Exact relativistic time evolution for a step potential barrier

Abstract

We derive an exact analytic solution to a Klein-Gordon equation for a step potential barrier with cutoff plane wave initial conditions, in order to explore wave evolution in a classical forbidden region. We find that the relativistic solution rapidly evanesces within a depth $2x_p$ inside the potential, where $x_p$ is the penetration length of the stationary solution. Beyond the characteristic distance $2x_p$, a Sommerfeld-type precursor travels along the potential at the speed of light, $c$. However, no spatial propagation of a main wavefront along the structure is observed. We also find a non-causal time evolution of the wavefront peak. The effect is only an apparent violation of Einstein causality.