Group Theory: And its Application to the Quantum Mechanics of Atomic Spectra

Group Theory: And Its Application To The Quantum Mechanics Of Atomic Spectra aims to describe the application of group theoretical methods to problems of quantum mechanics with specific reference to atomic spectra. Chapters 1 to 3 discuss the elements of linear vector theory, while Chapters 4 to 6 deal more specifically with the rudiments of quantum mechanics itself. Chapters 7 to 16 discuss the abstract group theory, invariant subgroups, and the general theory of representations. These chapters are mathematical, although much of the material covered should be familiar from an elementary course in quantum theory. Chapters 17 to 23 are specifically concerned with atomic spectra, as is Chapter 25. The remaining chapters discuss topics such as the recoupling (Racah) coefficients, the time inversion operation, and the classical interpretations of the coefficients. The text is recommended for physicists and mathematicians who are interested in the application of group theory to quantum mechanics. Those who are only interested in mathematics can choose to focus on the parts more devoted to that particular area of the subject.

Table of Contents:
ContentsAuthor's Preface Translator's Preface 1. Vectors and Matrices Linear Transformations Linear Independence of Vectors2. 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 Product17. The Characteristics of Atomic Spectra Eigenvalues and Quantum Numbers The Vector Addition Model Appendix to 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 Unitarity of Rotation Operators 21. The Total Angular Momentum Quantum Number 22. The Fine Structure of Spectral Lines23. Selection and Intensity Rules with Spin The Hönl-Kronig Intensity Formulas The Lande g-Formula The Interval Rule 24. Racah Coefficients Conjugate Complex Representations Symmetric Form of the Vector Coupling Coefficients Co variant and Contra variant 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