An exact analytical solution to the time-dependent Schrödinger equation with cutoff wave initial conditions is used to investigate the fast tunneling response of a rectangular potential barrier. We find that just across the tunneling region, the probability density exhibits at short times a transient behavior that may be characterized by a peak t p and a width ⌬t. We show that t p provides the earliest tunneling response of the system and that the top-barrier S-matrix poles play an important role in the process. As a function of the barrier width, t p exhibits two regimes. Along the first regime, t p remains almost a constant; as the barrier width increases, a second regime appears where t p grows linearly with the barrier width.M (x,Ϯk;t) and M (x,k n ;t) are defined as M ͑ x,q;t ͒ϭ 1 2 e (imx 2 /2បt) e y q 2 erfc͑ y q ͒, ͑3͒where the argument y q is given by y q ϵe Ϫi/4 ͩ m 2បt ͪ 1/2 ͫ xϪ បq m t ͬ .
We examine an analytical expression for the survival probability for the time evolution of quantum decay to discuss a regime where quantum decay is nonexponential at all times. We find that the interference between the exponential and nonexponential terms of the survival amplitude modifies the usual exponential decay regime in systems where the ratio of the resonance energy to the decay width, is less than 0.3. We suggest that such regime could be observed in semiconductor doublebarrier resonant quantum structures with appropriate parameters.
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.
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