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 a positive incoming energy that tends to the limit point of the so-called Klein energy zone, i.e., $E\rightarrow V_{0}-\mathrm{m}c^{2}$, for a given $V_{0}$. In such a case, the particle is actually colliding with an impenetrable barrier. In fact, $V_{0}\rightarrow E+\mathrm{m}c^{2}$, 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 note 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 impenetrable wall on the particle in this eigenstate and taking its nonrelativistic limit, we recover the required result. We use two different approaches to obtain the latter two results. In one of these approaches, the corresponding force on the particle is a type of boundary quantum force. Throughout the article, various issues related to the Klein energy zone, the transmitted solutions to this problem, and impenetrable barriers related to boundary conditions are also discussed. In particular, if the negative-energy transmitted solution is used, the lower component of the scattering solution satisfies the Dirichlet boundary condition at the barrier, but the mean value of the external force when $V_{0}\rightarrow E+\mathrm{m}c^{2}$ does not seem to be compatible with the existence of the impenetrable barrier.