Computationally efficient analytic representation of relativistic transition matrix elements in the Lamb shift calculations for hydrogenic atoms

TL;DR

This paper introduces a new explicit finite series representation of relativistic transition matrix elements for hydrogenic atoms, simplifying numerical calculations of the Lamb shift.

Contribution

It provides a novel finite series expression using hypergeometric functions for relativistic matrix elements, enhancing computational efficiency.

Findings

Facilitates direct numerical calculation of Lamb shift contributions

Uses plane-wave expansion for electromagnetic vector potential

Expresses matrix elements as finite hypergeometric series

Abstract

By using the plane-wave expansion for the electromagnetic-field vector potential, transition matrix elements between the relativistic bound and unbound states of hydrogenic atoms were expressed explicitly in terms of finite series made of hypergeometric functions. This representation for the above mentioned matrix elements proved very convenient for direct numerical calculation of their contributions to the Lamb shift in hydrogenic atoms.