Maple procedures for the coupling of angular momenta II. Sum rule evaluation

Abstract

In a previous paper (S. Fritzsche, Comput. Phys. Commun. 103 (1997) 51), we defined data structures to deal with typical expressions from Racah algebra within the framework of Maple. Such expressions arise very frequently in various fields, for instance, by treating composite wave functions and tensor operators in many-particle physics. Often, these Racah expressions are written in terms of Clebsch-Gordan coefficients and Wigner n-j symbols. Our previous set of Maple procedures mainly concerned numerical computations on such symbols, the simplification by special values as well as the use of recursion relations. The full elegance of applying Racah algebra techniques in daily research work is, however, only revealed by the analytic simplification of more complex expressions. In practise, this even requires the major effort in dealing with these techniques. Its success closely depends on the knowledge of sum rules which typically include a number of dummy summation indices. The application of these sum rules is a rather straightforward task but may become very tedious for more difficult expressions due to the large number of symmetries of the Clebsch-Gordan coefficients and Wigner n-j symbols. We therefore extended the Racah program to facilitate sum rule evaluations in the given framework. A set of new and revised procedures now supports the evaluation of Racah algebra expressions by applying the orthogonality properties of the Wigner symbols and a variety of sum rules. More than 40 sum rules known from the literature and involving products of up to six Wigner n-j symbols have been implemented and are available for interactive use. The applicability of this new tool will be demonstrated by three examples from many-particle physics.