2 00 1 Clebsch - Gordan coefficients and the binomial distribution

There appear to be two standard ways of calculation C-G coefficients in quantum mechanics. One method is to combine the “ladder” operator approach with orthogonality conditions. The second method is to use some type of closed form [2] which allows the coefficients to be calculated directly. However, the latter approach is at times considered “tedious” [2] and does not reveal any new information into the nature of these coefficients. In this paper we prove a theorem which not only permits a special class of C-G coefficients to be calculated from a simple formula but also directly connects them to both the hypergeometric and binomial distributions of classical probability. Before formulating and proving the theorem, we first define some notation. Let L = (L1, L2, L3) denote the angular momentum operator and define L = Lx ± iLy. (1)