SEMICLASSICAL COMPUTATIONS OF TIME-DEPENDENT TUNNELING

In this paper we consider the time evolution of wave-packet tunneling through a potential barrier. Using a path-summation approach, we can describe the quantum evolution in terms of particle trajectories both in the allowed and forbidden regions. Near the barrier edges, where the potential changes rapidly, the wave aspects dominate. These make the trajectories branch into infinite families of paths, which have to be included in the summation. Combining these features, we treat correctly the complementary aspects of the full quantum process, and this allows us to obtain a computational method that gives accurate numerical results with considerably less expenditure of computer time than a direct integration of the Schrödinger equation. We test the method on a simple rectangular potential by comparing the calculations with both conventional wave scattering results and computations from the Schrödinger equation. A general potential can be approximated by a staircase, which allows the application of our method. A simple adiabatic extension is shown to work excellently for the case of a periodically modulated barrier. © 1995 The American Physical Society.