6 Examples of tunnelling events 2
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4.6 Examples of tunnelling events -
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4.6 Examples of tunnelling events t
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4.6 Examples of tunnelling events P
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4.6 Examples of tunnelling events T
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Interlude Wave functions in graphic
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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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7.3 Implementation of OscInt and re
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Page 213 and 214: 8.2 Outline of the program structur
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Page 217 and 218: 8.3 Implementation of the quadratur
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Page 221 and 222: 8.4 Test of the package 8.4 Test of
Page 223 and 224: 8.4 Test of the package -200 -150 -
Page 225 and 226: 8.4 Test of the package 0.4 0.2 -0.
Page 227 and 228: 8.4 Test of the package 8.4.2 Forma
Page 229 and 230: 8.4 Test of the package Out[24]= 0
Page 231 and 232: A.1 Numerical quadrature Asymptotic
Page 233 and 234: A.1 Numerical quadrature WynnDegree
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Page 239 and 240: A.2 Solutions for the step potentia
Page 245 and 246: A.3 Solutions for the square barrie
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Page 249 and 250: A.4 Electromagnetic waves function
Page 251 and 252: A.5 Utilities for displaying points
Page 253 and 254: Bibliography Bibliography [1] James
Page 255 and 256: Bibliography [29] Kurt Edmund Oughs
Page 257 and 258: Bibliography [59] Ch. Spielmann, R.
Page 259 and 260: Bibliography [91] C. R. Leavens and
Page 261 and 262: Bibliography [123] T. O. Espelid an
Page 263 and 264: Index Index absorption, 7, 9 accura
Page 265 and 266: Index saddle point integration, 11,
Page 267: Curriculum vitae Dipl.-Ing. Thilo S
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