Solution of the Dirac Equation using the Lanczos Algorithm

TL;DR

This paper demonstrates an efficient method for solving the Dirac Equation for relativistic electrons in Coulomb potentials using the Lanczos Algorithm, enabling accurate eigenvalue computation and identification of spurious solutions.

Contribution

It introduces a novel iterative approach employing the Lanczos Algorithm to solve the Dirac Equation, improving convergence and solution accuracy over traditional methods.

Findings

Successfully computed convergent eigenvalues for the Dirac Equation.

Effectively identified and excluded spurious solutions.

Provided a practical iterative framework for relativistic quantum problems.

Abstract

Covergent eigensolutions of the Dirac Equation for a relativistic electron in an external Coulomb potential are obtained using the Lanczos Algorithm. A tri-diagonal matrix representation of the Dirac Hamiltonian operator is constructed iteratively and diagonalized after each iteration step to form a sequence of convergent eigenvalue solutions. Any spurious solutions which arise from the presence of continuum states can easily be identified.