Chapter 10 - The Three-dimensional Rotation Groups

Abstract

This chapter discusses various aspects of the three-dimensional rotation groups. The conjugacy classes of the groups are studied, and the irreducible representations of the isomorphic Lie algebras su(2) and so(3) are derived. The intimate connection between the isomorphic Lie algebras su(2) and so(3), and the algebra of quantum mechanical angular momentum operators is noted. The main conclusions concerning the irreducible representations of su(2) are summarized in the form of a theorem. The results have immediate application in the theory of angular momentum and the theory of isotopic spin, as well as forming the basis of the representation theory of semisimple Lie algebras. The direct products of irreducible representations and the Clebsch–Gordan coefficients are analyzed. It is found that the eigenfunctions of the time-independent Schrodinger equation are the basic functions of the irreducible representations of this group. The components of the operator grad transform as irreducible tensor operators of the representation of the 0(3) are also elaborated.