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Quantum-wave evolution in a step potential barrier

Jorge Villavicencio·Roberto Romo·Sukey Sosa y Silva
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Abstract

By using an exact solution to the time-dependent Schrödinger 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, $τ$. 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 {\it forerunner}, and to analyze the role played by the time of arrival, $3^{-1/2}τ$, found recently by Muga and Büttiker [Phys. Rev. A {\bf 62}, 023808 (2000)].

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Jorge Villavicencio, Roberto Romo, Sukey Sosa y Silva. 2002-10-02. Quantum-wave evolution in a step potential barrier. https://doi.org/10.1103/physreva.66.042110

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