Traversal, Reflection and Dwell Time for Quantum Tunneling

Abstract

Many interesting questions in physics concern the calculation of time scales, i. e. elastic and inelastic scattering rates, and the comparison of such scales to find the most important process for the problem under consideration. Here we are specifically interested in characterizing the time scales for quantum tunneling. Most of the elementary textbooks on quantum mechanics discuss the time it takes a population of carriers to escape from a long lived state, such as a metastable state in a local potential valley or a resonant state in a double or multiple barrier structure. The textbooks, however, are silent when it comes to time scales which are not associated with a long lived state, such as tunneling through a single barrier or through a double barrier away from the resonant condition. A popular but unjustified method to investigate the time scales under the latter conditions, is the use of wave packets and the calculation and extrapolation of the peak motion. Buttiker and Landauer (1982) have criticized this technique, since the shape of a wave packet can differ appreciably from that of an incident packet. Moreover, this approach invariably invokes also an extrapolation procedure. The time of incidence of a packet is calculated by extrapolating the motion of the wave packet far from the barrier to the incident point. Such an extrapolation procedure is unwarranted as the motion of an incident wave packet near a barrier is strongly distorted by the interference with precursors of the packet which have already been reflected. Leavens and Aers (1989a) give a simple but striking example of this effect. Therefore, alternative methods which do not depend on wave packet motion, have been developed (Buttiker and Landauer, 1982; Buttiker, 1983). Below, a very brief review of these methods and their generalizations by Leavens and Aers (1987, 1988a) are presented. Some basic relations between different characteristic time scales are derived analyzing a simple model which describes absorption of particles in a tunneling barrier.