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

Abstract

We consider exact time-dependent analytic solutions to the Schrödinger equation for tunneling in one dimension with cut off wave initial conditions at $t=0$. We obtain that as soon as $t\ne 0$ 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.

4 pages, 1 figure

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