Semiclassical physics / Matthias Brack, Rajat K. Bhaduri.

Format:

Book

Author/Creator:

Brack, Matthias.

Contributor:

Bhaduri, Rajat K.

Series:

Frontiers in physics ; v. 96.

Frontiers in physics ; v. 96

Language:

English

Subjects (All):

Quantum theory.

Path integrals.

Thomas-Fermi theory.

Physical Description:

xii, 444 pages : illustrations ; 24 cm.

Place of Publication:

Reading, Mass. : Addison-Wesley Publishing, [1997]

Contents:

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

2 Quantization of integrable systems 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

2.6.1 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

3 The single-particle level density 111

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

4 The extended Thomas-Fermi model 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 169

4.5 Bose-Einstein condensation in a trap 176

4.5.1 BEC in an ideal trapped bose gas 177

4.5.2 Inclusion of interactions in a dilute gas 179

4.6 H expansion for cavities and billiards 180

4.6.1 The Euler-MacLaurin expansion 180

4.6.2 The Weyl expansion 183

4.6.3 Black-body radiation in a small cavity 186

4.7 The Strutinsky method 188

4.7.1 The energy averaging method 189

4.7.2 The shell-correction method 195

4.7.3 Relation between ETF and Strutinsky averaging 197

5 Gutzwiller's trace formula for isolated orbits 213

5.1 The semiclassical Green's function 215

5.2 Taking the trace of G[subscript scl] (r, r'; E) 220

5.3 The trace formula for isolated orbits 224

5.4 Stability of periodic orbits 227

5.5 Convergence of the periodic orbit sum 229

5.6.1 Applications to chaotic systems 235

5.6.2 The irrational anisotropic harmonic oscillator 235

5.6.3 The inverted harmonic oscillator 236

5.6.4 The Henon-Heiles potential 238

6 Extensions of the Gutzwiller theory 249

6.1 Trace formulae for degenerate orbits 251

6.1.1 Two-dimensional systems, singly degenerate orbits 252

6.1.2 Example 1: The equilateral triangular billiard 253

6.1.3 Example 2: The two-dimensional disk billiard 262

6.1.4 More general treatment of continuous symmetries 266

6.1.5 Example 3: The 2-dimensional rectangular billiard 273

6.1.6 Example 4: The three-dimensional spherical cavity 275

6.2 The problem of symmetry breaking 278

6.2.1 A trace formula for broken symmetry 279

6.2.2 Example 1: The two-dimensional elliptic billiard 280

6.2.3 Example 2: Inclusion of weak magnetic fields 287

6.2.4 Example 3: The quartic Henon-Heiles potential 293

6.3 Uniform approximations 300

6.3.1 U(1) symmetry breaking 300

6.3.2 An example of SU(2) symmetry breaking 303

6.3.3 Uniform approximations for bifurcations 306

7 Quantization of nonintegrable systems 311

7.1 The Riemann zeta function 312

7.1.1 The zeros of the Riemann zeta function 312

7.1.2 A trace formula for the zeros 315

7.1.3 Nearest-neighbor spacings and chaos 318

7.1.4 The Riemann-Siegel relation 319

7.2 The quantization condition 322

7.2.1 The Selberg zeta function 322

7.2.2 Pseudo-orbits and the Selberg zeta function 326

7.3 The scattering matrix method 329

7.4 The transfer-matrix method of Bogomolny 335

7.5 Diffractive Corrections to the Trace Formula 343

7.5.2 Quantum theory of scattering 345

7.5.3 Scattering by a hard disk 347

7.5.4 The scattering amplitude and the Green's function 354

7.5.5 Modification to the trace formula 357

7.5.6 The circular annulus billiard 362

8 Shells and periodic orbits in finite fermion systems 377

8.1 Shells and shapes in atomic nuclei 377

8.1.1 Nuclear ground-state deformations 379

8.1.2 The double-humped fission barrier 384

8.1.3 The mass asymmetry in nuclear fission 388

8.2 Shells and supershells in metal clusters 394

8.3 Conductance oscillations in a circular quantum dot 403

A The self-consistent mean field approach 419

A.1 Hartree-Fock theory 420

A.2 Density functional theory 423

A.3 The Strutinsky energy theorem 425

B Inverse Laplace transforms 429

C More about the monodromy matrix 431

C.1 Linear differential equations with periodic coefficients 431

C.2 Hamiltonian equations 433

C.2.1 Example: Two-dimensional harmonic oscillator 433

C.3 Non-linear systems and the Poincare variational equations 434

C.4 Calculation of the monodromy matrix M 435

C.5 Calculation of M for two-dimensional billiards 436

C.5.1 Example: Elliptic billiard 439

D Calculation of Maslov indices for isolated orbits 443

D.1 Isolated orbits in smooth potentials 444

D.1.1 Unstable orbits 444

D.1.2 Stable orbits 445

D.1.3 Example: Two-dimensional harmonic oscillator 447

D.2 Isolated orbits in billiards 449.

Notes:

"The Advanced Book Program."

Includes bibliographical references and index.

ISBN:

0201483513

OCLC:

36862376