Chapter 7 - Some Applications to Physical Problems

Summary

Introductory Remarks

The concepts and techniques of angular momentum theory are by now all but ubiquitous in present-day quantum physics; there is scarcely a single issue of a physics journal that does not in some article directly use angular momentum constructs. It would accordingly be rather fatuous, if not indeed quite impossible, to attempt any definitive (or even very detailed) survey of applications.

The purpose of this chapter is therefore rather more modest; we shall focus attention on several typical applications of the theory, in widely different contexts, with an emphasis toward illustrating the basic principles. In discussing any particular field of application, however, we have attempted not to be extensive, but instead have chosen to present a novel, or, at least, unfamiliar, aspect of the subject in some detail. (The discussion in Section 6g – of a result due to Casimir – provides a good example of our intentions.)

Basic Principles Underlying the Applications

Before turning to specific examples, it will be useful to review briefly the principles underlying the applications. We have seen that a symmetry in quantum physics is a transformation on state vectors (active view – vectors in Hilbert space are transformed and operators are kept fixed) and on observables (passive view – vectors in Hilbert space are kept fixed and operators are transformed) such that all probabilities are preserved (Chapter 3, Section 14). Using the Wigner theorem, we found that this led to either a unitary, or an antiunitary, realization of the symmetry by operators.