Non-Relativistic Quantum Theory: Dynamics, Symmetry and Geometry Book

Preface vii

1 How Did Schrodinger Get His Equation? 1

2 Heisenberg's Matrix Mechanics and Dirac's Re-creation of it 11

3 Dirac's Derivation of the Quantum Conditions 17

4 The Equivalence between Matrix Mechanics and Wave Mechanics 21

5 The Dirac Delta Function 27

6 Why Do We Need Hilbert Space? 33

7 The Dirac Bra Ket Notation and the Riesz Theorem 37

8 Self-Adjoint Operators in Hilbert Space 45

9 The Spectral Theorem, Discrete and Continuous Spectra 53

10 Coordinate and Momentum Representations of Quantum States, Fourier Transforms 59

11 The Uncertainty Principle 63

12 Commutator Algebra 73

13 Ehrenfest's Theorem 77

14 The Simple Harmonic Oscillator 81

15 Complete Set of Commuting Observables 95

16 Solving Schrodinger's Equation 99

17 Symmetry, Invariance, and Conservation in Quantum Mechanics 113

18 Why is Group Theory Useful in Quantum Mechanics? 125

19 50(3) and SU(2) 131

20 The Spectrum of the Angular Momentum Operators 145

21 Whence the Spherical Harmonics? 151

22 Irreducible Representations of SU(2) and 50(3), Rotation Matrices 159

23 Direct Product Representations, Clebsch-Gordon Coefficients 169

24 Transformations of Wave Functions and Vector Operators under 50(3) 173

25 Irreducible Tensor Operators and the Wigner-Eckart Theorem 179

26 Reduction of Direct Product Representations of 50(3): The Addition of Angular Momenta 183

27 The Calculation of Clebsch-Gordon Coefficients: The 3-j Symbols 189

28 Applications of the Wigner-Eckart Theorem 199

29 The Symmetric Groups 209

30 The Lie Algebra of 50(4) and the Hydrogen Atom 233

31 Stationary Perturbations 243

32 The Fine Structure of Hydrogen: Application ofDegenerate Perturbation Theory 255

33 Time-Dependent Perturbation Theory 263

34 Interaction of Matter with the Classical Radiation Field: Application of Time-Dependent Perturbation Theory 275

35 Potential Scattering Theory 285

36 Analytic Properties of the 5-Matrbc: Bound States and Resonances 307

37 Non-Perturbative Bound-State and Scattering-State SoJutions: Radiation-Induced Bound-Continuum Interactions 317

38 Geometric Phases: The Aharonov-Bohm Effect and the Magnetic Monopole 333

39 The Berry Phase in Molecular Dynamics 339

40 The Dynamic Phase: Riemann Surfaces in the Semiclassical Theory of Non-Adiabatic Collisions; Homotopy and Homology 349

41 ”The Connection is the Gauge Field and the Curvature is the Force“: Some Differential Geometry 367

42 Topological Quantum (Chern) Numbers: The Integer Quantum Hall Effect 385

43 de Rham Cohomology and Chern Classes: Some More Differential Geometry 405

44 Chern-Simons Forms: The Fractional Quantum Hall Effect, Anyons and Knots 413

References 429

Index 433