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CRC Press eBooks

Angular Momentum

Ahmad Amer

Published Nov 5, 2025• 0 citations
Publisher

Abstract

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

DOI
10.1201/9781003677109-11

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The Future of Quantum Theory of Angular Momentum

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THE QUANTUM THEORY OF ANGULAR MOMENTUM began with Bohr’s enunciation of the rules for the quantization of orbital angular momentum in 1913, and the refinement of these rules by Sommerfeld [ 1 ] and Landé [ 2 ] within the framework of the old quantum theory. A basic result for angular momentum in modern quantum theory, which we call The First Fundamental Result, was given in the third foundational paper in 1926 by Born, Heisenberg, and Jordan [ 3 ], who not only gave the basic commutation relations for the components of the total angular momentum J, but all finite-dimensional Hermitian matrix representations of these components. Dirac [ 4 ] simultaneously deduced these basic commutation relations using his algebraic approach to quantum mechanics, and later gave a marvelously brief derivation of their matrix representations in his book [ 5 ]. Further developments, including the famous Pauli matrices and results relating to angular momentum selection rules for transitions in atomic systems, are given in the papers by Heisenberg and Jordan [ 6 ], Pauli [ 7 ], and Güttinger and Pauli [ 8 ]. These works, and the books by Born and Jordan [ 9 ] and Condon and Shortley [ 10 ], fixed the framework for much of the developments in angular momentum theory, which was hardly distinguishable from the general quantum theory in the context of which it was developed for the purpose of giving a consistent description of the spectra of atoms. Little emphasis was given to the universal and distinctive role of symmetry.

Birkhäuser Boston eBooks•Jan 1, 2000•0 citations
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 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.
This book offers a concise introduction to the angular momentum, one of the most fundamental quantities in all of quantum mechanics. Beginning with the quantization of angular momentum, spin angular momentum, and the orbital angular momentum, the author goes on to discuss the Clebsch-Gordan coefficients for a two-component system. After developing the necessary mathematics, specifically spherical tensors and tensor operators, the author then investigates the 3- j , 6- j , and 9- j symbols. Throughout, the author provides practical applications to atomic, molecular, and nuclear physics. These include partial-wave expansions, the emission and absorption of particles, the proton and electron quadrupole moment, matrix element calculation in practice, and the properties of the symmetrical top molecule.
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 chapter begins with the introduction of the two-particle Schrödinger wave equation (SWE) and the solution of this equation for the hydrogen atom. The orbital angular momentum of the electron results from the SWE solution. The Pauli spinors are introduced, and the SWE wavefunctions are modified to account for the spin of the electron. The structure of multielectron atoms is then discussed. The discussion is focused on low-Z atoms for which Russell–Saunders or LS coupling is appropriate. Alternate coupling schemes are briefly discussed. Angular momentum coupling algebra, the Clebsch–Gordan coefficients, and 3j symbols are then introduced. The Wigner–Eckart theorem is discussed, and the use of irreducible spherical tensors for evaluation of quantum mechanical matrix elements is discussed in detail.
Abstract Angular momentum in classical mechanics is given by a vector. The plane perpendicular to this vector, in accordance to central field theory, determines the space in which particle motion takes place. No such simple picture exists in quantum mechanics. The states of a particle in a central field are proportional to spherical harmonics which do not define any plane of motion. In the first part of this paper we discuss the angular distribution of particle position and compare it to the classical probabilistic approach. In the second part, the matter of addition of angular momenta is discussed. In classical mechanics this means addition of vectors, while in quantum mechanics Clebsch–Gordan coefficients have to be used. We have found classical approximations to quantum coefficients and the limit of their applicability. This analysis gives a basis for the so-called ‘vector addition model’ used in some elementary textbooks on atomic physics. It can help to better understand the addition of angular momenta in quantum mechanics.
THE QUANTUM THEORY OF ANGULAR MOMENTUM began with Bohr’s enunciation of the rules for the quantization of orbital angular momentum in 1913, and the refinement of these rules by Sommerfeld [ 1 ] and Landé [ 2 ] within the framework of the old quantum theory. A basic result for angular momentum in modern quantum theory, which we call The First Fundamental Result, was given in the third foundational paper in 1926 by Born, Heisenberg, and Jordan [ 3 ], who not only gave the basic commutation relations for the components of the total angular momentum J, but all finite-dimensional Hermitian matrix representations of these components. Dirac [ 4 ] simultaneously deduced these basic commutation relations using his algebraic approach to quantum mechanics, and later gave a marvelously brief derivation of their matrix representations in his book [ 5 ]. Further developments, including the famous Pauli matrices and results relating to angular momentum selection rules for transitions in atomic systems, are given in the papers by Heisenberg and Jordan [ 6 ], Pauli [ 7 ], and Güttinger and Pauli [ 8 ]. These works, and the books by Born and Jordan [ 9 ] and Condon and Shortley [ 10 ], fixed the framework for much of the developments in angular momentum theory, which was hardly distinguishable from the general quantum theory in the context of which it was developed for the purpose of giving a consistent description of the spectra of atoms. Little emphasis was given to the universal and distinctive role of symmetry.