Calculation of Matrix Elements in the Foldy-Wouthuysen Representation

TL;DR

This paper compares methods for calculating matrix elements of radial electron coordinates in the Foldy-Wouthuysen representation, confirming that wave functions can be simplified to one component for 1s-states of hydrogen-like and helium-like ions.

Contribution

It introduces a comparison of calculation methods in the Foldy-Wouthuysen representation and confirms the validity of using a single wave function component for matrix element calculations.

Findings

Wave function reduction is valid in the Foldy-Wouthuysen representation.

Matrix elements can be computed using only one component of the Dirac bispinor.

Analytical and numerical results support the method's accuracy.

Abstract

The paper compares the methods used to calculate matrix elements of the operator of radail electron coordinates in an arbitrary order in the Foldy-Wouthuysen representation and with the use of the Dirac equation for 1s-states of the hydrogen-like and helium-like ions of transuranic elements. The obtained analytical and numerical results for 1s-states of the hydrogen-like and helium-like ions proves that the wave function reduction requirement is met with transformation to the Foldy-Wouthuysen representation and confirms that matrix elements can be calculated using only one component (either upper, or lower) of the Dirac bispinor wave function.