Relativistic Kramers-Pasternack Recurrence Relations

TL;DR

This paper derives Kramers-Pasternack three-term recurrence relations for matrix elements involving Dirac operators in the relativistic Coulomb problem, expanding the mathematical tools for quantum relativistic calculations.

Contribution

It introduces Kramers-Pasternack recurrence relations for relativistic matrix elements, building on previous hypergeometric function evaluations in the Dirac Coulomb context.

Findings

Derived three-term vector recurrence relations for matrix elements.

Extended the mathematical framework for relativistic quantum calculations.

Connected hypergeometric function evaluations with recurrence relations.

Abstract

Recently we have evaluated the 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, in terms of generalized hypergeometric functions $_{3}F_{2}(1) $ for all suitable powers and established two sets of Pasternack-type matrix identities for these integrals. The corresponding Kramers--Pasternack three-term vector recurrence relations are derived here.