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International Journal of Quantum Chemistry

Derivation of the Clebsch–Gordan coefficients by means of projection operators

Jean‐Louis Calais

Published Sep 1, 1968• 5 citations
Publisher

Abstract

Abstract The Clebsch–Gordan coefficients for the coupling of two angular momenta are derived by using the projection operator technique, developed by Löwdin. The derivation is done in two steps; first for the so‐called principal case, i.e. k = m, then for the general case with an arbitrary m. Two different derivations are given for the principal case, a direct one and one based on a recursion procedure. The general case is obtained from the principal case with a step‐down operator.

DOI
10.1002/qua.560020511

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Abstract The Clebsch–Gordan coefficients for the coupling of two angular momenta are derived by using the projection operator technique, developed by Löwdin. The derivation is done in two steps; first for the so‐called principal case, i.e. k = m, then for the general case with an arbitrary m. Two different derivations are given for the principal case, a direct one and one based on a recursion procedure. The general case is obtained from the principal case with a step‐down operator.
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.
Abstract A general expression of Clebsch‐Gordan coefficients is derived by means of de Broglie‐like méthode de fusion: the second angular momentum is added to the first in n steps of half each, the maximum symmetric composition of the n spin functions being taken into account explicitly. The procedure is operatorial. Group‐theoretically speaking the irreducible representations of the direct product of two irreducible representations of the rotation group are realized via symmetric group Sn with the aid of projection operators (equivalent to Young's symmetrizers). The decomposition of the representations of Sn therein is performed inductively by utilizing the chain of subgroups: Sn ⊃ Sn−1 … ⊃ S2 ⊃ S1.
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.
This chapter discusses the addition of angular momenta in the physicist’s way. From the considerations it follows that there are several ways to construct complete systems of commuting angular momentum observables. Belonging to two complete sets (two different ‘couplings’) of commuting angular momentum observables, there must exist a unitary transformation. For more than two angular momenta there will be more sets and more unitary transformations between the different couplings. The matrix elements of these transformations are called ‘recoupling coefficients’. They can be expressed by sums over products of Clebsch-Gordan coefficients (CGCs); this is to be expected from the fact that repeated coupling of two angular momenta generates any desired coupling scheme. It also is to be expected that the transformations between different coupling schemes become rapidly more complicated with growing numbers of subsystems. A powerful and elegant graphical representation for CGCs exists which can be used in the calculations of coupling of angular momenta.