Quantum-wave evolution in a step potential barrier

Abstract

By using an exact solution to the time-dependent Schr\"{o}dinger equation with a point source initial condition, we investigate both the time and spatial dependence of quantum waves in a step potential barrier. We find that for a source with energy below the barrier height, and for distances larger than the penetration length, the probability density exhibits a {\it forerunner} associated with a non-tunneling process, which propagates in space at exactly the semiclassical group velocity. We show that the time of arrival of the maximum of the {\it forerunner} at a given fixed position inside the potential is exactly the traversal time, $\tau$. We also show that the spatial evolution of this transient pulse exhibits an invariant behavior under a rescaling process. This analytic property is used to characterize the evolution of the {\it forerunner}, and to analyze the role played by the time of arrival, $3^{-1/2}\tau$, found recently by Muga and B\"{u}ttiker [Phys. Rev. A {\bf 62}, 023808 (2000)].