Relativistic quantum bouncing particles in a homogeneous gravitational field

TL;DR

This paper investigates the relativistic effects on quantum bouncing particles in a gravitational field, analyzing Klein-Gordon and Dirac equations in Rindler coordinates, revealing differences in energy levels and wave functions compared to nonrelativistic models.

Contribution

It introduces a relativistic analysis of bouncing particles in gravity using Klein-Gordon and Dirac equations, highlighting differences from nonrelativistic results.

Findings

Dirac particles have higher transition frequencies than Klein-Gordon particles.

Both relativistic models show higher energies than the nonrelativistic limit.

Wave function behaviors differ near the boundary depending on particle type.

Abstract

In this paper, we study the relativistic effect on the wave functions for a bouncing particle in a gravitational field. Motivated by the equivalence principle, we investigate the Klein-Gordon and Dirac equations in Rindler coordinates with the boundary conditions mimicking a uniformly accelerated mirror in Minkowski space. In the nonrelativistic limit, all these models in the comoving frame reduce to the familiar eigenvalue problem for the Schr\"odinger equation with a fixed floor in a linear gravitational potential, as expected. We find that the transition frequency between two energy levels of a bouncing Dirac particle is greater than the counterpart of a Klein-Gordon particle, while both are greater than their nonrelativistic limit. The different corrections to eigen-energies of particles of different nature are associated with the different behaviors of their wave functions around the mirror boundary.