Chapter 5 - Representations of Groups — Developments

Abstract

This chapter describes developments that are particularly significant in the applications of theory of group representations to quantum mechanics. The basic functions of unitary irreducible representations are easily determined by a purely automatic process involving certain projection operators. The procedure for constructing basis functions requires an explicit knowledge of the matrix elements of the representations. The Wigner–Eckart theorem for groups of coordinate transformations is discussed. It is found that if the matrix of the Clebsch–Gordan coefficients is chosen to be unitary, there is still a degree of arbitrariness in the Clebsch–Gordan coefficients. The Wigner–Eckart theorem provides both the most succinct and the most powerful expression in the entire field of application of group theory in physical problems. Every finite Abelian group is either a finite cyclic group or is isomorphic to a direct product of a set of finite cyclic groups. It is observed that for one physically important type of group, the induced representation method not only produces irreducible representations of the group, but it also generates an entire set of such representations.