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Physica Scripta

Manifest Rotation Symmetric Expressions for Angular Momentum Eigenfunctions II

J. O. Eeg, J. Wroldsen

Published Oct 1, 1984• 3 citations
Publisher

Abstract

We consider expressions for eigenfunctions for spin s , orbital angular momentum l , and total angular momentum j in terms of (2 s + 1) × (2 j + 1) multipole transition matrices (MTM). In a previous paper explicit expressions were given for angular momenta s, j ⩽ 3. The present paper completes the previous work by giving expressions for arbitrary values of s, l and j .

DOI
10.1088/0031-8949/30/4/003

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We consider expressions for eigenfunctions for spin s , orbital angular momentum l , and total angular momentum j in terms of (2 s + 1) × (2 j + 1) multipole transition matrices (MTM). In a previous paper explicit expressions were given for angular momenta s, j ⩽ 3. The present paper completes the previous work by giving expressions for arbitrary values of s, l and j .
We give manifest rotation symmetric expressions for eigenfunctions for spin s , orbital angular momentum l and total angular momentum j = l + s ,...,| l - s | in terms of (2 j + 1) × (2 s + 1) multipole transition matrices (MTM). These matrices, which are irreducible tensor matrices, have an algebra together with ordinary spin matrices for spin s and spin j . Explicit expressions for MTM's and their algebra are given for angular momenta ⩽ 3. By means of some examples we show that within this formalism angular integrations in central field problems will be simplified considerably. Thus the formalism turns out to be very useful for instance for calculations within the MIT-bag and also within spin-spin interactions in atomic physics.
Synopsis: General operator properties of angular momenta are recalled, and a physical motivation for adding them is given. The product and total angular momentum bases are defined. It is shown how to express one basis in terms of the other, and an explicit example of adding two spin 1/2 angular momenta is given. A further example consists of adding a general orbital angular momentum value to spin 1/2. These considerations are applied to perturbations in the hydrogen atom energy spectrum, where the correct choice for calculations is the total angular momentum basis.
Synopsis: General operator properties of angular momenta are recalled, and a physical motivation for adding them is given. ƒe product and total angular momentum bases are defined. It is shown how to express one basis in terms of the other, and an explicit example of adding two spin 1/2 angular momenta is given. A further example consists of adding a general orbital angular momentum value to spin 1/2. ƒese considerations are applied to perturbations in the hydrogen atom energy spectrum, where the correct choice for calculations is the total angular momentum basis.
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.