A generalized Coulomb problem for a spin-1/2 fermion

TL;DR

This paper derives exact bound-state solutions for the Dirac equation with a generalized Coulomb potential involving scalar, vector, and tensor interactions, revealing how these parameters influence the energy spectrum and connecting to previous special cases.

Contribution

It provides a comprehensive analytical solution to a generalized Coulomb problem in Dirac theory, including new cases with broken symmetries and tensor potentials.

Findings

Exact energy spectrum derived and analyzed.

Wave functions expressed in generalized Laguerre polynomials.

New particular cases of Coulomb interactions with tensor potentials.

Abstract

We study the Dirac equation in 3+1 dimensions with a general combination of scalar, vector and tensor interactions with arbitrary strengths, all of them described by central Coulomb potentials acting on a particular plane of motion. For the tensor coupling a constant term is also included, since this gives rise to an effective Coulomb potential, which is necessary for the formation of bound states in a pure tensor coupling configuration. The exact bound-state solutions for this generalized Coulomb problem are computed by exploiting the freedom in choosing the coefficients of the \textit{Ans\"atze} for the radial functions, which leads to wave functions in terms of generalized Laguerre polynomials. From the quantization condition, the exact energy spectrum is also determined and its dependence on the parameters of the potentials is discussed. We show that similar features of the equations for the problem in the plane and the spherically symmetric problem allow a simple and direct mapping between them, thereby providing the solution to the spherical Coulomb problem. Our results are validated by showing that the solutions correctly encompass several previous solutions available in the literature for particular cases of this problem, for which we further develop the analysis of the parameters. We also derive two new particular cases not yet reported in the literature: the case of breaking of spin and pseudospin symmetries by the addition of a Coulomb plus constant tensor potential and the problem of a scalar plus tensor Coulomb potentials.