Accurate solution of the Dirac equation on Lagrange meshes

Daniel Baye; Livio Filippin; Michel Godefroid

doi:10.1103/physreve.89.043305

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arXiv · 1404.5409

Accurate solution of the Dirac equation on Lagrange meshes

Daniel Baye·Livio Filippin·Michel Godefroid

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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.

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Daniel Baye, Livio Filippin, Michel Godefroid. 2014-04-22. Accurate solution of the Dirac equation on Lagrange meshes. https://doi.org/10.1103/physreve.89.043305

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