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College Physics

Symmetry properties of Clebsch-Gordan coefficients

Qiong-Gui Lin

Published Jan 1, 2009• 0 citations
OpenAlex

Abstract

The freedom in the Clebsch-Gordan coefficients for the coupling between two angular momenta allowed by its definition is determined by the Schur's lemma.Then the symmetry properties of the Clebsch-Gordan coefficients are obtained by using their orthonormality and phase convention.

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Abstract The addition of angular momenta can be reduced to elementary coupling processes of spin‐ ‐particles. In this way, a method is developed which allows for a non‐recursive, simultaneous computation of all Clebsch‐Gordan coefficients concerning the addition of two angular momenta. The relevant equations can be interpreted easily, analogously to simple probabilistic considerations. They provide an improved understanding of the addition of angular momenta as well as a practicable evaluation of Clebsch‐Gordan coefficients in an easier way than within the well‐known methods.
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.
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.