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

Abstract

We consider exact time-dependent analytic solutions to the Schrodinger equation for tunneling in one dimension with cutoff wave initial conditions at t=0. We obtain that as soon as t not equal 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 xo 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. [S1050-2947(99)08903-9].