Group Theory

Group Theory

And Its Application to the Quantum Mechanics of Atomic Spectra

1st Edition - January 1, 1959

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  • Author: Eugene P. Wigner
  • eBook ISBN: 9781483275765

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Group Theory and its Application to the Quantum Mechanics of Atomic Spectra describes the applications of group theoretical methods to problems of quantum mechanics with particular reference to atomic spectra. The manuscript first takes a look at vectors and matrices, generalizations, and principal axis transformation. Topics include principal axis transformation for unitary and Hermitian matrices; unitary matrices and the scalar product; linear independence of vectors; and real orthogonal and symmetric matrices. The publication also ponders on the elements of quantum mechanics, perturbation theory, and transformation theory and the bases for the statistical interpretation of quantum mechanics. The book discusses abstract group theory and invariant subgroups, including theorems of finite groups, factor group, and isomorphism and homomorphism. The text also reviews the algebra of representation theory, rotation groups, three-dimensional pure rotation group, and characteristics of atomic spectra. Discussions focus on eigenvalues and quantum numbers, spherical harmonics, and representations of the unitary group. The manuscript is a valuable reference for readers interested in the applications of group theoretical methods.

Table of Contents

  • Contents

    Author's Preface

    Translator's Preface

    1. Vectors and Matrices

    Linear Transformations

    Linear Independence of Vectors

    2. Generalizations

    3. The Principal Axis Transformation

    Unitary Matrices and the Scalar Product

    The Principal Axis Transformation for Unitary and Hermitian Matrices

    Real Orthogonal and Symmetric Matrices

    4. The Elements of Quantum Mechanics

    5. Perturbation Theory

    6. Transformation Theory and the Bases for the Statistical Interpretation of Quantum Mechanics

    7. Abstract Group Theory

    Theorems for Finite Groups

    Examples of Groups

    Conjugate Elements and Classes

    8. Invariant Subgroups

    The Factor Group

    Isomorphism and Homomorphism

    9. The General Theory of Representations

    10. Continuous Groups

    11. Representations and Eigenfunctions

    12. The Algebra of Representation Theory

    13. The Symmetric Group

    Appendix to Chapter 13. A Lemma related to the Symmetric Group

    14. The Rotation Groups

    15. The Three-Dimensional Pure Rotation Group

    Spherical Harmonics

    The Homomorphism of the Two-Dimensional Unitary Group onto the Rotation Group

    The Representations of the Unitary Group

    The Representations of the Three-Dimensional Pure Rotation Group

    16. The Representations of the Direct Product

    17. The Characteristics of Atomic Spectra

    Eigenvalues and Quantum Numbers

    The Vector Addition Model

    Appendix t o Chapter 17. A Relationship Among Binomial Coefficients

    18. Selection Rules and the Splitting of Spectral Lines

    19. Partial Determination of Eigenfunctions from Their Transformation Properties

    20. Electron Spin

    The Physical Basis for the Pauli Theory

    Invariance of the Description under Spatial Rotation

    Connection with Representation Theory

    Appendix to Chapter 20. Linearity and unitary Rotation Operators

    21. The Total Angular Momentum Quantum Number

    22. The Fine Structure of Spectral Lines

    23. Selection and Intensity Rules with Spin

    The Hönl-Kronig Intensity Formulas

    The Landé G-Formula

    The Interval Rule

    24. Racah Coefficients

    Conjugate Complex Representations

    Symmetric Form of the Vector Coupling Coefficients

    Covariant and Contravariant Vector Coupling Coefficients

    Racah Coefficients

    Matrix Elements of Spin-Free Tensor Operators

    General Two-Sided Tensor Operators

    25. The Building-Up Principle

    26. Time Inversion

    Time Inversion and Antiunitary Operators

    Determination of the Time Inversion Operator

    Transformation of the Eigenfunctions under Antiunitary Operators

    Reduction of Corepresentations

    Determination of the Irreducible Corepresentations

    Consequences of Invariance under Time Inversion

    27. Physical Interpretation and Classical Limits of Representation Coefficients, Three- and Six-j Symbols

    Representation Coefficients

    Vector Coupling Coefficients

    Racah Coefficients

    Appendix A. Conventions

    Appendix B. Summary of Formulas

    Subject Index

Product details

  • No. of pages: 384
  • Language: English
  • Copyright: © Academic Press 1959
  • Published: January 1, 1959
  • Imprint: Academic Press
  • eBook ISBN: 9781483275765

About the Author

Eugene P. Wigner

About the Editor

H. S. W. Massey

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