Relativistic quantum bouncing particles in a homogeneous gravitational field.

We study the relativistic effect on the wavefunctions 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 non-relativistic 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 non-relativistic limit. The different corrections to eigen-energies of particles of different nature are associated with the different behaviors of their wavefunctions around the mirror boundary.