Title: Quantum coupling coefficients seen as discrete wave functions by Markus van Almsick Am Markuskreuz 6 45133 Essen Germany (Applications Consultant of Wolfram Research Inc.) markus@wri.com Work Shop: Stochastic Methods (Michael Trott) key words: application of CA to science, quantum mechanical probabilities, group theory. Abstract: Quantum mechanical coupling coefficients are conditional probabilities between different symmetry eigenstates. Computer algebra systems allow one to easily obtain quantum coupling coefficients, in particular SU(2)-coupling coefficients. We review three different ways to numerically, as well as symbolically, determine Clebsch-Gordan-, 3j-, and 6j-coefficients. We will derive them recursively, via generating functions, and explicitly exploiting symmetry relations between coupling coefficients. All three methods are suited for CA implementations. They incorporate a wide range of CA techniques; optimizing recursion relations of difference equations or putting generatingfunctionology to use. The possibilities to extend these techniques to other symmetry groups besides SU(2) will be discussed. With these CA implementations of coupling coefficients we investigate their properties. Coupling coefficients seen as functions of their magnetic or angular quantum numbers can be interpreted as quantum mechanical probability distributions. We thus obtain "discrete wave functions" in the configuration space of the vector model of angular momentum coupling. The classical limit of these "wave functions" is given by the Wigner estimate. Furthermore, the above vector model guides us in deriving and explaining relations between 3j-coefficients, 6j-coefficients and rotation matrices. Literature: Semiclassical Limit of Racah Coefficients, by G. Ponzano, T. Regge in Spectroscopic and group theoretical methods in physics, North-Holland Publ., Amsterdam, 1968 Drehimpulse in der Quantenmechanik, by A. Lindner Teubner Studienbuecher (Physik), Stuttgart 1984 Group Theory, by E.P. Wigner Academic Press, New York, 1959 Group Theory in Physics Vol. 1 + 2 (Techniques in Physics 7) by J.F. Cornwell Academic Press, New York, 1984 Exact recursive evaluation of 3j- and 6j-coefficients for quantum mechnical coupling of angular momenta J. Math. Phys. Vol 11, no. 2, 1976 by K.J. Schulten and G.R. Gordon North-Holland Publishing Company, Amsterdam