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Contents
1 Mathematics for the Semiclassicist 1
1.1 Single-Valued Analytic Functions 1
1.2 Method of Steepest Descent and Asymptotic Methods 2
1.2.1 Stationary-Phase Version 3
1.3 Generalized Variation and Perturbation Theories 4
1.4 Hypergeometric Series 6
1.5 Contour Integral Transforms 11
1.6 Combinatorics 14
1.6.1 Proof via Sister Celine's Technique 15
1.7 Generalized Hypergeometric Functions 16
1.8 Fourier and Laplace Transforms 19
1.8.1 Critical Fourier Transform Relation 19
1.8.2 Critical Laplace Transform Relation 20
2 Semiclassical Phase Integrals 21
2.1 Approximation 21
2.1.1 JWKB Approximation 21
2.1.2 Gans-Jeffreys Asymptotic Connection Formula 24
2.2 Phase Integrals 25
2.2.1 Stokes Phenomenon: One Transition Point 25
2.2.2 Application of JWKB to Coupled Wave Equations 29
2.3 Two and Four Transition Points: Crossing and Noncrossing 44
2.3.1 Introduction 44
2.3.2 Exact Resumming of Asymptotic Relations for Parabolic Cylinder Function...
2.3.3 The Crossing Parabolic Model 58
2.3.4 Connection to B á r á ny-Crothers Phase-Integral Nikitin-Mode...
2.3.5 Connections to Nakamura and Zhu Phase-Integral Analysis 62
2.3.6 Connections to the Frömans-Lundborg Phase-Integral Analysis 64
2.3.7 Conclusions 65
2.3.8 Curve Crossing Reflection Probabilities in One Dimension 66
2.4 Addition of a Simple Pole 71
2.4.1 Introduction 71
2.4.2 The Semiclassical Scattering Matrix 74
2.4.3 Phase-Integral Treatment 75
2.4.4 Comparison Equation 80
2.4.5 General Phase-Integral Abstraction 83
2.4.6 Discussion 83
2.5 Other Generalizations 85
2.5.1 Four Close Curve-Crossing Transition Points 85
2.5.2 Circuit-Dependent Adiabatic Phase Factors from Phase Integral Theory 88
3 Semiclassical Method for Hyperspherical Coordinate Systems 93
3.1 Wannier's Classical Treatment of Electron Correlation 93
3.2 Differential and Integrated Wannier Cross Sections 98
3.2.1 Conclusions 115
3.3 Doubly Excited States and Their Lifetimes 116
3.3.1 Results 123
3.3.2 Doubly Excited States of He 125
3.4 Divergent Exponents 128
3.4.1 Wannier's Theory 129
3.4.2 The Semiclassical JWKB Approximation 130
3.4.3 Semiclassical Theory when the Exponent Diverges 131
3.4.4 Results, Discussion, and Conclusions 137
4 Ion-Atom Collisions 139
4.1 The Semiclassical Impact Parameter Treatment 139
4.2 Traveling Atomic and Molecular Orbitals 144
4.2.1 Traveling Molecular H+ Orbitals 145
4.2.2 Traveling Molecular HeH 2+Orbitals 155
4.2.3 Traveling Atomic Orbitals 171
4.3 Continuum Distorted Waves and Their Generalizations 172
4.3.1 Introduction 172
4.3.2 Charge Transfer 173
4.3.3 Ionization 182
4.3.4 Fully differential cross sections for ionization 197
4.3.5 Generalized Continuum Distorted Waves 210
4.3.6 Double Ionization 215
4.4 Relativistic CDW 219
4.4.1 Antihydrogen Production 231
4.5 Semiclassical Acausality 234
4.5.1 Introduction 234
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Index 5-point Lobatto, 42
Aac field, 298
acausal, cybernetic effects, 234
action, 132
adiabatic parallel transport, 90
adiabatic perturbation, 88
adiabatic phase, 80, 86
adiabatic turning points, 68
after-effect solution, 296
alumina matrix, 282
analytic continuation, 8, 13, 50, 116, 120, 211
anisotropy energy, 245
anomalous rotational diffusion, 284
antihydrogen production, 231, 233
antiproton, 232
Appell functions, 16
arbitrary angular momentum, 120
Arrhenius law, 293
assembly of dipoles, 294
asymptotic expansion, 288
asymptotic methods, 2 atomic collision, 71
autocorrelation function, 251
BBaFeCoTiO, 278
barrier height parameter, 250, 260
Bates, 144
Bernoulli number, 87
Bessel functions, 128
Bohr-Sommerfeld, 70, 112, 116
Bohr-Sommerfeld quantization rule, 29
Boltzmann integral equation, 267
Boltzmann's constant, 244, 284
Born, 37
bound-state wave function, 110
Brown, 246
Brown's asymptotic formula, 265
Brownian motion, 249, 272
Brownian-type rotation, 244
Ccapture by pair production, 225
capture cross sections, 170
capture to continuum, 186
Cauchy Residue theorem, 287
CDW normalization problem, 218
Chapman-Kolmogorov equation, 267
charge-transfer, 176
circuit-dependent adiabatic phase, 88
classical mechanics, 93
classical trajectory, 71
classical treatment of Wannier, 111
Clebsch-Gordan, 10
Co, 278
cold atomic collisions, 44
Cole-Cole dielectric relaxation, 303
collision, 71
