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

• URL: http://arxiv.org/abs/2205.12588v3
• Date: Mon, 19 Sep 2022 14:37:04 GMT
• Title: The Dirac impenetrable barrier in the limit point of the Klein energy zone
• Authors: Salvatore De Vincenzo
• Abstract summary: We reanalyze the problem of a 1D Dirac single-particle colliding with the electrostatic potential step of height $V_0$ with an incoming energy. 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.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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., $E\rightarrow V_{0}-\mathrm{m}c^{2}$, for a given $V_{0}$. In this situation, 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 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 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 results.

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