Expectation Values in Relativistic Coulomb Problems

TL;DR

This paper computes matrix elements of Dirac operators in relativistic Coulomb systems using hypergeometric functions, revealing connections with special polynomials and deriving identities relevant to hydrogenlike atoms.

Contribution

It introduces new formulas for matrix elements involving Dirac operators and hypergeometric functions, and establishes their links with Chebyshev and Hahn polynomials, along with Pasternack-type identities.

Findings

Derived explicit formulas for <Or^{p}> in terms of hypergeometric functions.

Connected matrix elements with Chebyshev and Hahn polynomials.

Established Pasternack-type identities for these integrals.

Abstract

We evaluate the matrix elements <Or^{p}>, 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.