Group Theory in Quantum Mechanics

Group Theory in Quantum Mechanics: An Introduction to its Present Usage introduces the reader to the three main uses of group theory in quantum mechanics: to label energy levels and the corresponding eigenstates; to discuss qualitatively the splitting of energy levels as one starts from an approximate Hamiltonian and adds correction terms; and to aid in the evaluation of matrix elements of all kinds, and in particular to provide general selection rules for the non-zero ones. The theme is to show how all this is achieved by considering the symmetry properties of the Hamiltonian and the way in which these symmetries are reflected in the wave functions.
This book is comprised of eight chapters and begins with an overview of the necessary mathematical concepts, including representations and vector spaces and their relevance to quantum mechanics. The uses of symmetry properties and mathematical expression of symmetry operations are also outlined, along with symmetry transformations of the Hamiltonian. The next chapter describes the three uses of group theory, with particular reference to the theory of atomic energy levels and transitions. The following chapters deal with the theory of free atoms and ions; representations of finite groups; the electronic structure and vibrations of molecules; solid state physics; and relativistic quantum mechanics. Nuclear physics is also discussed, with emphasis on the isotopic spin formalism, nuclear forces, and the reactions that arise when the nuclei take part in time-dependent processes.
This monograph will be of interest to physicists and mathematicians.

Contenu

Preface
Notation
I. Symmetry Transformations

1. The Uses of Symmetry Properties
2. Expressing Symmetry Operations Mathematically
3. Symmetry Transformations of the Hamiltonian
4. Groups of Symmetry Transformations
5. Group Representations
6. Applications to Quantum Mechanics
   II. The Quantum Theory of a Free Atom
7. Some Simple Groups and Representations
8. The Irreducible Representations of the Full Rotation Group
9. Reduction of the Product Representation D(j) X D(j')
10. Quantum Mechanics of a Free Atom; Orbital Degeneracy
11. Quantum Mechanics of a Free Atom including Spin
12. The Effect of the Exclusion Principle
13. Calculating Matrix Elements and Selection Rules
    III. The Representations of Finite Groups
14. Group Characters
15. Product Groups
16. Point-Groups
17. the Relationship between Group Theory and the Dirac Method
    IV. Further Aspects of the Theory of Free Atoms and Ions
18. Paramagnetic Ions in Crystalline Fields
19. Time-Reversal and Kramers' Theorem
20. Wigner and Racah Coefficients
21. Hyperfine Structure
    V. The Structure and Vibrations of Molecules
22. Valence Bond Orbitals and Molecular Orbitals
23. Molecular Vibrations
24. Infra-Red and Raman Spectra
    VI. Solid State Physics
25. Brillouin Zone Theory of Simple Structures
26. Further Aspects of Brillouin Zone Theory
27. Tensor Properties of Crystals
    VII. Nuclear Physics
28. The Isotopic Spin Formalism
29. Nuclear Forces
30. Reactions
    VIII. Relativistic Quantum Mechanics
31. The Representations of the Lorentz Group
32. The Dirac Equation
33. Beta Decay
34. Positronium
    Appendices
    A. Matrix Algebra
    B. Homomorphism and Isomorphism
    C. Theorems on Vector Spaces and Group Representation
    D. Sohur's Lemma
    E. Irreducible Representations of Abelian Groups
    F. Momenta and Infinitesimal Transformations
    G. The Simple Harmonic Oscillator
    H. the Irreducible Representations of the Complete Lorentz Group
    I. Table of Wigner Coefficients (jj' mm'| JM)
    J. Notation For the Thirty-Two Crystal Point Groups
    K. Character Tables for the Crystal Point-Groups
    L. Character Tables for the Axial Rotation Group and Derived Groups
    List of General References, with Reviews
    Bibliography
    Subject Index