Contents of

Quantum Mechanics. A Modern Approach
L.E. Ballentine
World Scientific 1998

Preface xi

Introduction: The Phenomena of Quantum Mechanics 1

Chapter 1 Mathematical Prerequisites 7
1.1 Linear Vector Space 7
1.2 Linear Operators II
1.3 Self-Adjoint Operators 15
1.4 Hilbert Space and Rigged Hilbert Space 26
1.5 Probability Theory 29
Problems 38

Chapter 2 The Formulation of Quantum Mechanics 42
2.1 Basic Theoretical Concepts 42
2.2 Conditions on Operators 48
2.3 General States and Pure States 50
2.4 Probability Distributions 55
Problems 60

Chapter 3 Kinematics and Dynamics 63
3.1 Transformations of States and Observables 63
3.2 The Symmetries of Space-Time 66
3.3 Generators of the Galilei Group 68
3.4 Identification of Operators with Dynamical Variables 76
3.5 Composite Systems 85
3.6 [[ Quantizing a Classical System ]] 87
3.7 Equations of Motion 89
3.8 Symmetries and Conservation laws 92
Problems 94

Chapter 4 Coordinate Representation and Applications 97
4.1 Coordinate Representation 97
4.2 The Wave Equation and Its Interpretation 98
4.3 GaliIei Transformation of Schrodinger's Equation 102
4.4 Probability Flux 104
4.5 Conditions on Wave Functions 106
4.6 Energy Eigenfunctions for Free Particles 109
4.7 Tunneling 110
4.8 Path Integrals 116
Problems 123

Chapter 5 Momentum Representation ana Applications 126
5.1 Momentum Representation 126
5.2 Momentum Distribution in an Atom 128
5.3 Bloch's Theorem 131
5.4 Diffraction Scattering: Theory 133
5.5 Diffraction Scattering: Experiment 139
5.6 Motion in a Uniform Force Field 145
Problems 149

Chapter 6 The Harmonic Oscillator 151
6.1 Algebraic Solution 151
6.2 Solution in Coordinate Representation 154
6.3 Solution in H Representation 157
Problems 158

Chapter 7 Angular Momentum 160
7.1 Eigenvalues and Matrix Elements 160
7.2 Explicit Form of the Angular Momentum Operators 164
7.3 Orbital Angular Momentum 166
7.4 Spin 171
7.5 Finite Rotations 175
7.6 Rotation Through 2\pi 182
7.7 Addition of Angular Momenta 185
7.8 Irreducible Tensor Operators 193
7.9 Rotational Motion of a Rigid Body 200
Problems 203

Chapter 8 State Preparation and Determination 206
8.1 State Preparation 206
8.2 State Determination 210
8.3 States of Composite Systems 216
8.4 Indeterminacy Relations 223
Problems 227

Chapter 9 Measurement and the Interpretation of States 230
9.1 An Example of Spin Measurement 230
9.2 A General Theorem of Measurement Theory 232
9.3 The Interpretation of a State Vector 234
9.4 Which Wave Function? 238
9.5 Spin Recombination Experiment 241
9.6 Joint and Conditional Probabilities 244
Problems 254

Chapter 10 Formation of Bound States 258
10.1 Spherical Potential Well 258
10.2 The Hydrogen Atom 263
10.3 Estimates from Indeterminacy Relations 271
10.4 Some Unusual Bound States 273
10.5 Stationary State Perturbation Theory 276
10.6 Variational Method 290
Problems 304

Chapter 11 Charged Particle in a Magnetic Field 307
11.1 Classical Theory 307
11.2 Quantum Theory 309
11.3 Motion in a Uniform Static Magnetic Field 314
11.4 The Aharonov-Bohm Effect 321
11.5 The Zeeman Effect 325
Problems 330

Chapter 12 Time-Dependent Phenomena 332
12.1 Spin Dynamics 332
12.2 Exponential and Nonexponential Decay 338
12.3 Energy-Time Indeterminacy Relations 343
12.4 Quantum Beats 347
12.5 Time-Dependent Perturbation Theory 349
12.6 Atomic Radiation 356
12.7 Adiabatic Approximation 363
Problems 367

Chapter 13 Discrete Symmetries 370
13.1 Space Inversion 370
13.2 Parity Nonconservation 374
13.3 Time Reversal 377
Problems 386

Chapter 14 The Classical Limit 388
14.1 Enrenfest's Theorem and Beyond 389
14.2 The Hamilton-Jacobi Equation and the
Quantum Potential 394
14.3 Quantal Trajectories 398
14.4 The Large Quantum Number Limit 400
Problems 404

Chapter 15 Quantum Mechanics in Phase Space 406
15.1 Why Phase Space Distributions? 406
15.2 The Wigner Representation 407
15.3 The Husimi Distribution 414
Problems 420

Chapter 16 Scattering 421
16.1 Cross Section 421
16.2 Scattering by a Spherical Potential 427
16.3 General Scattering Theory 433
16.4 Born Approximation and DWBA 441
16.5 Scattering Operators 447
16.6 Scattering Resonances 458
16.7 Diverse Topics 462
Problems 468

Chapter 17 Identical Particles 470
17.1 Permutation Symmetry 470
17.2 Indistinguishability of Particles 472
17.3 The Symmetrization Postulate 474
17.4 Creation and Annihilation Operators 478
Problems 492

Chapter 18 Many-Fermion Systems 493
18.1 Exchange 493
18.2 The Hartree-Fock Method 499
18.3 Dynamic Correlations 506
18.4 Fundamental Consequences for Theory 513
18.5 BCS Pairing Theory 514
Problems 525

Chapter 19 Quantum Mechanics of the
Electromagnetic Field 526
19.1 Normal Modes of the Field 526
19.2 Electric and Magnetic Field Operators 529
19.3 Zero-Point Energy and the Casimir Force 533
19.4 States of the EM Field 539
19.5 Spontaneous Emission 548
19.6 Photon Detectors 551
19.7 Correlation Functions 558
19.8 Coherence 566
19.9 Optical Homodyne Tomography -
Determining the Quantum State of the Field 578
Problems 581

Chapter 20 Bell's Theorem and Its Consequences 583
20.1 The Argument of Einstein, Podolsky, and Rosen 583
20.2 Spin Correlations 585
20.3 Bell's Inequality 587
20.4 A Stronger Proof of Bell's Theorem 591
20.5 Polarization Correlations 595
20.6 Bell's Theorem Without Probabilities 602
20.7 Implications of Bell's Theorem 607
Proolems 610

Appendix A Schur's Lemma 613

Appendix B Irreducibility of Q and P 615

Appendix C Proof of Wick's Theorem 616

Appendix D Solutions to Selected Problems 618

Bibliography 639

Index 651