Dynamic transmission and reflection phenomena for a time-dependent rectangular potential

Abstract

The time-dependent Schr\"odinger equation is solved for a one-dimensional rectangular potential whose height changes sinusoidally in time. By using standard wave-matching techniques, dynamic transmission and reflection probabilities are found. It is shown that the flux transmitted through the barrier region is modulated in time at integer multiples of ${\ensuremath{\omega}}_{b}$, the angular frequency at which the potential oscillates. The results obtained for a delocalized plane wave incident on the barrier are extended by considering the interaction of a Gaussian wave packet with the oscillating potential barrier. The modulation of the transmitted and reflected probability density by the oscillating barrier is clearly demonstrated by a series of calculations performed to illustrate this interaction. These calculations also show that the probability density leaving the barrier region is composed of individual wave packets having different average velocities, a result which is a direct consequence of the transitions induced by the time-dependent potential barrier under consideration. Calculations are also performed which show that the flux transmitted across the potential barrier depends on the phase of the oscillating barrier at the time the incident Gaussian wave packet begins to interact with it. The results of these model calculations suggest that experiments properly designed to measure the transmission probability of a quantum particle as it escapes from a metal surface are likely to provide information about the time formation of the dynamic image potential.