Low-energy relativistic effects and nonlocality in time-dependent tunneling

Abstract

We consider exact time-dependent analytic solutions to the Schr\"odinger equation for tunneling in one dimension with cutoff wave initial conditions at $t=0.$ We obtain that as soon as $t\ensuremath{\ne}0$ the transmitted probability density at any arbitrary distance rises instantaneously with time in a linear manner. Using a simple model we find that the above nonlocal effect of the time-dependent solution is suppressed by consideration of low-energy relativistic effects. Hence at a distance ${x}_{0}$ from the potential the probability density rises after a time ${t}_{0}{=x}_{0}/c$ restoring Einstein causality. This implies that the tunneling time of a particle can never be zero.