Advanced Quantum Mechanics Book

Preface

1 Math Tools 1

1.1 Introduction 2

1.2 Integer Functions 2

1.3 Complex Numbers 6

1.4 Elementary Functions 8

1.5 Quadratic and Cubic Equations 15

1.6 Vectors and Matrices 16

1.7 Differentiation and Differential Equations 23

1.8 Taylor Series Expansions 27

1.9 Basic Integrals 29

1.10 Stationary-Phase Approximation 33

1.11 Heaviside's Step Function 38

1.12 Dirac's Delta Function 40

1.13 Fourier Transforms and Fourier Sums 43

1.14 Poisson's Summation Formula 47

1.15 Summary 49

2 Special Functions 53

2.1 Introduction 53

2.2 Gamma Function 54

2.3 Orthogonal Polynomials 58

2.4 Elliptic Integrals 61

2.5 Hypergeometric Function 63

2.6 Airy Functions 65

2.7 Hassel Functions 68

2.8 Summary 70

3 Newton's Particle Mechanics 73

3.1 Introduction 73

3.2 Newton's Equation 74

3.3 Energy 76

3.4 Motion in Force Fields 78

3.5 Bounded Motion 81

3.6 Classical Scattering 85

3.7 Gravitational Pendulum 92

3.8 Phase Space 97

3.9 Summary 102

4 Schr b) sdinger's Wave Mechanics 105

4.1 Introduction 105

4.2 Schr b) sdinger's Equation 106

4.3 Observables 109

4.4 Probability Density and Current 113

4.5 Bound States 116

4.6 Quantum Scattering 121

4.7 Spectral Staircase and Density of States 130

4.8 Summary 134

5 Semiclassics 137

5.1 Introduction 137

5.2 A First Classical-Quantum Connection 138

5.3 Bohr-Sommerfeld Quantization 140

5.4 WKB Quantization 143

5.5 Generalized WKB 151

5.6 Semiclassics Versus Numerics 152

5.7 Summary 154

6 Test Potentials 159

6.1 Introduction 159

6.2 Degrees of Solvability 160

6.3 Globally Confining, Smooth Potentials 164

6.4 Globally Confining, Singular Potentials 171

6.5 Partially Confining, Smooth Potentials 178

6.6 Partially Confining, Singular Potentials 185

6.7 Summary 191

7 Periodic-Orbit Quantization 195

7.1 Introduction 195

7.2 Exact Formula for the Density of States 196

7.3 The Classical-Quantum Link 200

7.4 Individual Energy Levels 204

7.5 Summary 207

8 Quantum Graphs 211

8.1 Introduction 211

8.2 From Graphs to Quantum Graphs 213

8.3 Three-Pronged Star Graph 219

8.4 Scattering Quantization of Quantum Graphs 231

8.5 Summary 235

9 Advanced Mathematical Techniques 237

9.1 Introduction 237

9.2 Principal Value 238

9.3 Calculus of Complex Functions 240

9.4 Multivariable Calculus 249

9.5 Multi-Dimensional Stationary Phase 254

9.6 Summary 256

10 Analytical Mechanics 259

10.1 Introduction 259

10.2 Lagrange's Formulation 260

10.3 Hamilton's Formulation 270

10.4 Canonical Transformations 275

10.5 Constants of the Motion 282

10.6 Integrable Motion 285

10.7 EBK Quantization 287

10.8 Summary 288

11 Chaos 291

11.1 Introduction 291

11.2 Orders of Infinity 292

11.3 Fractals 296

11.4 One-Dimensional Mappings 300

11.5 Logistic Mapping 302

11.6 Area-Preserving Mappings 309

11.7 Poincar b1 s Sections 314

11.8 Hamiltonian Chaos 319

11.9 Summary 321

12 Advanced Concepts in Quantum Mechanics 325

12.1 Introduction 325

12.2 Dirac Notation 326

12.3 Time-Dependent Propagators 331

12.4 Energy-Dependent Propagators 338

12.5 Lippmann-Schwinger Equations 347

12.6 Feynman's Path Integrals 350

12.7 Summary 354

13 Semiclassical Propagators 357

13.1 Introduction 357

13.2 Trajectory Bundles 359

13.3 'rime-Dependent Van Vleck-Gutswiller Propagator 364

13.4 Energy-Dependent Van Vleck-Gutswiller Propagator 368

13.5 Simplifications 375

13.6 Summary 381

14 Applications 383

14.1 Introduction 383

14.2 Exact Trace Formula 384

14.3 Weyl Formulas 387

14.4 WKB in a Different Bight 390

14.5 Gutswiller's Trace Formula 395

14.6 Summary 404

Index 407