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Manuscript Title: Maple procedures for the coupling of angular momenta. I. Data structures and numerical computations.
Authors: S. Fritzsche
Program title: Racah
Catalogue identifier: ADFV_v1_0
Distribution format: gz
Journal reference: Comput. Phys. Commun. 103(1997)51
Programming language: Maple.
Computer: IBM Workstation.
Operating system: AIX 3.2.5.
RAM: 100K words
Keywords: General purpose, Computer algebra, Rotation group, Angular momentum, Vector coupling, Wigner n-j symbols, Racah algebra techniques, Spherical operators.
Classification: 4.1, 5.

Nature of problem:
Computer algebra (CA) is used to evaluate Wigner n-j symbols and vector coupling coefficients and to simplify typical expressions which appear by using Racah algebra techniques. A set of Maple commands for interactive work is presented.

Solution method:
The simplification of typical Racah algebra expressions is based on the numerical computation of Wigner n-j symbols and the explicit knowledge of symmetries, special values, orthogonality properties and sum rules. To apply these rules by means of CA we first define the term Racah expression and also proper data structures for the internal representation of typical expressions. These structures form the basis to set up a general scheme for the simplification of Racah expressions in a series of steps. All symmetric forms of the Wigner n-j symbols are taken into account. This paper, in particular, deals with the implementation of numerical computations for Racah algebra expressions, recursion formulas and with simplifications due to special values.

The Racah package is mainly based on the properties which are known for Wigner 3-j and 6-j symbols. For 9-j symbols, we will only implement a few sum rules. The numerical computation of such symbols, however, is supported. There is currently a maximal number of four internal summation variables which are accepted in the explicit numerical evaluation of Racah algebra expressions. Higher n-j symbols (n=12,15,...) could be defined in different ways; they are not included in this program. In the present version, we only implement recursion relations for 3-j symbols. Furthermore, special values are restricted to a set of 3-j and 6-j symbols as they are given in the tabulation of Edmonds [2], i.e. with quantum numbers which differ by no more than 2. The full support for the simplification of general expressions due to orthogonality and sum rules which are known for the Wigner symbols, however, will be the subject of forthcoming work.

Unusual features:
All commands of the Racah package are available for interactive work. The program is based on data structures which are suitable for almost any complexity of Racah algebra expressions. More enhanced expressions which include the summation over various quantum numbers, weight factors, some phase as well as any number of n-j symbols are built up from simpler data structures. Wigner n-j symbols with all arguments having integer or half-integer values can be computed both, as floating point number (with a precision due to the global Digits variable) or as algebraic expression. Three appendices summarize the most important mathematical relations for the manipulation and computation of typical expressions, the internal data structures, as well as all commands at user's level for quick reference. There is also some on-line help available. In addition to the classical symmetric forms of Wigner symbols, the package enables to deal with the full extended range of symmetries due to Regge [3].

Running time:
All examples of the long write-up takes about 20 seconds on an IBM workstation.

[1] Maple is an established computer algebra program and a registered trademark of Waterloo Maple Inc.
[2] A.R. Edmonds, Angular Momentum in Quantum Mechanics (Princeton University Press, New York, 1957).
[3] T. Regge, Nuovo cimento 10, 544 (1958).