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Physica Scripta

Manifest Rotation Symmetric Expressions for Angular Momentum Eigenfunctions

J. O. Eeg, J. Wroldsen

Published Dec 1, 1983• 9 citations
Publisher

Abstract

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.

DOI
10.1088/0031-8949/28/6/001

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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.
In Section 3 we saw how to combine the intrinsic spin s and the orbital angular momentum L to form eigenstates of J 2 where J = L + s . The coefficients in ( 3.30 ) and ( 3.31 ) are special cases of a general class of transformation coefficients. The states on the r.h.s. of ( 3.30 ) and ( 3.31 ) are eigenstates of s 2 , s z , L 2 , L z . On the left hand side there are eigenstates of s 2 , L 2 , J 2 and J z . In general, we may start with any two sets ψ j 1 m 1 , ψ j 2 m 2 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203739716/ff034a11-a369-4bc6-a470-f8cf1794dcb5/content/eq354.tif"/> of two different systems. The spins j 1 and j 2 may be either orbital angular momenta or total spins of two particles, two intrinsic spins or one orbital angular momentum and one intrinsic spin. The latter case was considered in Section 3. In other cases j 1 may be the rank of an irreducible tensor operator and j 2 the angular momentum of a wave function or both j 1 and j 2 may be the ranks of two irreducible tensor operators. To simplify the notation we will use, for the transformation of the sets ψ j 1 m 1 , ψ j 2 m 2 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203739716/ff034a11-a369-4bc6-a470-f8cf1794dcb5/content/eq355.tif"/> under infinitesimal rotations, the expressions ( 6.20 ). This will not detract from the applicability of the following results to other tensorial sets. As pointed out several times above, the following mathematical derivations have general validity and are independent of the physical nature of ψ j 1 m 1 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203739716/ff034a11-a369-4bc6-a470-f8cf1794dcb5/content/eq356.tif"/> and ψ j 2 m 2 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203739716/ff034a11-a369-4bc6-a470-f8cf1794dcb5/content/eq357.tif"/> . They are due only to the transformation properties of the two sets under rotations.
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.