Accurate solution of the Dirac equation on Lagrange meshes

TL;DR

This paper demonstrates that the Lagrange-mesh method can accurately solve the Dirac equation for hydrogenic atoms and Yukawa potentials, achieving high precision with relatively few mesh points.

Contribution

It introduces an application of the Lagrange-mesh method to the Dirac equation, providing highly accurate solutions even with small numbers of mesh points.

Findings

Exact energies and wave functions for hydrogenic atoms with few mesh points

High-precision mean values of radial coordinate powers

Agreement with benchmark energies for Yukawa potential

Abstract

The Lagrange-mesh method is an approximate variational method taking the form of equations on a grid because of the use of a Gauss quadrature approximation. With a basis of Lagrange functions involving associated Laguerre polynomials related to the Gauss quadrature, the method is applied to the Dirac equation. The potential may possess a $1/r$ singularity. For hydrogenic atoms, numerically exact energies and wave functions are obtained with small numbers $n+1$ of mesh points, where $n$ is the principal quantum number. Numerically exact mean values of powers $-2$ to 3 of the radial coordinate $r$ can also be obtained with $n+2$ mesh points. For the Yukawa potential, a 15-digit agreement with benchmark energies of the literature is obtained with 50 mesh points or less.