Clebsch-Gordan coefficients and the binomial distribution

TL;DR

This paper derives a specific class of Clebsch-Gordan coefficients using properties of the binomial distribution and conditional probability, revealing a new mathematical relationship in angular momentum coupling.

Contribution

It introduces a novel derivation of Clebsch-Gordan coefficients based on binomial distribution properties and conditional probability, expanding the mathematical understanding of angular momentum addition.

Findings

Derived a class of Clebsch-Gordan coefficients from binomial distribution

Established a specific formula relating coefficients when l=l1+l2

Connected probability theory with angular momentum coupling mathematics

Abstract

A class of Clebsch-Gordan coefficients are derived from the properties of conditional probability using the binomial distribution. In particular, in the case of $l=l_1+l_2$ it is shown that $$[<l_1/2-k_1, l_2/2-k_2|l/2, k=k_1+k_2]>^2 =\frac{(\begin{array}{c} l_1 k_1\end{array}) (\begin{array}{c}l_2 k_2\end{array})}{(\begin{array}{c}l k \end{array})}$$