Non-Relativistic Quantum Theory

Dynamics, Symmetry, and Geometry

Kai S Lam

• Publisher: WSPC
• Year: 2009
• eBook ISBN: 9789813107489
• Topic: Physical Sciences
• Subtopic: Mathematical & Computational Physics

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.

Table of contents

• Contents
• Preface
• 1 HowDid Schr¨odinger Get His Equation?
• 2 Heisenberg’s Matrix Mechanics and Dirac’s Re-creation of it
• 3 Dirac’s Derivation of the Quantum Conditions
• 4 The Equivalence between Matrix Mechanics and Wave Mechanics
• 5 The Dirac Delta Function
• 6 Why Do We Need Hilbert Space?
• 7 The Dirac Bra Ket Notation and the Riesz Theorem
• 8 Self-Adjoint Operators in Hilbert Space
• 9 The Spectral Theorem, Discrete and Continuous Spectra
• 10 Coordinate and Momentum Representations of Quantum States, Fourier Transforms
• 11 The Uncertainty Principle
• 12 Commutator Algebra
• 13 Ehrenfest’s Theorem
• 14 The Simple Harmonic Oscillator
• 15 Complete Set of Commuting Observables
• 16 Solving Schr¨odinger’s Equation
• 17 Symmetry, Invariance, and Conservation in Quantum Mechanics
• 18 Why is Group Theory Useful in Quantum Mechanics?
• 19 SO(3) and SU(2)
• 20 The Spectrum of the Angular Momentum Operators
• 21 Whence the Spherical Harmonics?
• 22 Irreducible Representations of SU(2) and SO(3), Rotation Matrices
• 23 Direct Product Representations, Clebsch-Gordon Coefficients
• 24 Transformations of Wave Functions and Vector Operators under SO(3)
• 25 Irreducible Tensor Operators and the Wigner-Eckart Theorem
• 26 Reduction of Direct Product Representations of SO(3): The Addition of Angular Momenta
• 27 The Calculation of Clebsch-Gordon Coe.cients: The 3-j Symbols
• 28 Applications of the Wigner-Eckart Theorem
• 29 The Symmetric Groups
• 30 The Lie Algebra of SO(4) and the Hydrogen Atom
• 31 Stationary Perturbations
• 32 The Fine Structure of Hydrogen: Application of Degenerate Perturbation Theory
• 33 Time-Dependent Perturbation Theory
• 34 Interaction of Matter with the Classical Radiation Field: Application of Time-Dependent Perturbation Theory
• 35 Potential Scattering Theory
• 36 Analytic Properties of the S-Matrix: Bound States and Resonances
• 37 Non-Perturbative Bound-State and Scattering-State Solutions: Radiation-Induced Bound-Continuum Interactions
• 38 Geometric Phases: The Aharonov-Bohm Effect and the Magnetic Monopole
• 39 The Berry Phase in Molecular Dynamics
• 40 The Dynamic Phase: Riemann Surfaces in the Semiclassical Theory of Non-Adiabatic Collisions; Homotopy and Homology
• 41 “The Connection is the Gauge Field and the Curvature is the Force”: Some Differential Geometry
• 42 Topological Quantum (Chern) Numbers: The Integer Quantum Hall Effect
• 43 de Rham Cohomology and Chern Classes: Some More Differential Geometry
• 44 Chern-Simons Forms: The Fractional Quantum Hall Effect, Anyons and Knots
• References
• Index