Group Theory and Its Applications in Physics [E-Book] / by Teturo Inui, Yukito Tanabe, Yositaka Onodera.
Inui, Teturo, (author)
Onodera, Yositaka, (author) / Tanabe, Yukito, (author)
Berlin, Heidelberg : Springer, 1990
XV, 397 p. online resource.
englisch
9783642800214
10.1007/978-3-642-80021-4
Springer Series in Solid-State Sciences ; 78
Full Text
Table of Contents:
  • 1. Symmetry and the Role of Group Theory
  • 1.1 Arrangement of the Book
  • 2. Groups
  • 2.1 Definition of a Group
  • 2.1.1 Multiplication Tables
  • 2.1.2 Generating Elements
  • 2.1.3 Commutative Groups
  • 2.2 Covering Operations of Regular Polygons
  • 2.3 Permutations and the Symmetric Group
  • 2.4 The Rearrangement Theorem
  • 2.5 Isomorphism and Homomorphism
  • 2.5.1 Isomorphism
  • 2.5.2 Homomorphism
  • 2.5.3 Note on Mapping
  • 2.6 Subgroups
  • 2.7 Cosets and Coset Decomposition
  • 2.8 Conjugate Elements; Classes
  • 2.9 Multiplication of Classes
  • 2.10 Invariant Subgroups
  • 2.11 The Factor Group
  • 2.11.1 The Kernel
  • 2.11.2 Homomorphism Theorem
  • 2.12 The Direct-Product Group
  • 3. Vector Spaces
  • 3.1 Vectors and Vector Spaces
  • 3.1.1 Mathematical Definition of a Vector Space
  • 3.1.2 Basis of a Vector Space
  • 3.2 Transformation of Vectors
  • 3.3 Subspaces and Invariant Subspaces
  • 3.4 Metric Vector Spaces
  • 3.4.1 Inner Product of Vectors
  • 3.4.2 Orthonormal Basis
  • 3.4.3 Unitary Operators and Unitary Matrices
  • 3.4.4 Hermitian Operators and Hermitian Matrices
  • 3.5 Eigenvalue Problems of Hermitian and Unitary Operators
  • 3.6 Linear Transformation Groups
  • 4. Representations of a Group I
  • 4.1 Representations
  • 4.1.1 Basis for a Representation
  • 4.1.2 Equivalence of Representations
  • 4.1.3 Reducible and Irreducible Representations
  • 4.2 Irreducible Representations of the Group C?v
  • 4.3 Effect of Symmetry Transformation Operators on Functions
  • 4.4 Representations of the Group C3v Based on Homogeneous Polynomials
  • 4.5 General Representation Theory
  • 4.5.1 Unitarization of a Representation
  • 4.5.2 Schur’s First Lemma
  • 4.5.3 Schur’s Second Lemma
  • 4.5.4 The Great Orthogonality Theorem T
  • 4.6 Characters
  • 4.6.1 First and Second Orthogonalities of Characters
  • 4.7 Reduction of Reducible Representations
  • 4.7.1 Restriction to a Subgroup
  • 4.8 Product Representations
  • 4.8.1 Symmetric and Antisymmetric Product Representations
  • 4.9 Representations of a Direct-Product Group
  • 4.10 The Regular Representation
  • 4.11 Construction of Character Tables
  • 4.12 Adjoint Representations
  • 4.13 Proofs of the Theorems on Group Representations
  • 4.13.1 Unitarization of a Representation
  • 4.13.2 Schur’s First Lemma
  • 4.13.3 Schur’s Second Lemma
  • 4.13.4 Second Orthogonality of Characters
  • 5. Representations of a Group II
  • 5.1 Induced Representations
  • 5.2 Irreducible Representations of a Group with an Invariant Subgroup
  • 5.3 Irreducible Representations of Little Groups or Small Representations
  • 5.4 Ray Representations
  • 5.5 Construction of Matrices of Irreducible Ray Representations
  • 6. Group Representations in Quantum Mechanics
  • 6.1 Symmetry Transformations of Wavefunctions and Quantum-Mechanical Operators
  • 6.2 Eigenstates of the Hamiltonian and Irreducibility
  • 6.3 Splitting of Energy Levels by a Perturbation
  • 6.4 Orthogonality of Basis Functions
  • 6.5 Selection Rules
  • 6.5.1 Derivation of the Selection Rule for Diagonal Matrix Elements
  • 6.6 Projection Operators
  • 7. The Rotation Group
  • 7.1 Rotations
  • 7.2 Rotation and Euler Angles
  • 7.3 Rotations as Operators; Infinitesimal Rotations
  • 7.4 Representation of Infinitesimal Rotations
  • 7.4.1 Rotation of Spin Functions
  • 7.5 Representations of the Rotation Group
  • 7.6 SU(2), SO(3) and O(3)
  • 7.7 Basis of Representations
  • 7.8 Spherical Harmonics
  • 7.9 Orthogonality of Representation Matrices and Characters
  • 7.9.1 Completeness Relation for XJ(?)
