Petrashen M.I., Trifonov E.D. Applications of Group Theory in Quantum Mechanics

Dover Publications, Inc., 2009. — 478 p. — (Dover Books on Physics). — ISBN 9780486172729.Geared toward postgraduate students, theoretical physicists, and researchers, this advanced text explores the role of modern group-theoretical methods in quantum theory. The authors based their text on a physics course they taught at a prominent Soviet university. Readers will find it a lucid guide to group theory and matrix representations that develops concepts to the level required for applications.
The text's main focus rests upon point and space groups, with applications to electronic and vibrational states. Additional topics include continuous rotation groups, permutation groups, and Lorentz groups. A number of problems involve studies of the symmetry properties of the Schroedinger wave function, as well as the explanation of "additional" degeneracy in the Coulomb field and certain subjects in solid-state physics. The text concludes with an instructive account of problems related to the conditions for relativistic invariance in quantum theory.Abstract Groups
Representations of Point Groups
Composition of Representations and the Direct Products of Groups
Wigner’s Theorem
Point Groups
Decomposition of a Reducible Representation into an Irreducible Representation
Space Groups and Their Irreducible Representations
Classification of the Vibrational and Electronic States of a Crystal
Continuous Groups
Irreducible Representations of the Three—Dimensional Rotation Group
The Properties of Irreducible Representations of the Rotation Group
Some Applications of the Theory of Representation of the Rotation Group in Quantum Mechanics
Additional Degeneracy in a Spherically Symmetric Field
Permutation Groups
Symmetrized Powers of Representations
Symmetry Properties of Multi-Electron Wave Functions
Symmetry Properties of Wave Functions for a System of Identical Particles with Arbitrary Spins
Classification of the States of a Multi-Electron Atom
Applications of Group Theory To Problems Connected With the Perturbation Theory
Selection Rules
The Lorentz Group and its Irreducible Representations
The Dirac Equation