Group Theory and Its Applications in Physics

This book has been written to introduce readers to group theory and its ap­ plications in atomic physics, molecular physics, and solid-state physics. The first Japanese edition was published in 1976. The present English edi­ tion has been translated by the authors from the revised and enlarged edition of 1980. In translation, slight modifications have been made in. Chaps. 8 and 14 to update and condense the contents, together with some minor additions and improvements throughout the volume. The authors cordially thank Professor J. L. Birman and Professor M. Car­ dona, who encouraged them to prepare the English translation. Tokyo, January 1990 T. Inui . Y. Tanabe Y. Onodera Preface to the Japanese Edition As the title shows, this book has been prepared as a textbook to introduce readers to the applications of group theory in several fields of physics. Group theory is, in a nutshell, the mathematics of symmetry. It has three main areas of application in modern physics. The first originates from early studies of crystal morphology and constitutes a framework for classical crystal physics. The analysis of the symmetry of tensors representing macroscopic physical properties (such as elastic constants) belongs to this category. The sec­ ond area was enunciated by E. Wigner (1926) as a powerful means of handling quantum-mechanical problems and was first applied in this sense to the analysis of atomic spectra. Soon, H.

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.