Group Theory and Its Applications in Physics / by Teturo Inui, Yukito Tanabe, Yositaka Onodera.

Format:

Book

Author/Creator:

Inui, Teturo., Author.

Tanabe, Yukito., Author.

Onodera, Y. (Yositaka), Author.

Series:

Springer Series in Solid-State Sciences, 2197-4179 ; 78

Language:

English

Subjects (All):

Mathematical physics.

Crystallography.

Atoms.

Molecules.

Mathematical Methods in Physics.

Theoretical, Mathematical and Computational Physics.

Crystallography and Scattering Methods.

Atomic, Molecular and Chemical Physics.

Local Subjects:

Mathematical Methods in Physics.

Theoretical, Mathematical and Computational Physics.

Crystallography and Scattering Methods.

Atomic, Molecular and Chemical Physics.

Physical Description:

1 online resource (XV, 397 p.)

Edition:

1st ed. 1990.

Place of Publication:

Berlin, Heidelberg : Springer Berlin Heidelberg : Imprint: Springer, 1990.

Language Note:

English

Summary:

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.

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 ofReducible 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

7.14Electronic 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.

Notes:

"With 72 Figures."

Includes bibliographical references and index.

ISBN:

3-642-80021-1