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 or fewer mesh points.
[1]
Thorsten Gerber,et al.
Handbook Of Mathematical Functions
,
2016
.
[2]
I. P. Grant,et al.
Relativistic quantum theory of atoms and molecules
,
2006
.
[3]
Rene F. Swarttouw,et al.
Orthogonal Polynomials
,
2005,
Series and Products in the Development of Mathematics.
[4]
S. Karshenboim,et al.
Precision physics of simple atomic systems
,
2003
.
[5]
V. Shabaev.
Virial Relations for the Dirac Equation and Their Applications to Calculations of Hydrogen-Like Atoms
,
2002,
physics/0211087.