Mathematical Structure of Relativistic Coulomb Integrals

TL;DR

This paper reveals that certain relativistic Coulomb matrix elements can be viewed as difference analogs of radial wave functions, enabling new elementary methods for their evaluation and deriving recurrence relations.

Contribution

It introduces a novel perspective on Coulomb matrix elements as difference analogs, providing elementary derivations and recurrence relations without explicit integral calculations.

Findings

Derived three-term recurrence relations for expectation values.

Established transformation formulas for hypergeometric series.

Provided an alternative method for calculating relativistic Coulomb integrals.

Abstract

We show that the diagonal matrix elements $< Or^{p} >,$ where $O$ $={1,\beta,i\mathbf{\alpha n}\beta}$ are the standard Dirac matrix operators and the angular brackets denote the quantum-mechanical average for the relativistic Coulomb problem, may be considered as difference analogs of the radial wave functions. Such structure provides an independent way of obtaining closed forms of these matrix elements by elementary methods of the theory of difference equations without explicit evaluation of the integrals. Three-term recurrence relations for each of these expectation values are derived as a by-product. Transformation formulas for the corresponding generalized hypergeometric series are discussed.