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Journal of Mathematical Physics

Exact Eigenfunctions of Angular Momentum by Rotational Projection

Jerome K. Percus, A. Rotenberg

Published Sep 1, 1962• 70 citations
Publisher

Abstract

By an appropriate rotational procedure, a projection operator can be constructed to give exact eigenfunctions of angular momentum. Applications to cases involving spin and orbital angular momentum are given and, in particular, all spin multiplets of the alternant molecular structure are obtained. The connection with the group theoretical approach is discussed.

DOI
10.1063/1.1724308

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By an appropriate rotational procedure, a projection operator can be constructed to give exact eigenfunctions of angular momentum. Applications to cases involving spin and orbital angular momentum are given and, in particular, all spin multiplets of the alternant molecular structure are obtained. The connection with the group theoretical approach is discussed.
The conventional method of constructing wave functions of pure angular momentum for a composite system is based on the idea of coupling the angular momenta of the constituents. Instead of this synthetic method, an analytical approach is here introduced. It is pointed out that an arbitrary trial wave function for the total system in a unique way must be resolvable into orthogonal components of pure angular momentum associated with different quantum numbers. A particular component can be selected by means of certain 'projection operators', which annihilate all components except the one desired. Such a projection operator is simply a product of commuting factors, each one of which annihilates a specific eigenfunction to the angular momentum under consideration. Physically this idea is of importance, since one can now start out from a rough model wave function based essentially on qualitative arguments, for instance of the independent-particle type, and then obtain a mathematically and physically much better trial function by selecting the particular component of the original function which has the correct symmetry type desired. The projection operators are studied in some detail both as products and in expanded form. The case of degenerate subspaces of the same angular momentum quantum numbers is discussed, and the problem of constructing an orthogonal subset of functions is solved by a simple elimination procedure; the connection with the conventional seniority idea is briefly discussed.
The Frobenius covariant is used to construct a projection operator onto the spin-S eigenspaces associated with the squares of the total and orbital angular momenta. The covariant admits two equivalent representations: as a polynomial in powers of the scalar product of the spin and orbital angular momentum operators, and as a finite expansion in terms of their respective polarization operators. A correspondence is established with Villars' angular momentum projection, used in nuclear structure studies.
The Frobenius covariant is used to construct a projection operator onto the spin-S eigenspaces associated with the squares of the total and orbital angular momenta. The covariant admits two equivalent representations: as a polynomial in powers of the scalar product of the spin and orbital angular momentum operators, and as a finite expansion in terms of their respective polarization operators. A correspondence is established with Villars’ angular momentum projection, used in nuclear structure studies.
Abstract In this chapter we introduce the concept of orbital angular momentum into the mathematical language of quantum mechanics. We discuss the commutation relations of the corresponding operators and their roles as generators of rotations in the context of the theory of Lie groups and Lie algebras, focusing on the three-dimensional rotation group SO(3). We further examine how scalar and vector-valued operators transform under spatial rotations. We then turn to the derivation of the eigenfunctions and eigenvalues of the angular-momentum operators, for which we follow a dual approach in analogy to that employed for the harmonic oscillator in Chapter 5. Using both the algebraic ladder-operator method and an analytical method using spherical polar coordinates we discover the relevant azimuthal and magnetic quantum numbers and their relation to the spherical harmonics and the Legendre polynomials