Group Theory in Physics: An Introduction with a Focus on Solid State Physics

✍ Scribed by Jörg Bünemann

• Publisher: Springer International Publishing
• Year: 2024
• Tongue: English
• Leaves: 233
• Category: Library

✦ Table of Contents

Preface
Acknowledgements
Contents
1 Introduction
1.1 A Brief Historical Overview
1.2 Symmetries
1.2.1 Symmetries of Bodies
1.2.2 Symmetries in Classical Physics
1.2.3 Symmetries in Quantum Mechanics
Reference
2 Groups: Definitions and Properties
2.1 Definition of Groups and Simple Properties
2.2 Examples
2.2.1 The Group upper D 2D2
2.2.2 The Cyclic Group upper C 4C4
2.2.3 The Group upper D 3D3
2.3 Classes of Conjugate Elements, Subsets, and Cosets
2.3.1 The Rearrangement Theorem
2.3.2 Definition and Properties of Classes
2.3.3 Class Multiplication
2.3.4 Sub-groups and Cosets
2.3.5 Normal Sub-groups
2.3.6 Factor Groups
2.4 Product Groups
3 Point Groups
3.1 Definition of Point Groups
3.2 The Point Groups of the First Kind
3.2.1 The Groups upper C Subscript nCn
3.2.2 The Groups upper D Subscript nDn
3.2.3 The Tetrahedral Group upper TT
3.2.4 The Cubic Group upper OO
3.2.5 Icosahedron Group upper YY
3.3 Point Groups in Solids
3.4 The Point Groups of the Second Kind
3.4.1 Improper Point Groups Without the Inversion
3.4.2 Improper Point Groups which Include the Inversion
3.5 The 3232 Point Groups in Solids
3.6 The Seven Crystal Systems
4 Representations and Characters
4.1 Matrix Groups
4.1.1 Equivalent and Irreducible Matrix Groups
4.1.2 Schur's Lemma
4.2 Representations
5 Orthogonality Theorems
5.1 The Fundamental Theorem in the Theory of Representations
5.2 Consequences
5.2.1 Theorem 4: Orthogonality of the Characters
5.2.2 Proof of Theorems 1–4
5.2.3 Clear Criterion for the Irreducibility of a Representation
6 Quantum Mechanics and Group Theory
6.1 Representation Spaces
6.1.1 Definition of Representation Spaces
6.1.2 Representation Functions of Irreducible Representations
6.1.3 Representation Spaces and Invariant Sub-spaces
6.1.4 Irreducibility of Representation Spaces
6.1.5 The Expansion Theorem
6.2 Projection Operators
6.2.1 Theorem on the Orthogonality of Representation Spaces
6.3 Hamiltonians with Symmetries
6.3.1 Reminder: Degeneracies in Quantum Mechanics
6.3.2 Group Theoretical Treatment
6.3.3 Irreducibility Postulate
6.3.4 Example: A Particle in a One-Dimensional Potential
6.3.5 Diagonalization of Hamiltonians
Reference
7 Irreducible Representations of the Point Groups in Solids
7.1 Character Table with Representation Functions
7.2 Example: A Particle in a Cubic Box
8 Group Theory in Stationary Perturbation Theory Calculations
8.1 Reminder: Rayleigh-Schrödinger Perturbation Theory
8.2 Subduced Representations
8.3 Degenerate Perturbation Theory
8.4 Application: Splitting of Atomic Orbitals in Crystal Fields
8.4.1 The Atomic Problem
8.4.2 Splitting of Orbital Energies in Crystal Fields
8.5 Matrix Elements in Perturbation Theory
Reference
9 Material Tensors and Tensor Operators
9.1 Material Tensors
9.1.1 Physical Motivation
9.1.2 Transformation of Tensors
9.2 Product Representations
9.3 Independent Tensor Components
9.4 Tensor Operators
9.4.1 Definition of Tensor Operators
9.4.2 Irreducible Tensor Components
Reference
10 Matrix Elements of Tensor Operators: The Wigner-Eckart Theorem
10.1 Clebsch-Gordan Coefficients and the Wigner-Eckart Theorem for Angular Momenta
10.1.1 Clebsch-Gordan Coefficients
10.1.2 The Wigner-Eckart Theorem for Angular Momenta
10.2 Matrix Elements in the Time-Dependent Perturbation Theory
10.3 Coupling or Clebsch-Gordan Coefficients
10.4 The Wigner-Eckhart Theorem
11 Double Groups and Their Representations
11.1 Particles with Spin 1/2
11.2 Definition of Double Groups
11.3 The Algebra of the Double Groups
11.4 The Classes of the Double Groups
11.5 The Irreducible Representations of the Double Groups
11.5.1 Symmetric Representations
11.5.2 Extra Representations
Reference
12 Space Groups
12.1 Definitions
12.1.1 The Real Affine Group
12.1.2 Space Groups
12.2 Symmorphic and Non-symmorphic Space Groups
12.2.1 Non-primitive Translations
12.2.2 Difference Between Symmorphic and Non-symmorphic Space Groups
12.3 Inequivalent Space Groups
12.3.1 Matrix Space Groups
12.3.2 The 1414 Inequivalent Bravais Lattices
12.3.3 Classification of Space Groups
13 Representations of Space Groups
13.1 Irreducible Representations of the Translation Group
13.2 The Irreducible Representations of Symmorphic Space Groups
13.2.1 Irreducible Representations of a Star in General Position
13.2.2 Irreducible Representations of a Star in Non-general Position
13.3 Spectrum of a Hamiltonian with Space Group Symmetry
14 Particles in Periodic Potentials
14.1 Schrödinger Equation, Bloch Theorem
14.2 Irreducible Part of the Brillouin Zone
14.3 Compatibility Conditions
14.4 Solution of the Eigenvalue Problem with Plane Waves
14.5 Tight-Binding Models
14.5.1 Derivation of Tight-Binding Models
14.5.2 The Slater Koster Parameters
References
Appendix A The Schoenflies and the International Notation
A.1 The Schoenflies Notation
A.2 The International Notation
Appendix B Solutions to the Exercises
Index