© 2026 Aqora Quantum S.A.S.

Terms Privacy Legal Notice

CRC Press eBooks

Clebsch-Gordan Coefficients

Victor Anisimov, James J. P. Stewart

Published Dec 6, 2019• 0 citations

Abstract

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.

DOI: 10.1201/9780429063862-6

Related papers

Angular Momentum

Ahmad Amer

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

CRC Press eBooks•Nov 5, 2025•0 citations

Addition of Angular Momenta

Масуд Чайчиан, Rolf Hagedorn

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.

CRC Press eBooks•Jul 14, 2023•0 citations

Addition theorems for spin spherical harmonics. I Preliminaries

Antonio O. Bouzas

We develop a systematic approach to deriving addition theorems for, and some other bilocal sums of, spin spherical harmonics. In this first part we establish some necessary technical results. We discuss the factorization of orbital and spin degrees of freedom in certain products of Clebsch-Gordan coefficients, and obtain general explicit results for the matrix elements in configuration space of tensor products of arbitrary rank of the position and angular-momentum operators. These results are the basis of the addition theorems for spin spherical harmonics obtained in part II.

•Jan 1, 2013•4 citations

Addition theorems for spin spherical harmonics: I. Preliminaries

Antonio O Bouzas

We develop a systematic approach to deriving addition theorems for, and some other bilocal sums of, spin spherical harmonics. In this first part we establish some necessary technical results. We discuss the factorization of orbital and spin degrees of freedom in certain products of Clebsch–Gordan coefficients, and obtain general explicit results for the matrix elements in configuration space of tensor products of arbitrary rank of the position and angular-momentum operators. These results are the basis of the addition theorems for spin spherical harmonics obtained in part II (2011 J. Phys. A: Math. Theor. 44 165302 ).

arXiv:1103.2982•Journal of Physics A Mathematical and Theoretical•Mar 24, 2011•5 citations

Angular Momentum

A. C. Sharma

This chapter covers various aspects of orbital, total, and spin angular momenta and the Clebsch-Gordon coefficients. An approach that is not found in many books, is used to derive eigenvalues of orbital angular momentum. The Pauli theory of spin half systems, addition of angular momenta, eigenfunctions of orbital angular momentum, computation of Clebsch-Gordon coefficients, solved examples, and unsolved exercises are included in the chapter. The basic purpose of the chapter has been to bring out the quantum mechanical importance of the theory of angular momenta with the use of essential mathematical derivations, which are self-consistent, and to adopt an elegant modern approach. Special mention on why orbital angular momentum cannot have non-integer values is made. The concept of spin angular momentum, which is purely a quantum mechanical quantity, is introduced in a simple and most convincing manner.

CRC Press eBooks•Jun 18, 2021•0 citations

Efficient Computation of Clebsch-Gordan Coefficients

William O. Straub

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.

•Mar 1, 2014•1 citation

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 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

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.

We develop a systematic approach to deriving addition theorems for, and some other bilocal sums of, spin spherical harmonics. In this first part we establish some necessary technical results. We discuss the factorization of orbital and spin degrees of freedom in certain products of Clebsch-Gordan coefficients, and obtain general explicit results for the matrix elements in configuration space of tensor products of arbitrary rank of the position and angular-momentum operators. These results are the basis of the addition theorems for spin spherical harmonics obtained in part II.

We develop a systematic approach to deriving addition theorems for, and some other bilocal sums of, spin spherical harmonics. In this first part we establish some necessary technical results. We discuss the factorization of orbital and spin degrees of freedom in certain products of Clebsch–Gordan coefficients, and obtain general explicit results for the matrix elements in configuration space of tensor products of arbitrary rank of the position and angular-momentum operators. These results are the basis of the addition theorems for spin spherical harmonics obtained in part II (2011 J. Phys. A: Math. Theor. 44 165302 ).

This chapter covers various aspects of orbital, total, and spin angular momenta and the Clebsch-Gordon coefficients. An approach that is not found in many books, is used to derive eigenvalues of orbital angular momentum. The Pauli theory of spin half systems, addition of angular momenta, eigenfunctions of orbital angular momentum, computation of Clebsch-Gordon coefficients, solved examples, and unsolved exercises are included in the chapter. The basic purpose of the chapter has been to bring out the quantum mechanical importance of the theory of angular momenta with the use of essential mathematical derivations, which are self-consistent, and to adopt an elegant modern approach. Special mention on why orbital angular momentum cannot have non-integer values is made. The concept of spin angular momentum, which is purely a quantum mechanical quantity, is introduced in a simple and most convincing manner.

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.