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Fortschritte der Physik

An Operator Method for the Addition of Two Angular Momenta by Symmetrizing the Product of n Spin Functions

S. M. Razaullah Ansari

Published Jan 1, 1967• 4 citations
Publisher

Abstract

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.

DOI
10.1002/prop.19670151202

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

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A Spinor Approach to the SU(2) Clebsch-Gordan Coefficients*

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Clebsch-Gordan Coefficients

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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.

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Products of generalized equivalent operators in angular momentum theory

Serge Luryi

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Combinatorial computation of Clebsch‐Gordan coefficients

Klaus Schertler, Markus H. Thoma

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.

arXiv:quant-ph/9504015•Annalen der Physik•Jan 1, 1996•1 citation
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.
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.
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.
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.