colour of the quantum noise, 284
common translation factor, 148, 157
commutator relationships, 163
comparison equation, 80
comparison equation method, 66
complex r-plane, 78
complex Bohr-Sommerfeld, 123
complex Coulomb phase, 72, 83
complex dielectric susceptibility, 294
complex Newton-Raphson, 290
complex Nikitin model, 65
complex susceptibility, 294, 305
complex-contour integral representation, 284, 288
computer simulations, 197
connected kernel, 210
constant of proportionality, 111
continued fraction, 248, 261, 285, 305
continuity equation, 132
continuum distorted wave, 136, 172, 173, 185, 210
continuum-distorted-wave eikonal-initial-state, 192
coordinates, 158
coplanar asymmetric geometry, 100, 103
coplanar geometry, 99
Copper, 229
correlation, 98
correlation function, 315, 317
correlation time, 249
coupled equations, 31, 174
cross sections, 31
crossing, 44
crossing symmetries, 227
crystalline anisotropy, 249
current probability density, 135
curve crossing, 66, 68, 71, 87
curve-crossing parabolic model, 58
cusp, 190
DDarwin correction terms, 223
de Broglie wavelength, 23
Debye relaxation process, 249
decay modes, 295
degeneracies, 143
degenerate Appell function, 306
delta function, 120
Demkov-Kunicke model, 76
dependent variable, 119
detailed balance, 143
dielectric, 294
dielectric relaxation, 260
dielectric spectroscopy, 302
differential cross section, 176, 177
differential recurrence relations, 297
diffusion coefficient, 270
diffusion in liquids and solids, 243
dipole interaction, 109
Dirac matrices, 227
Dirac sea theory, 232
distorted wave, 211
distorted-wave, 37, 38
distorted-wave Born, 186
distortion, 143
distribution function, 267
divergent exponents, 128
divergent series, 256
divergent tail, 48
dominant, 53
double ionization, 215
doubly differential cross section, 189
doubly excited states, 116
drift coefficient, 270
dynamic Kerr effect, 299, 301, 313
dynamic molecular theory, 210
dynamical couplings, 157
Eeffective charge, 112
effective frequency, 57
eigenvalue, 245, 301
eikonal initial state, 187
eikonal method, 144
elastic divergence free, 210
elastic scattering phaseshifts, 89
electric birefringence of nematics, 319
electron capture to the continuum, 195
electron transfer, 173
electron translation factor, 66, 141, 160, 200
electron-electron correlation, 215
equilibrium correlation function, 296
ergodicity, 97
escape rate, 257, 318
Euler-Lagrange variational technique, 161
evolution equation, 275
excess energy, 98
exchange, 151
excitation, 151
exponential law, 129
exponential model, 71
FFaá di Bruno's formula, 310
ferrofluid, 294
first Born cross section, 240
Fokker-Planck equation, 260, 267, 276
Fokker-Planck operator, 245 forced-common turning point, 29, 38
four close curve-crossing transition points, 85
Fourier transform, 19, 239
Fox functions, 19
fractal Mittag-Leffler functions, 284 fractional diffusion relaxation, 303 fractional...
fractional Fokker-Planck operator, 284 fractional Klein-Kramers equation, 284 free di...
fully differential cross section, 198
Furry, 27
GGalilean invariance, 141, 172, 182
Gans-Jeffreys, 24, 123
gauge invariance, 172, 182
gauge transformation, 234
Gauss hypergeometric, 122
Gauss hypergeometric function, 253
generalized CDW-EIS theory, 213
generalized complex susceptibilities, 315
generalized continuum distorted wave, 173, 210
generalized hypergeometric functions, 16
generalized hyperspherical coordinates, 93
generalized impact parameter treatment, 235
generalized nonorthogonal coordinates, 175
generating function, 48
geometric phase, 88
Gilbert equation, 244, 282
Glauber, 183
gold, 229
Green function, 35, 298, 314
Green-Liouville, 21
gyromagnetic ratio, 245
HHamilton-Jacobi equation, 119, 132
Hankel transform, 288
Heaviside step function, 286
heavy particles, 144
heavy-particle collisions, 144
Hermitian equations, 146
Heun equations, 122
highly relativistic atomic collisions, 225
hypergeometric function, 7
hyperspherical coordinate systems, 93
Iimpact parameter, 58, 70
impact-parameter method, 235
impact-parameter treatment, 235
incomplete gamma function, 286
independent-event model, 216
individual nanoparticles, 282
inertia-corrected Debye model, 284
insulating BaFeCoTiO particle, 280, 282
integral equation, 6
intermediate to high damping, 280
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