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Manuscript Title: Maple procedures for the coupling of angular momenta. IX. Wigner D - functions and rotation matrices.
Authors: J. Pagaran, S. Fritzsche, G. Gaigalas
Program title: RACAH
Catalogue identifier: ADFV_v9_0 Distribution format: tar.gz
Journal reference: Comput. Phys. Commun. 174(2006)616
Programming language: MAPLE, Release 8 and 9.
Computer: All computers with a licence for the computer algebra package Maple [1].
Operating system: Linux 8.2+.
RAM: 10-50 MB
Keywords: Angular momentum, Clebsch-Gordan expansion, finite rotation matrix, Racah algebra techniques, reduced rotation matrix, Wigner D-function rotation matrix, Wigner n-j symbol.
PACS: 3.65Fd, 2.90+p.
Classification: 4.1, 5.
Does the new version supersede the previous version?: Yes, in addition to the spherical harmonics and recoupling coefficients, the program now supports also the occurance of the Wigner rotation matrices in the algebraic expressions to be evaluated.
Nature of problem: The Wigner D-functions and (reduced) rotation matrices occur very frequently in physical applications. They are known not only as the (infinite) representation of the rotation group but also to obey a number of integral and summation rules, including those for their orthogonality and completeness. Instead of the direct computation of these matrices, therefore, one first often wishes to find algebraic simplifications before the computations can be carried out in practice.
Solution method: In a revised version of the RACAH program [2], we now also support the occurence of the Wigner D-functions and reduced rotation matrices. By following our previous design, the (algebraic) properties of these rotation matrices as well as a number of summation and integration rules are implemented to facilitate the algebraic simplification of expressions from the theories of angular momentum and the spherical tensor operators.
Reasons for new version: The RACAH program has been found an efficient tool during recent years, in order to evaluate and simplify expressions from Racah's algebra. Apart from the Wigner n-j symbols (j = 3, 6, 9) and spherical harmonics, we have now extended the code to allow for Wigner rotation matrices. This extension will support the study of those quantum processes especially where different axis of quantization occur in the course of the theoretical deviations.
Summary of revisions: In a revised version of the RACAH program [2], we now also support the occurence of Wigner D-functions and reduced rotation matrices. By following our previous design, the (algebraic) properties of these rotation matrices as well as a number of summation and integration rules are implemented to facilitate the algebraic simplification of expressions from the theories of angular momentum and the spherical tensor operators.
Restrictions: The definition as well as the properties of the rotation matrices, as used in our implementation, are based mainly on the book of Varshalovich et al. [3], Chapter 4. From this monograph, most of the relations involving the Wigner D-functions and rotation matrices are taken into account although, in practice, only a rather selected set was needed to be implemented explicitly owing to the symmetries of these functions. In the integration over the rotation matrices, products up to three Wigner D-functions or reduced matrices (with the same angular arguments) are recognised and simplified properly; for the integration over a solid angle, however, the domain of integration must be specified for the Euler angles α and γ. This restriction arose because MAPLE [1] does not generate a constant of integration when the limits in the integral are omitted. For any integration over the angle β the range of integration, if omitted, is always taken from 0 to π.
Unusual features: The RACAH program is designed for interactive use which allows a quick algebraic evaluation of (complex) expressions from Racah's algebra. It is based on a number of well-defined data structures which are extended now to incorporate also the Wigner rotation matrices. For these matrices, the transformation properties, sum rules, recursion relations, as well as a variety of special function expansions were incorporated in addition to our previous functionality of the RACAH program [2]. Moreover, the knowledge of orthogonality as well as the completeness of the Wigner D-functions is also implemented.
Running time: All examples presented in Section 4 take only a few seconds on a 1.5 GHZ Pentium Pro computer.
References:
[1]	Maple is a registered trademark of Waterloo Maple Inc.
[2]	S. Fritsche, Comp. Phys. Commun. 103, 51 (1997); S. Fritzsche, T. Inghoff and M. Tomaselli, Comp. Phys. Commun. 153, 424 (2003).
[3]	D. A. Varshalovich, A.N. Moskalev, V. K. Khersonskii, Quantum Theory of Angular Momentum (World Scientific, Singapore, 1988).