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American Journal of Physics

Simple Derivation of the Clebsch-Gordan Coefficients

R. T. Sharp

Published Feb 1, 1960• 8 citations
Publisher

Abstract

The three-dimensional angular momentum operator J takes a simple form when the operand is a scalar function f(ξ,η) of the two complex variables ξ, η; so do the angular momentum eigen-states fmi(ξ,η). Exploitation of this circumstance leads to a simple derivation of Racah's formula for the Clebsch-Gordan (vector addition) coefficients.

DOI
10.1119/1.1935073

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The three-dimensional angular momentum operator J takes a simple form when the operand is a scalar function f(ξ,η) of the two complex variables ξ, η; so do the angular momentum eigen-states fmi(ξ,η). Exploitation of this circumstance leads to a simple derivation of Racah's formula for the Clebsch-Gordan (vector addition) coefficients.
The addition theorem of angular momentum states that the angular momentum operator of a coupled state is the vector sum of the angular momentum operators of the uncoupled states. Given that, the eigenfunction of the angular momentum of a system in the coupled state can be represented as a linear combination of eigenfunctions of the system in its uncoupled state and, in turn, the eigenfunctions of the uncoupled state are a direct product of eigenfunctions of the individual particles. Clebsch-Gordan coefficients emerge as an overlap integral in the series expansion of the coupled state in the basis of uncoupled states. This chapter provides a detailed derivation of the addition theorem of angular momentum, including obtaining the lower and upper boundaries of the quantum numbers, deriving the orthonormality conditions for Clebsch-Gordan coefficients, and using shift operators to generate the members of a multiplet. It also derives a general expression for the evaluation of Clebsch-Gordan coefficients, provides a detailed analysis of the addition of angular momenta and electron spin to build the resulting Clebsch-Gordan matrix, and shows that a rotation transformation of the direct product of eigenfunctions of the angular momentum and spin leads to the recurrence relations for the Wigner matrix.
The problem of angular momentum addition requires the calculation of Clebsch-Gordan coefficients. While systems involving small values of momenta and spin present no special problem, larger systems require extensive computational effort. This paper describes a straightforward method for computing the coefficients for any two-particle problem exactly by means of a simplified form of the recursion formula in a notation that is particularly accessible to the third- or fourth-year student. The method is summarized in a brief BASIC program.
Abstract We review the irreducible representation of an angular momentum vector operator constructed in terms of spinor algebra. We generalize the idea of spinor approach to study the coupling of the eigenstates of two independent angular momentum vector operators. Utilizing the spinor algebra, we are able to develop a simple way for calculating the SU(2) Clebsch-Gordan (CG) coefficients. The explicit expression for the SU(2) CG coefficients is worked out, and some simple physical examples are presented to illustrate the spinor approach.