Solving the radial Dirac equations: a numerical odyssey

Abstract

We discuss, in a pedagogical way, how to solve for relativistic wavefunctions from the radial Dirac equations. We first solve the equations for a linear Lorentz scalar potential, Vs(r), that provides for confinement of a quark; the case of massless u and d quarks is necessarily relativistic. We use an iterative ‘shooting and matching’ procedure to find the eigenenergies and the upper and lower component wavefunctions. Solutions for the massive quarks (s, c, and b) are also presented. We then consider the Coulomb potential [Vv(r), 0]. We re-derive, numerically, the (analytically well-known) relativistic hydrogen atom eigenenergies and wavefunctions, and later extend that to the cases of heavier one-electron atoms and muonic atoms. Finally, we solve for a combination of the Vs and Vv potentials, when both potentials are linearly confining and Vv has a color Coulombic component. We establish when these potentials give a vanishing spin–orbit interaction (as is approximately the case in quark models of the baryonic spectrum).