Relativistic particle in a box: Klein-Gordon vs Dirac Equations

TL;DR

This paper compares solutions of a relativistic particle in a box using Klein-Gordon and Dirac equations, highlighting spin effects on energy spectra and the non-relativistic limit.

Contribution

It provides a detailed comparison of Klein-Gordon and Dirac equation solutions for a particle in a box, emphasizing relativistic spin effects on energy spectra.

Findings

Spin influences the relativistic energy spectrum.

Non-relativistic limits are consistent for both equations.

Differences and similarities in solutions are analyzed.

Abstract

The problem of a particle in a box is probably the simplest problem in quantum mechanics which allows for significant insight into the nature of quantum systems and thus is a cornerstone in the teaching of quantum mechanics. In relativistic quantum mechanics this problem allows also to highlight the implications of special relativity for quantum physics, namely the effect that spin has on the quantized energy spectra. To illustrate this point, we solve the problem of a spin zero relativistic particle in a one- and three-dimensional box using the Klein-Gordon equation in the Feshbach-Villars formalism. We compare the solutions and the energy spectra obtained with the corresponding ones from the Dirac equation for a spin one-half relativistic particle. We note the similarities and differences, in particular the spin effects in the relativistic energy spectrum. As expected, the non-relativistic limit is the same for both kinds of particles, since, for a particle in a box, the spin contribution to the energy is a relativistic effect.