Group Theory and Quantum Mechanics
1. Introduction
1.1.
The Nature of the Problem
1.2.
The Role of Symmetry
2. Abstract Group Theory
2.1.
Definitions and Nomenclature
2.2.
Illustrative Examples
2.3.
2.4.
2.5.
Rearrangement Theorem
Cyclic Groups
Subgroups and Cosets
2.6.
2.7.
2.8.
2.9.
Example Groups of Finite Order
Conjugate Elements and Class Structure
Normal Divisors and Factor Groups
Class Multiplication
3. Theory of Group Representations
3.1.
Definitions
3.2.
3.3.
3.4.
3.5.
3.6.
3.7.
3.8.
3.9.
3.10.
Proof of the Orthogonality Theorem
The Character of a Representation
Construction of Character Tables
Decomposition of Reducible Representations
Application of Representation Theory in Quantum Mechanics
Illustrative Representations of Abelian Groups
Basis Functions for Irreducible Representations
Direct-product Groups
Direct-product Representations within a Group
4. Physical Applications of Group Theory
4.1.
Crystal-symmetry Operators
4.2.
The Crystallographic Point Groups
4.3.
Irreducible Representations of the Point Groups
4.4.
Elementary Representations of the Three-dimensional Rotation Group
4.5.
Crystal-field Splitting of Atomic Energy Levels
4.6.
Intermediate Crystal-field-splitting Case
4-7.
Weak-crystal-field Case and Crystal Double Groups
4.8.
4.9.
4.10.
4.11.
4.12.
Introduction of Spin Effects in the Medium-field Case
Group-theoretical Matrix-element Theorems
Selection Kules and Parity
Directed Valence
Application of Group Theory to Directed Valence
5. Full Rotation Group and Angular Momentum
5.1.
Rotational Transformation Properties and Angular Momentum
5.2.
Continuous Groups
5.3.
Representation of Rotations through Eulerian Angles
5.4.
5.5.
5.6.
Homomorphism with the Unitary Group
Representations of the Unitary Group
Representation of the Rotation Group by Representations of the Unitary
Group
5.7.
5.8.
5.9.
5.10.
5.11.
5.12.
Application of the Rotation-representation Matrices
Vector Model for Addition of Angular Momenta
The Wigner or Clebsch-Gordan Coefficients
Notation, Tabulations, and Symmetry Properties of the Wigner Coefficients
Tensor Operators
The Wigner-Eckart Theorem
5.13.
5.14.
5.15.
5.16.
5.17.
The Racah Coefficients
Application of Racah Coefficients
The Rotation-Inversion Group
Time-reversal Symmetry
More General Invariances
6. Quantum Mechanics of Atoms
6.1.
Review of Elementary Atomic Structure and Nomenclature
6.2.
The Hamiltonian
6.3.
6.4.
6.5.
6.6.
6.7.
6.8.
6.9.
6.10.
Approximate Eigenfunctions
Calculation of Matrix Elements between Determinantal Wavefunctions
Hartree-Fock Method
Calculation of L-S-term Energies
Evaluation of Matrix Elements of the Energy
Eigenfunctions and Angular-momentum Operations
Calculation of Fine Structure
Zeeman Effect
6.11.
Magnetic Hyperfine Structure
6.12.
Electric Hyperfine Structure
7. Molecular Quantum Mechanics
7.1.
Born-Oppenheimer Approximation
7.2.
Simple Electronic Eigenfunctions
7.3.
Irreducible Representations for Linear Molecules
7.4.
The Hydrogen Molecule
7.5.
Molecular Orbitals
7.6.
Heitler-London Method
7.7.
Orthogonal Atomic Orbitals
7.8.
7.9.
7.10.
Group Theory and Molecular Orbitals
Selection Rules for Electronic Transitions
Vibration of Diatomic Molecules
7.11.
7.12.
7.13.
7.14.
7.15.
7.16.
7.17.
Normal Modes in Polyatomic Molecules
Group Theory and Normal Modes
Selection Rules for Vibrational Transitions
Molecular Rotation
Effect of Nuclear Statistics on Molecular Rotation
Asymmetric Rotor
Vibration-Rotation Interaction
7.18.
Rotation-Electronic Coupling
8. Solid-state Theory
8.1.
Symmetry Properties in Solids
8.2.
The Reciprocal Lattice and Brillouin Zones
8.3.
Form of Energy-band Wavefunctions
8.4.
Crystal Symmetry and the Group of the k Vector
8.5.
Pictorial Consideration of Eigenfunctions
8.6.
Formal Consideration of Degeneracy and Compatibility
8.7.
8.8.
8.9.
8.10.
8.11.
8.12.
8.13.
8.14.
Group Theory and the Plane-wave Approximation
Connection between Tight- and Loose-binding Approximations
Spin-orbit Coupling in Band Theory
Time Reversal in Band Theory
Magnetic Crystal Groups
Symmetries of Magnetic Structures
The Landau Theory of-Second-order Phase Transitions
Irreducible Representations of Magnetic Groups
Appendix
A.
Review of Vectors, Vector Spaces, and Matrices
B.
Character Tables for Point-symmetry Groups
C.
Tables of ck and ak Coefficients for s, p, and d Electrons