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Journal of Mathematical Physics

Projected Angular Momentum States

Nick Karayianis, Carole A. Morrison

Published Jun 1, 1965• 4 citations
Publisher

Abstract

Known results regarding the projection of total spin states from product wavefunctions involving spin-½ particles are generalized. It is shown that total spin states projected from product wavefunctions involving particles with arbitrary spin, but otherwise restricted to having either maximum or minimum z component of spin, are no more complicated in form than for the spin-½ case.

DOI
10.1063/1.1704346

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Known results regarding the projection of total spin states from product wavefunctions involving spin-½ particles are generalized. It is shown that total spin states projected from product wavefunctions involving particles with arbitrary spin, but otherwise restricted to having either maximum or minimum z component of spin, are no more complicated in form than for the spin-½ case.
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.
Spin-s systems Angular momentum is one of the fundamental quantities of Newtonian physics, and in quantum physics its importance is at least as great. In quantum mechanics we often distinguish between two types of angular momentum: orbital angular momentum , which a system of particles possesses due to particle motion through space; and spin angular momentum , which is an intrinsic property of a particle. The distinction will be important later, but for now we will ignore it. We will here refer to angular momentum of any sort as “spin” and develop general-purpose mathematical tools for its description. We have already dealt with spin systems, particularly the example of a spin-½ particle. Our approach began with the empirical observation that a measurement of any spin component of a spin-½ particle could yield only the results +ħ/2 or −ħ/2. We introduced the basis states | z ±〉 for the two-dimensional Hilbert space ℋ. We also gave other basis states such as {| x ±〉} and {| y ±〉} in terms of the | z ±〉 states. From basis states and measurement values we constructed operators for the spin components S x , S y , and S z . With the operators in hand, we could then examine the algebraic relations between them (such as the commutation relation in Exercise 3.56). Our job here is to generalize our analysis to systems of arbitrary spin. To do this, we will reverse our chain of logic.
In the first part of this chapter we shall show how it is possible to represent dynamical variables by matrices instead of differential operators without affecting the predicted results of physically significant quantities. We shall see that, although this representation is often more complicated and cumbersome than the methods used earlier, it has a particularly simple form when applied to problems involving angular momentum, and has the great advantage that it can be used in the case of spin where no differential operator representation exists. We shall use such spin matrices to analyze experiments designed to measure the components of spin in various directions and find that these provide an important and illustrative example of the quantum theory of measurement.