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Journal of Physics B Atomic Molecular and Optical Physics

Angular momentum and Heisenberg's correspondence principle. II. More complex matrix elements

S C McFarlane

Published Jul 14, 1993• 3 citations
Publisher

Abstract

For pt.I see ibid., vol.25, no.20, p.4045-57 (1992). Previous work on the correspondence principle limit of matrix elements of tensor operators is extended to more complex matrix elements. Limiting forms of several such matrix elements are obtained; each case yields an approximate form for an angular momentum coupling coefficient: two for the 6-j symbol and one for the 9-j symbol. These are similar to existing approximations but are more symmetrical in form and, as a result, more accurate.

DOI
10.1088/0953-4075/26/13/011

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For pt.I see ibid., vol.25, no.20, p.4045-57 (1992). Previous work on the correspondence principle limit of matrix elements of tensor operators is extended to more complex matrix elements. Limiting forms of several such matrix elements are obtained; each case yields an approximate form for an angular momentum coupling coefficient: two for the 6-j symbol and one for the 9-j symbol. These are similar to existing approximations but are more symmetrical in form and, as a result, more accurate.
Heisenberg's form of the correspondence principle is applied to the matrix elements of tensor operators. A correspondence limit of the Wigner-Eckart theorem is obtained. It is shown that the Clebsch-Gordan coefficient tends in this limit to the product of a reduced rotation matrix element with a simple multiplicative factor. A similar, but not identical, expression has long been known as a formal limit obtaining when two of the three angular momenta involved become large. It is characteristic of correspondence principle expressions that the initial and final quantal states must be represented by some 'mean' classical orbit, the choice of which remains ambiguous. Here recursion relations are used to arrive at the optimal choice. In this way, a correspondence principle approximation to the Clebsch-Gordan coefficient is obtained which may be used with success far from the limit to which it nominally refers. This approximation is tested numerically against both exact and semiclassical computations.
Heisenberg's form of the correspondence principle for non-relativistic matrix elements has been used to evaluate matrix elements for various potentials. These are compared with available quantum mechanical results to check the general validity of the use of classical mechanics and correspondence principles in the derivation of quantum mechanical expressions. Using a one dimensional harmonic oscillator potential matrix elements of the form (n mod qm mod n+s) and (n mod pm mod n+s) are worked out. A Morse potential is then considered and the matrix elements for position, momentum, and kinetic energy are evaluated. Using a Coulomb potential a similar procedure is used for functions of various position coordinates, from which dipole and quadrupole moments from transitions between n, l states are calculated. The agreement with quantum mechanics is found to be generally good, and in some cases identical results are obtained.
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.
Reduced matrix elements of a tensorial product of n tensor operators in the basis of N-particle angular-momentum eigenstates are expanded into a sum of products of n 1-particle reduced matrix elements and a single (N + n)-particle recoupling coefficient. Application of the formula is illustrated by specific examples. The method leaves the coupling schemes of N-particle states undisturbed, which allows summation over intermediate states in a product of matrix elements to be made for any number of factors. A formula is given for the sum of products of two matrix elements and the extension to a greater number of matrix elements is illustrated by an example which is reduced to a form suitable for numerical evaluation. Such summations over products of matrix elements occur in the perturbation theory of configuration interaction and have hitherto been discussed in terms of effective operators. The connection of the method used here with the effective-operator approach is demonstrated.
The semiclassical formula for matrix elements, sometimes called the Heisenberg correspondence principle, relates matrix elements (operators in energy representation) to phase space functions (Weyl representatives). The formula does not make sense for arbitrary operators; when it is valid it implicitly fixes the relative phases of the eigenfunctions of the Hamiltonian. The conventions, which have to be used for the wavefunctions in position or momentum representation, are given here in explicit form, and we present a class of operators related to coherent states of high energy for which the Heisenberg correspondence principle holds.