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Journal of Physics B Atomic and Molecular Physics

Determination of operator equivalents from irreducible tensors

L L Boyle, L C Jones

Published Jan 1, 1971• 4 citations
Publisher

Abstract

A new method of derivation of operator equivalents is proposed, in which individual irreducible tensor components are expressed in terms of both real and complex unit tensors derived from real and complex (or bi) vectors. The method is extended to irreducible tensor operators which do not have full permutation symmetry and it is found that these are equivalent to totally symmetric tensor operators of rank equal to the angular momentum quantum number.

DOI
10.1088/0022-3700/4/1/006

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A new method of derivation of operator equivalents is proposed, in which individual irreducible tensor components are expressed in terms of both real and complex unit tensors derived from real and complex (or bi) vectors. The method is extended to irreducible tensor operators which do not have full permutation symmetry and it is found that these are equivalent to totally symmetric tensor operators of rank equal to the angular momentum quantum number.
The angular momentum operators define a set of irreducible tensors which are unique except for a normalization constant. The normalization is conveniently defined in terms of statistical tensors which describe oriented states. The properties of the tensors discussed here include: (1) the trace of products of components of such tensors, (2) symmetry properties of the traces, and (3) the expansion of products of components of these tensors into a sum of irreducible tensors. The corresponding expansion of commutators and anticommutators of these components is also discussed briefly.
Formulas are obtained which give the matrix representation, relative to a product basis, of the projection operator for the total angular momentum of a system. If the individual angular momenta are not too large, the matrix elements depend upon a small number of parameters, independent of the number of angular momenta coupled. Recurrence relations between elements and symmetry properties of elements are derived. These results enable one to perform the vector coupling of a large number of angular momenta in a relatively simple fashion. The connection between the matrix elements and vector coupling coefficients is discussed. Several important special cases are treated in detail.