Non-Relativistic Quantum Theory: Dynamics, Symmetry, and Geometry (Hardcover)

This textbook is mainly for physics students at the advanced undergraduate and beginning graduate levels, especially those with a theoretical inclination. Its chief purpose is to give a systematic introduction to the main ingredients of the fundamentals of quantum theory, with special emphasis on those aspects of group theory (spacetime and permutational symmetries and group representations) and differential geometry (geometrical phases, topological quantum numbers, and Chern–Simons Theory) that are relevant in modern developments of the subject. It will provide students with an overview of key elements of the theory, as well as a solid preparation in calculational techniques.

Contents

Preface vii
1 HowDid Schr‥odinger 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 Schr‥odinger’s Equation 99
17 Symmetry, Invariance, and Conservation
in Quantum Mechanics 113
18 Why is Group Theory Useful in Quantum Mechanics? 125
19 SO(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 SO(3),
Rotation Matrices 159
23 Direct Product Representations,
Clebsch-Gordon Coefficients 169
24 Transformations of Wave Functions and
Vector Operators under SO(3) 173
25 Irreducible Tensor Operators and
the Wigner-Eckart Theorem 179
26 Reduction of Direct Product Representations of SO(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 SO(4) and the Hydrogen Atom 233
31 Stationary Perturbations 243
32 The Fine Structure of Hydrogen:
Application of Degenerate 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 S-Matrix:
Bound States and Resonances 307
37 Non-Perturbative Bound-State and Scattering-State
Solutions: 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