Dirac particle in a spherical scalar potential well

TL;DR

This paper solves the Dirac equation for a spin-1/2 particle in a spherical scalar potential well, calculating energy levels and wave functions, and compares results with the MIT bag model, offering insights into nuclear modeling.

Contribution

It provides analytical solutions for the Dirac equation in a spherical scalar potential well, including finite potential cases, extending previous models like the MIT bag model.

Findings

Energy eigenvalues for specific states calculated

Wave functions analyzed at the potential boundary

Results agree with the MIT bag model at infinite potential limit

Abstract

In this paper we investigate a solution of the Dirac equation for a spin-$\frac{1}2$ particle in a scalar potential well with full spherical symmetry. The energy eigenvalues for the quark particle in $s_{1/2}$ states (with $\kappa=-1$) and $p_{1/2}$ states (with $\kappa=1$) are calculated. We also study the continuous Dirac wave function for a quark in such a potential, which is not necessarily infinite. Our results, at infinite limit, are in good agreement with the MIT bag model. We make some remarks about the sharpness value of the wave function on the wall. This model, for finite values of potential, also could serve as an effective model for the nucleus where $U(r)$ is the effective single particle potential.