Expectation Values in Relativistic Coulomb Problems

We evaluate the matrix elements , where O ={1, \beta, i\alpha n \beta} are the standard Dirac matrix operators and the angular brackets denote the quantum-mechanical average for the relativistic Coulomb problem, in terms of the generalized hypergeometric functions_{3}F_{2} for all suitable powers. Their connections with the Chebyshev and Hahn polynomials of a discrete variable are emphasized. As a result, we derive two sets of Pasternack-type matrix identities for these integrals, when p->-p-1 and p->-p-3, respectively. Some applications to the theory of hydrogenlike relativistic systems are reviewed.Comment: 16 pages, one table, two appendices, no figures; to appear in J. Phys. B: At. Mol. Opt. Phy

MULTILINEARITY OF SKETCHES

ABSTRACT. We give a precise characterization for when the models of the tensor product of sketches are structurally isomorphic to the models of either sketch inthe models of the other. For each basecategoryK call the just mentioned property (sketch) K-multilinearity. Saythattwo sketches are K-compatible with respect to base category K just in case in each K-model, the limits for each limit speci cation in each sketch commute with the colimits for each colimit speci cation in the other sketch and all limits and colimits are pointwise. Two sketches are K-multilinear if and only if the two sketches are K-compatible. This property then extends to strong Colimits of sketches. We shall use the technically useful property of limited completeness and completeness of every category of models of sketches. That is, categories of sketch modelshave all limits commuting with the sketched colimits and and all colimits commuting with the sketched limits. Often used implicitly, the precise statement of this property andits proof appears here. 1