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Journal of Mathematical Physics

Construction of Orthonormal Angular-Momentum Operators

Paul L. Corio

Published Jul 1, 1968• 12 citations
Publisher

Abstract

The problem of constructing an orthonormal basis for the (2j + 1)2-dimensional space of angular-momentum operators is reduced to the problem of constructing a sequence of orthogonal polynomials of a discrete real variable. A number of properties of the polynomials are obtained, including a pure recurrence formula and several mixed recurrence relations. Three explicit representations of the polynomials are given, together with a table of orthonormal angular-momentum operators.

DOI
10.1063/1.1664678

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The problem of constructing an orthonormal basis for the (2j + 1)2-dimensional space of angular-momentum operators is reduced to the problem of constructing a sequence of orthogonal polynomials of a discrete real variable. A number of properties of the polynomials are obtained, including a pure recurrence formula and several mixed recurrence relations. Three explicit representations of the polynomials are given, together with a table of orthonormal angular-momentum 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.
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 theory of angular momentum performance a significant position in the classical and quantum mechanical study of physical properties, such as studies into nuclear, atomic, and molecular processes, as well as other quantum problems, including spherical symmetry. In this analysis, angular momentum operators are described in multiple ways, based on the angular momentum operator's commutator, matrix, and geometric representation, The eigenvalue and eigenvector were also known for operatorsJ ̂_±,J ⃑ ̂^2, J ̂_x,J ̂_y and J ̂_zwithin the |j,┤ ├ m⟩ basis. Furthermore, in quantum mechanics, angular momentum is called quantized variable, meaning that it comes in discrete quantities. In contrast to the macroscopic system case where a continuous variable is angular momentum.
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.