Matthias Brack, Rajat K. Bhaduri,
Perseus Books Group | ISBN: 0201483513 | 1997-02 | djvu (ocr) | 444 pages | 3.81 Mb
Semiclassical Physics emphasizes the close connection between the shorter classical periodic orbits, and the partially resolved quantum fluctuations in the level density and response of an autonomous finite quantum system. Particular care is taken to present a detailed derivation of Gutzwiller's trace formula, and its extensions to continuous symmetries, zeta function techniques, and diffractive orbits.
Contents:
1 Introduction 1
1.1 The quantum propagator 4
1.2 Old quantum theory 9
1.2.1 A ball bouncing off a moving wall 10
1.2.2 A pendulum with variable string length 11
1.2.3 The phase space of a simple harmonic oscillator 13
1.2.4 Three-dimensional anisotropic harmonic oscillator 16
1.3 Wave packets in Rydberg atoms 19
1.3.1 The large-n limit in the Bohr atom 19
1.3.2 Where are the periodic orbits in quantum mechanics? 20
1.4 Chaotic motion: atoms in a magnetic field 27
1.4.1 Scaling of classical Hamiltonian and chaos 27
1.4.2 Quasi-Landau resonances in atomic photoabsorption 32
1.5 Chaos and periodic orbits in mesoscopic systems 36
1.5.1 Ballistic magnetoresistance in a cavity 37
1.5.2 Scars in the wave function 39
1.5.3 Tunneling in a quantum diode with a tilted magnetic field ... 42
1.5.4 Electron transport in a superlattice of antidots 44
1.6 Problems 49
2 Quantization of integrable systems 57
2.1 Introduction 57
2.2 Hamiltonian formalism and the classical limit 59
2.3 Hamilton-Jacobi theory and wave mechanics 63
2.4 The WKB method 67
2.4.1 WKB in one dimension 68
2.4.2 WKB for radial motion 75
2.5 Torus quantization: from WKB to EBK 78
2.6 Examples 83
2.6.1 The two-dimensional hydrogen atom 83
2.6.2 The three-dimensional hydrogen atom 86
2.6.3 The two-dimensional disk billiard 88
2.7 Connection to classical periodic orbits 89
2.7.1 Example: The two-dimensional rectangular billiard 94
2.8 Transition from integrability to chaos 98
2.8.1 Destruction of resonant tori 98
2.8.2 The model of Walker and Ford 100
2.9 Problems 106
3 The single-particle level density 111
3.1 Introduction Ill
3.1.1 Level density and other basic tools 112
3.1.2 Separation of g(E) into smooth and oscillating parts 117
3.2 Some exact trace formulae 118
3.2.1 The linear harmonic oscillator 118
3.2.2 General spectrum depending on one quantum number 120
3.2.3 One-dimensional box 122
3.2.4 More-dimensional spherical harmonic oscillators 122
3.2.5 Harmonic oscillators at finite temperature 124
3.2.6 Three-dimensional rectangular box 126
3.2.7 Equilateral triangular billiard 128
3.2.8 Cranked or anisotropic harmonic oscillator 132
3.3 Problems 136
4 The extended Thomas-Fermi model 143
4.1 Introduction 143
4.2 The Wigner distribution function 148
4.3 The Wigner-Kirkwood expansion 151
4.4 The extended Thomas-Fermi model 155
4.4.1 The ETF model at zero temperature 155
4.4.2 The ETF density variational method 162
4.4.3 The finite-temperature ETF model 170
4.5 h expansion for cavities and billiards 176
4.5.1 The Euler-MacLaurin expansion 177
4.5.2 The Weyl expansion 180
4.5.3 Black-body radiation in a small cavity 183
4.6 The Strutinsky method 185
4.6.1 The energy averaging method 186
4.6.2 The shell-correction method 192
4.6.3 Relation between ETF and Strutinsky averaging 194
4.7 Problems 197
5 Gutzwiller's trace formula for isolated orbits 207
5.1 The semiclassical Green's function 209
5.2 Taking the trace of G,cl(r,r';E) 214
5.3 The trace formula for isolated orbits 218
5.4 Monodromy matrix and stability of periodic orbits 221
5.5 Convergence of the periodic orbit sum 223
5.6 Examples 228
5.6.1 Applications to chaotic systems 228
5.6.2 The irrational anisotropic harmonic oscillator 229
5.6.3 The inverted harmonic oscillator 230
5.6.4 The Henon-Heiles potential 232
5.7 Problems 237
6 Extensions of the Gutzwiller theory 243
6.1 Trace formulae for degenerate orbits 245
6.1.1 Two-dimensional systems, singly degenerate orbits 246
6.1.2 Example 1: The equilateral triangular billiard 247
6.1.3 Example 2: The two-dimensional disk billiard 256
6.1.4 More general treatment of continuous symmetries 260
6.1.5 Example 3: The 2-dimensional rectangular billiard 267
6.1.6 Example 4: The three-dimensional spherical cavity 269
6.2 The problem of symmetry breaking 272
6.2.1 A trace formula for broken symmetry 273
6.2.2 Example 1: The two-dimensional elliptic billiard 274
6.2.3 Example 2: Inclusion of weak magnetic fields 281
6.2.4 Example 3: The quartic Henon-Heiles potential 288
6.3 Problems 292
7 Quantization of nonintegrable systems 297
7.1 The Riemann zeta function 298
7.1.1 The zeros of the Riemann zeta function 298
7.1.2 A trace formula for the zeros 301
7.1.3 Nearest-neighbor spacings and chaos 304
7.1.4 The Riemann-Siegel relation 305
7.2 The quantization condition 308
7.2.1 The Selberg zeta function 308
7.2.2 Pseudo-orbits and the Selberg zeta function 312
7.3 The scattering matrix method 315
7.4 The transfer-matrix method of Bogomolny 321
7.5 Diffractive Corrections to the Trace Formula 329
7.5.1 Introduction 329
7.5.2 Quantum theory of scattering 331
7.5.3 Scattering by a hard disk 333
7.5.4 The scattering amplitude and the Green's function 340
7.5.5 Modification to the trace formula 343
7.5.6 The circular annulus billiard 348
7.6 Problems 356
8 Shells and periodic orbits in finite fermion systems 363
8.1 Shells and shapes in atomic nuclei 363
8.1.1 Nuclear ground-state deformations 365
8.1.2 The double-humped fission barrier 370
8.1.3 The mass asymmetry in nuclear fission 374
8.2 Shells and supershells in metal clusters 380
8.3 Conductance oscillations in a circular quantum dot 389
9 Concluding remarks 401
A The self-consistent mean field approach 405
A.I Hartree-Fock theory 406
A.2 Density functional theory 409
A.3 The Strutinsky energy theorem 411
B Inverse Laplace transforms 415
C More about the monodromy matrix 417
C.I Linear differential equations with periodic coefficients 417
C.2 Hamiltonian equations 419
C.2.1 Example: Harmonic oscillator 419
C.3 Non-linear systems and the Poincare variational equations 420
C.4 Numerical calculation of M for smooth potentials 421
C.5 Calculation of M for two-dimensional billiards 422
C.5.1 Elliptic billiard 425
C.6 Problems 426
D Calculation of Maslov indices for isolated orbits 429
D.I Isolated orbits in smooth potentials 429
D.I.I Unstable orbits 430
D.1.2 Stable orbits 431
D.1.3 Example: Two-dimensional harmonic oscillator 433
D.2 Isolated orbits in billiards 434
Uploading.com