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

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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 ≠ 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 x₀ from the potential the probability density rises after a time t₀=x₀/c restoring Einstein causality. This implies that the tunneling time of a particle can never be zero.