Quantum-wave evolution in a step potential barrier

By using an exact solution to the time-dependent Schroedinger 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 forerunner associated with a nontunneling process, which propagates in space at exactly the semiclassical group velocity. We show that the time of arrival of the maximum of the forerunner at a given fixed position inside the potential is exactly the traversal time τ. 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 forerunner, and to analyze the role played by the time of arrival 3-1/2τ found recently by Muga and Buettiker [Phys. Rev. A 62, 023808 (2000)]