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Communications in Theoretical Physics

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

Ling-Di Yang, Famin Chen

Published Jun 1, 2019• 0 citations
Publisher

Abstract

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.

DOI
10.1088/0253-6102/71/6/663

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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.
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.
This chapter presents key results on angular momentum in quantum mechanics, beginning with a theorem on the simultaneous diagonalization of commuting operators. The eigenvalues and eigenstates of both the total angular momentum operator and its z-component are derived. For a quantum rotator in the p-state with zero magnetic quantum number, it is shown that the most probable trajectory lies in a plane containing the z-axis, and the angular momentum vector is perpendicular to it. The procedure for angular momentum addition is explained in detail, including the use of Clebsch–Gordan coefficients to perform the explicit decomposition of tensor products of irreducible representations. The Bloch sphere representation of qubit spin states is introduced. The chapter then provides an experimental background to the discovery of spin, introduces the Pauli matrices, and demonstrates how they act on spin states. It concludes with the formulation of the Pauli equation, which incorporates spin into the framework of nonrelativistic quantum mechanics. Metadata/Keywords: angular momentum, simultaneous diagonalization, magnetic quantum number, Clebsch–Gordan coefficients, spin, Pauli matrices, Bloch sphere, spin states, tensor product, Pauli equation
The approach to quantum mechanics which we have used to derive the matrix treatment of spin from first principles is now employed to treat systems of compounded angular momentum. A general treatment is first given, which is then applied to the concrete cases of a spin-0 and a spin-1 system obtained by adding the spins of two spin-1/2 systems. Thus the probability amplitudes for measurements on the systems are derived, as well as the matrix vectors and operators corresponding to the systems. The matrix operators and states obtained are different from the standard forms and are much more generalized. The new results are applied to the case of joint measurements on the subsystems of such a system; this is a problem that has been made very topical by the high level of interest in the foundations of quantum mechanics. As a consequence of the insights arising from this treatment, we show that the Clebsch-Gordan coefficients are amenable to generalization, and we give the generalized forms for these cases.
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.