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Journal of Mathematical Physics

Projection of Exact Spin Eigenfunctions

Vedene H. Smith, Frank E. Harris

Published Apr 1, 1969• 39 citations
Publisher

Abstract

The relationships defining spin projectors are reviewed and the explicit formulations of the resulting spin-dependent (Sanibel) coefficients are shown to be consistent. Recurrence relations among Sanibel coefficients are derived. It is shown how the formal results here presented are useful in the calculation of matrix elements of spin-free and spin-dependent operators.

DOI
10.1063/1.1664904

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The relationships defining spin projectors are reviewed and the explicit formulations of the resulting spin-dependent (Sanibel) coefficients are shown to be consistent. Recurrence relations among Sanibel coefficients are derived. It is shown how the formal results here presented are useful in the calculation of matrix elements of spin-free and spin-dependent operators.
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.
General, closed-form expressions are presented for the matrix elements of ssSK{\mathit{ssS}}_{\mathit{K}}ssSK​(si{\mathbf{s}}_{\mathit{i}}si​,sj{\mathbf{s}}_{\mathit{j}}sj​) (rank K=0,1,2), tensors in the single-particle spin operators si{\mathbf{s}}_{\mathit{i}}si​, in the unitary-group approach. Factorization of the matrix elements in terms of 6-j and 9-j coefficients is achieved through methods of spin-graphical analysis. Results are also obtained for the rank-2 spin-spin operator.
By mapping the perturbation scheme onto the ladder operator formalism, the field of application of the Schrodinger-Infeld-Hull factorisation method is enlarged. It is shown how, at each order of the perturbation, perturbed ladder operators can be constructed. Thus, without having to calculate explicitly either the excited unperturbed functions or any matrix element, one obtains analytical expressions of the perturbed eigenvalues in terms of the quantum numbers of the factorisable unperturbed problem. A three-terms recurrence relation, valid at any rank of the perturbation, is derived and leads to closed form expressions of the perturbed eigenfunctions. Consequently, a closed form expression of any matrix element on the basis of the perturbed eigenfunctions is easily obtained from the calculation of one unique particular integral.
Abstract Some compositions of the addition and subtraction operators and recurrence relations for the Sanibel‐type coefficients cu, v (n, s, M) generated by these compositions are studied. A local representation of the fermion creation–annihilation operators via the addition and subtraction operators is obtained. Operators of single excitations, coupling, and decoupling operators, in terms of which the unitary group generators can be expressed are defined. The resulting representation of the nonelementary unitary group generators is much more simple than in the Gelfand–Tzetlin basis and in the most general case contains only six logically different terms, each of them possessing quite transparent physical significance.