The Dirac impenetrable barrier in the limit point of the Klein energy zone

Abstract

We reanalyze the problem of a 1D Dirac single-particle colliding with the electrostatic potential step of height V 0 with an incoming energy that tends to the limit point of the so-called Klein energy zone, i.e., E → V 0 − mc 2 , for a given V 0 . In this situation, the particle is actually colliding with an impenetrable barrier. In fact, V 0 → E + mc 2 , for a given relativistic energy E (< V 0 ), is the maximum value that the height of the step can reach and that ensures the perfect impenetrability of the barrier. Nevertheless, we notice that, unlike the nonrelativistic case, the entire eigensolution does not completely vanish, either at the barrier or in the region under the step, but its upper component does satisfy the Dirichlet boundary condition at the barrier. More importantly, by calculating the mean value of the force exerted by the wall on the particle in this eigenstate and taking its nonrelativistic limit, we recover the required result. The presentation is suitable for the advanced graduate level.