  • 7.10 Wigner Coefficients
  • 7.11 Tensor Operators
  • 7.12 Operator Equivalents
  • 7.13 Addition of Three Angular Momenta;Racah Coefficients
  • 7.14 Electronic Wavefunctions for the Configuration (nl)x
  • 7.15 Electrons and Holes
  • 7.16 Evaluation of the Matrix Elements of Operators
  • 8. Point Groups
  • 8.1 Symmetry Operations in Point Groups
  • 8.2 Point Groups and Their Notation
  • 8.3 Class Structure in Point Groups
  • 8.4 Irreducible Representations of Point Groups
  • 8.5 Double-Valued Representations and Double Groups
  • 8.6 Transformation of Spin and Orbital Functions
  • 8.7 Constructive Derivation of Point Groups Consisting of Proper Rotations
  • 9. Electronic States of Molecules
  • 9.1 Molecular Orbitals
  • 9.2 Diatomic Molecules: LCAO Method
  • 9.3 Construction of LCAO-MO: The ?-Electron Approximation for the Benzene Molecule
  • 9.3.1 Further Methods for Determining the Basis Sets
  • 9.4 The Benzene Molecule (Continued)
  • 9.5 Hybridized Orbitals
  • 9.5.1 Methane and sp3-Hybridization
  • 9.6 Ligand Field Theory
  • 9.7 Multiplet Terms in Molecules
  • 9.8 Clebsch - Gordan Coefficients for Simply Reducible Groups and the Wigner-Eckart Theorem
  • 10. Molecular Vibrations
  • 10.1 Normal Modes and Normal Coordinates
  • 10.2 Group Theory and Normal Modes
  • 10.3 Selection Rules for Infrared Absorption and Raman Scattering
  • 10.4 Interaction of Electrons with Atomic Displacements
  • 10.4.1 Kramers Degeneracy
  • 11. Space Groups
  • 11.1 Translational Symmetry of Crystals
  • 11.2 Symmetry Operations in Space Groups
  • 11.3 Structure of Space Groups
  • 11.4 Bravais Lattices
  • 11.5 Nomenclature of Space Groups
  • 11.6 The Reciprocal Lattice and the Brillouin Zone
  • 11.7 Irreducible Representations of the Translation Group…
  • 11.8 The Group of the Wavevector k and Its Irreducible Representations
  • 11.9 Irreducible Representations of a Space Group
  • 11.10 Double Space Groups
  • 12. Electronic States in Crystals
  • 12.1 Bloch Functions and E(k) Spectra
  • 12.2 Examples of Energy Bands: Ge and TIBr
  • 12.3 Compatibility or Connectivity Relations
  • 12.4 Bloch Functions Expressed in Terms of Plane Waves
  • 12.5 Choice of the Origin
  • 12.5.1 Effect of the Choice on Bloch Wavefunctions
  • 12.6 Bloch Functions Expressed in Terms of Atomic Orbitals
  • 12.7 Lattice Vibrations
  • 12.8 The Spin-Orbit Interaction and Double Space Groups….
  • 12.9 Scattering of an Electron by Lattice Vibrations
  • 12.10 Interband Optical Transitions
  • 12.11 Frenkel Excitons in Molecular Crystals
  • 12.12 Selection Rules in Space Groups
  • 12.12.1 Symmetric and Antisymmetric Product Representations
  • 13. Time Reversal and Nonunitary Groups
  • 13.1 Time Reversal
  • 13.2 Nonunitary Groups and Corepresentations
  • 13.3 Criteria for Space Groups and Examples
  • 13.4 Magnetic Space Groups
  • 13.5 Excitons in Magnetic Compounds; Spin Waves
  • 13.5.1 Symmetry of the Hamiltonian
  • 14. Landau’s Theory of Phase Transitions
  • 14.1 Landau’s Theory of Second-Order Phase Transitions
  • 14.2 Crystal Structures and Spin Alignments
  • 14.3 Derivation of the Lifshitz Criterion
  • 14.3.1 Lifshitz’s Derivation of the Lifshitz Criterion
  • 15. The Symmetric Group
  • 15.1 The Symmetric Group (Permutation Group)
  • 15.2 Irreducible Characters
  • 15.3 Construction of Irreducible Representation Matrices
  • 15.4 The Basis for Irreducible Representations
  • 15.5 The Unitary Group and the Symmetric Group
  • 15.6 The Branching Rule
  • 15.7 Wavefunctions for the Configuration (nl)x
  • 15.8 D(J) as Irreducible Representations of SU(2)
  • 15.9 Irreducible Representations of U(m)
  • Appendices
  • A. The Thirty-Two Crystallographic Point Groups
  • B. Character Tables for Point Groups
  • Answers and Hints to the Exercises
  • Motifs of the Family Crests
  • References.