Solving the radial Dirac equations: a numerical odyssey

TL;DR

This paper presents a comprehensive numerical approach to solving the radial Dirac equations for various potentials, including linear and Coulomb, for different quark masses and configurations, demonstrating applications to quark confinement and atomic systems.

Contribution

It introduces an iterative numerical method for solving the radial Dirac equations across multiple potential types and quark mass scenarios, extending to combined potentials relevant in hadronic physics.

Findings

Successfully computed eigenenergies and wave functions for massless and massive quarks.

Re-derived relativistic hydrogen atom solutions numerically.

Analyzed combined scalar and vector potentials for spin-orbit effects.

Abstract

We discuss, in a pedagogical way, how to solve for relativistic wave functions from the radial Dirac equations. After an brief introduction, in Section II we solve the equations for a linear Lorentz scalar potential, V_s(r), that provides for confinement of a quark. The case of massless u and d quarks is treated first, as these are necessarily quite relativistic. We use an iterative procedure to find the eigenenergies and the upper and lower component wave functions for the ground state and then, later, some excited states. Solutions for the massive quarks (s, c, and b) are also presented. In Section III we solve for the case of a Coulomb potential, which is a time-like component of a Lorentz vector potential, V_v(r). We re-derive, numerically, the (analytically well-known) relativistic hydrogen atom eigenenergies and wave functions, and later extend that to the cases of heavier one-electron atoms and muonic atoms. Finally, Section IV finds solutions for a combination of the V_s and V_v potentials. We treat two cases. The first is one in which V_s is the linear potential used in Sec. II and V_v is Coulombic, as in Sec. III. The other is when both V_s and V_v are linearly confining, and we establish when these potentials give a vanishing spin-orbit interaction (as has been shown to be the case in quark models of the hadronic spectrum).