Addition of Angular Momenta—The Wigner-Eckart Theorem

Abstract

In Section V.1 the elementary rotator is defined as the system described by an irreducible representation of the algebra of angular momentum. Section V.2 discusses then the direct product of two irreducible representations of angular momentum and its reduction with respect to the total angular momentum. The Clebsch-Gordan coefficients are introduced, their recursion relations are derived, and their most frequently used values are tabulated. In Section V.3 tensor operators are introduced and the Wigner-Eckart theorem for the rotation group is stated without derivation. In Section V.4 a new observable, parity, is introduced. Parity is then applied to discuss the spectrum of diatomic symmetric-top molecules. In an Appendix to Section V.3 the irreducible representations of the algebras of SO(3, 1), SO(4), and E(3) are derived.