Relativistic time-dependent quantum dynamics across supercritical barriers for Klein-Gordon and Dirac particles

Abstract

We investigate wavepacket dynamics across supercritical barriers for the Klein-Gordon and Dirac equations. Our treatment is based on a multiple scattering expansion (MSE). For spin-0 particles, the MSE diverges, rendering invalid the use of the usual connection formulas for the scattering basis functions. In a time-dependent formulation, the divergent character of the MSE naturally accounts for charge creation at the barrier boundaries. In the Dirac case, the MSE converges and no charge is created. We show that this time-dependent charge behavior dynamics can adequately explain the Klein paradox in a first quantized setting. We further compare our semi-analytical wavepacket approach to exact finite-difference solutions of the relativistic wave equations.

Relativistic spin-0 particle in a box: bound states, wavepackets, and the disappearance of the Klein paradox

“…5) results obtained from solving numerically the KGE equation through a finite difference scheme. The numerical method employed is described elsewhere [8] -here its use is only to testify about the exactness of our MSE based wavepacket approach.…”

“…This will be done by considering multiple scattering expansions inside the well. Such expansions were recently employed to investigate relativistic dynamics across supercritical barriers [7,8]. We will see below that Klein tunneling, that is prominent for a supercritical potential well reasonably higher than the particle energy placed inside, disappears as the well's depth V is increased.…”