Solution to the Dirac equation using the finite difference method

In this study, single-particle energy was examined using the finite difference method by taking 208\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$^{208}$$\end{document}Pb as an example. If the first derivative term in the spherical Dirac equation is discretized using a three-point formula, a one-to-one correspondence occurs between the physical and spurious states. Although these energies are exactly the same, the wave functions of the spurious states exhibit a much faster staggering than those of the physical states. Such spurious states can be eliminated when applying the finite difference method by introducing an extra Wilson term into the Hamiltonian. Furthermore, it was also found that the number of spurious states can be reduced if we improve the accuracy of the numerical differential formula. The Dirac equation is then solved in a momentum space in which there is no differential operator, and we found that the spurious states can be completely avoided in the momentum space, even without an extra Wilson term.