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

The Wigner D-functions, image, are known for their frequent use in quantum mechanics. Defined as the matrix elements of the rotation operator image in image 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 image symbols, and the spherical harmonics. Moreover, the application of the program has been illustrated below by means of three examples.