Electron Scattering at a Potential Temporal Step Discontinuity

TL;DR

This paper investigates electron scattering at a potential temporal step discontinuity, revealing the necessity of the Dirac equation over Schrödinger's, and explores relativistic effects and smooth potential transitions.

Contribution

It demonstrates the importance of the Dirac equation for temporal discontinuities and derives scattering probabilities, including relativistic backward waves, extending classical and spatial step analyses.

Findings

Dirac equation is required for temporal step scattering.

Relativistic backward waves are observed in scattering.

Smooth potential steps replicate sharp discontinuity physics when transition time is short.

Abstract

We solve the problem of electron scattering at a potential temporal step discontinuity. We show that the Schrodinger equation cannot account for scattering in this problem, necessitating resort to the Dirac equation, and that breaking gauge symmetry requires a vector potential, a scalar potential inducing only Aharonov-Bohm type energy transitions. We derive the scattering probabilities, of later forward and backward nature, with the later-backward wave being a relativistic effect, and compare the results with those for the spatial step and classical electromagnetic counterparts of the problem. Given the unrealizability of an infinitely sharp temporal discontinuity - which is of the same nature as its spatial counterpart! - we also provide solutions for a smooth potential step and demonstrate that the same physics as for the infinitely sharp case is obtained when the duration of the potential transition is sufficiently smaller than the de Broglie period of the electron (or deeply sub-period).