Maple procedures for the coupling of angular momenta. IX. Wigner D-functions and rotation matrices

The Wigner D  -functions, Dpqj(α,β,γ), are known for their frequent use in quantum mechanics. Defined as the matrix elements of the rotation operator Rˆ(α,β,γ) in R3R3 and parametrized in terms of the three Euler angles α, β, and γ, these functions arise not only in the transformation of tensor components under the rotation of the coordinates, but also as the eigenfunctions of the spherical top. In practice, however, the use of the Wigner D-functions is not always that simple, in particular, if expressions in terms of these and other functions from the theory of angular momentum need to be simplified before some computations can be carried out in detail.To facilitate the manipulation of such Racah expressions, here we present an extension to the Racah program [S. Fritzsche, Comput. Phys. Comm. 103 (1997) 51] in which the properties and the algebraic rules of the Wigner D  -functions and reduced rotation matrices are implemented. Care has been taken to combine the standard knowledge about the rotation matrices with the previously implemented rules for the Clebsch–Gordan coefficients, Wigner n−jn−j symbols, and the spherical harmonics. Moreover, the application of the program has been illustrated below by means of three examples.Program summaryTitle of program:RacahCatalogue identifier:ADFv_9_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/ADFv_9_0Program obtainable from: CPC Program Library, Queen's University of Belfast, N. IrelandCatalogue identifier of previous version: ADFW, ADHW, title RacahJournal reference of previous version(s): S. Fritzsche, Comput. Phys. Comm. 103 (1997) 51; S. Fritzsche, S. Varga, D. Geschke, B. Fricke, Comput. Phys. Comm. 111 (1998) 167; S. Fritzsche, T. Inghoff, M. Tomaselli, Comput. Phys. Comm. 153 (2003) 424.Does the new version supersede the previous one: Yes, in addition to the spherical harmonics and recoupling coefficients, the program now supports also the occurrence of the Wigner rotation matrices in the algebraic expressions to be evaluated.Licensing provisions:NoneComputer for which the program is designed and others on which it is operable: All computers with a license for the computer algebra package Maple [Maple is a registered trademark of Waterloo Maple Inc.]Installations:University of Kassel (Germany)Operating systems under which the program has been tested: Linux 8.2+Program language used:Maple, Release 8 and 9Memory required to execute with typical data:10–50 MBNo. of lines in distributed program, including test data, etc.:52 653No. of bytes in distributed program, including test data, etc.:1 195 346Distribution format:tar.gzipNature of the physical 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.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−jn−j symbols (j=3,6,9j=3,6,9) and spherical harmonics, we 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 occurs in course of the theoretical deviations.Summary of revisions: In a revised version of the Racah program [S. Fritzsche, Comput. Phys. Comm. 103 (1997) 51; S. Fritzsche, T. Inghoff, M. Tomaselli, Comput. Phys. Comm. 153 (2003) 424], we now also support the occurrence 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.Restrictions onto the complexity of the problem: 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. [D.A. Varshalovich, A.N. Moskalev, V.K. Khersonskii, Quantum Theory of Angular Momentum, World Scientific, Singapore, 1988], 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 of up to three Wigner D-functions or reduced matrices (with the same angular arguments) are recognized 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 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 the integration, if omitted, is always taken from 0 to π.Unusual features of the program: The Racah program is designed for interactive use that allows a quick and algebraic evaluation of (complex) expression from Racah's algebra. It is based on a number of well-defined data structures that are now extended to incorporate the Wigner rotation matrices. For these matrices, the transformation properties, sum rules, recursion relations, as well as a variety of special function expansions have been added to the previous functionality of the Racah program. Moreover, the knowledge about the orthogonality as well as the completeness of the Wigner D-functions is also implemented.Typical running time:All the examples presented in Section 4 take only a few seconds on a 1.5 GHz Pentium Pro computer.