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Physical Review A

Note on the traces of the angular momentum operators

Nazakat Ullah

Published Mar 1, 1980• 1 citation
Publisher

Abstract

It is shown that angular momentum operator identities between ladder operators and rotation operators can be used to study the trace problem. Using an explicit representation of the rotation matrices, the author evaluates the trace of the operator ${J}_{\ensuremath{-}}^{k}{J}_{+}^{k}$ in closed form. The identities are found to be particularly useful in providing checks for the sum of the traces.

DOI
10.1103/physreva.21.1069

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It is shown that angular momentum operator identities between ladder operators and rotation operators can be used to study the trace problem. Using an explicit representation of the rotation matrices, the author evaluates the trace of the operator J\ensuremath−kJ+k{J}_{\ensuremath{-}}^{k}{J}_{+}^{k}J\ensuremath−k​J+k​ in closed form. The identities are found to be particularly useful in providing checks for the sum of the traces.
The method of self-adjoint ladder operators, developed in Parts I and II, is applied to the solution of the generalized angular-momentum problem. This reveals many interesting aspects of this approach to eigenvalue problems and, in particular, its relationship to addition of angular momentum. The complete set of irreducible unitary representations of the underlying algebra is obtained and also the corresponding Clebsch-Gordan (Wigner) coefficients for the addition of spin and angular momentum in a space of arbitrary dimension.
For pedagogical purposes it is demonstrated that with little effort the customary discussion of rotations in introductory courses can be extended to naturally produce a number of angular momentum operator properties, if coordinate-free rotation operators are used in their exponential form. This form gives rise to a particularly useful coordinate-free representation of the generators of rotations in three-space, from which the student can derive the usual angular momentum formalism for the case j=1 by elementary vector algebra alone. Since in this approach the j=1 formalism appears as a set of relations among simple geometric objects in ordinary space, it admits of a geometric interpretation and will leave the student better prepared for a more profound appreciation of the abstract concepts of the full theory.