Time dependence of the probability density in the transient regime for tunneling

Abstract

An exact analytical solution to the time-dependent Schrodinger 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 Deltat. 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.