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Semiclassical Dynamics and Relaxation : Springer Series on Atomic, Optical, and Plasma Physics - D.S.F. Crothers

Semiclassical Dynamics and Relaxation

Springer Series on Atomic, Optical, and Plasma Physics


Published: 4th December 2007
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Condensed-matter physics plays an ever increasing role in photonics, electronic and atomic collisions research. Dispersion (Dynamics and Relaxation) includes scattering/collisions in the gaseous phase. It also includes thermal agitation, tunneling and relaxation in the liquid and solid phases. Classical mechanics, classical statistical mechanics, classical relativity and quantum mechanics are all implicated. 'Semiclassical' essentially means that there is a large or asymptotic real parameter. 'Semiclassical' can also mean 'classical with first-order quantal correction', based on an exponentiated Liouville series commencing with a simple pole in the -plane, being Planck's reduced constant and coming with all the attendant connection problems associated with the singularity at the turning or transition point and with the Stokes phenomenon. Equally, ' semiclassical' can mean 'electrons described quantally and the heavy particles classically'. This latter gives rise to the so-called impact parameter method based on a pre-assigned classical trajectory.

With evermore sophisticated experiments, it has become equally more important to test theory over a wider range of parameters. For instance, at low impact energies in heavy-particle collisions, the inverse velocity is a large parameter; in single-domain ferromagnetism, thermal agitation (including Brownian motion and continuous-time random walks) is faced with a barrier of height 'sigma', a possibly large parameter. Methods of solution include phase-integral analysis, integral transforms and change-of-dependent variable. We shall consider the Schrodinger time-independent and time-dependent equations, the Dirac equation, the Fokker Planck equation, the Langevin equation and the equations of Einstein's classical general relativity equations.

There is an increasing tendency among physicists to decry applied mathematics and theoretical physics in favour of computational blackboxes. One may say applied mathematics concerns hard sums and products (and their inverses) but unless one can simplify and sum infinite series of products of infinite series, can one believe the results of a computer program? The era of the polymath has passed; this book proposal aims to show the relevance to, and impact of theory on, laboratory scientists.

From the reviews: "This book is based on some lectures given by the author to postgraduate Ph.D. students at the Centre for Atomic, Molecular and Optical Physics, School of Mathematics and Physics, of Queen's University Belfast, and reflects the author's research interests. ... At the end of the book, the reader will find a very rich list of references. The book is very instructive, both to physicists and mathematicians, in that it gives many important instances of the use of the fundamental technique of semiclassical approximation." (Alberto Parmeggiani, Mathematical Reviews, Issue 2011 d)

Mathematics for the Semiclassicistp. 1
Single-Valued Analytic Functionsp. 1
Method of Steepest Descent and Asymptotic Methodsp. 2
Stationary-Phase Versionp. 3
Generalized Variation and Perturbation Theoriesp. 4
Hypergeometric Seriesp. 6
Contour Integral Transformsp. 11
Combinatoricsp. 14
Proof via Sister Celine's Techniquep. 15
Generalized Hypergeometric Functionsp. 16
Fourier and Laplace Transformsp. 19
Critical Fourier Transform Relationp. 19
Critical Laplace Transform Relationp. 20
Semiclassical Phase Integralsp. 21
Approximationp. 21
JWKB Approximationp. 21
Gans-Jeffreys Asymptotic Connection Formulap. 24
Phase Integralsp. 25
Stokes Phenomenon: One Transition Pointp. 25
Application of JWKB to Coupled Wave Equationsp. 29
Two and Four Transition Points: Crossing and Noncrossingp. 44
Introductionp. 44
Exact Resumming of Asymptotic Relations for Parabolic Cylinder Functions of Large Order and Argumentp. 45
The Crossing Parabolic Modelp. 58
Connection to Bárány-Crothers Phase-Integral Nikitin-Model Analysisp. 61
Connections to Nakamura and Zhu Phase-Integral Analysisp. 62
Connections to the Frömans-Lundborg Phase-Integral Analysisp. 64
Conclusionsp. 65
Curve Crossing Reflection Probabilities in One Dimensionp. 66
Addition of a Simple Polep. 71
Introductionp. 71
The Semiclassical Scattering Matrixp. 74
Phase-Integral Treatmentp. 75
Comparison Equationp. 80
General Phase-Integral Abstractionp. 83
Discussionp. 83
Other Generalizationsp. 85
Four Close Curve-Crossing Transition Pointsp. 85
Circuit-Dependent Adiabatic Phase Factors from Phase Integral Theoryp. 88
Semiclassical Method for Hyperspherical Coordinate Systemsp. 93
Wannier's Classical Treatment of Electron Correlationp. 93
Differential and Integrated Wannier Cross Sectionsp. 98
Conclusionsp. 115
Doubly Excited States and Their Lifetimesp. 116
Resultsp. 123
Doubly Excited States of Hep. 125
Divergent Exponentsp. 128
Wannier's Theoryp. 129
The Semiclassical JWKB Approximationp. 130
Semiclassical Theory when the Exponent Divergesp. 131
Results, Discussion, and Conclusionsp. 137
Ion-Atom Collisionsp. 139
The Semiclassical Impact Parameter Treatmentp. 139
Traveling Atomic and Molecular Orbitalsp. 144
Traveling Molecular H2 Orbitalsp. 145
Traveling Molecular HeH2+ Orbitalsp. 155
Traveling Atomic Orbitalsp. 171
Continuum Distorted Waves and Their Generalizationsp. 172
Introductionp. 172
Charge Transferp. 173
Ionizationp. 182
Fully differential cross sections for ionizationp. 197
Generalized Continuum Distorted Wavesp. 210
Double Ionizationp. 215
Relativistic CDWp. 219
Antihydrogen Productionp. 231
Semiclassical Acausalityp. 234
Introductionp. 234
Generalized Impact-Parameter Treatmentp. 236
Perturbation Theoryp. 238
Discussion and Conclusionsp. 240
Diffusion in Liquids and Solidsp. 243
Single-Domain Ferromagnetic Particlesp. 243
The Fokker-Planck and Langevin Equationsp. 267
Drift and Diffusion Coefficientsp. 273
Dieletric Relaxation, Anomalous Diffusion, Fractals, and After Effectsp. 284
Numerical Calculation and Physical Understandingp. 289
Nonlinear Response of Permanent Dipoles and After Effectsp. 292
Complex Susceptibility for the Debye and Debye-Fröhlich Models of Relaxationp. 294
Linear Dielectric Responsep. 297
Dynamic Kerr Effectp. 299
Nonlinear Dielectric Relaxationp. 300
Approximate Analytical Formula for the Dynamic Kerr Effect for a Pure Cosinusoidp. 301
Continued Fraction Solutions of Eq. (5.301)p. 305
Mittag-Leffler Functionsp. 309
Properties of Mittag-Leffler Functionsp. 309
Asymptotics of Mittag-Leffler functionsp. 309
Check on Norm of x2(t)p. 311
Nonlinear Response to Alternating Fieldsp. 313
Referencesp. 321
Indexp. 337
Table of Contents provided by Publisher. All Rights Reserved.

ISBN: 9780387743127
ISBN-10: 038774312X
Series: Springer Series on Atomic, Optical, and Plasma Physics
Audience: Tertiary; University or College
Format: Hardcover
Language: English
Number Of Pages: 344
Published: 4th December 2007
Publisher: Springer-Verlag New York Inc.
Country of Publication: US
Dimensions (cm): 23.5 x 15.6  x 2.54
Weight (kg): 1.49