Group Theory and Its Applications in Physics [E-Book] / by Teturo Inui, Yukito Tanabe, Yositaka Onodera.
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 editi...
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Full text |
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Personal Name(s): | Inui, Teturo, author |
Onodera, Yositaka, author / Tanabe, Yukito, author | |
Imprint: |
Berlin, Heidelberg :
Springer,
1990
|
Physical Description: |
XV, 397 p. online resource. |
Note: |
englisch |
ISBN: |
9783642800214 |
DOI: |
10.1007/978-3-642-80021-4 |
Series Title: |
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Springer Series in Solid-State Sciences ;
78 |
Subject (LOC): |
- 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.