Real time quantum tunneling through a double square barrier

The propagator of a one-dimensional double square barrier is obtained analytically in terms of the relevant eigenfunctions. With the aid of the propagator the wavefunction evolution for a particle, initially in a state of wave packet, located on the left-hand side of the barrier, with ignorable overlap with the barrier, is obtained. There follows study of the probability and current densities stemming from the particle initial state. In particular, the space distributions at given times of the above densities are given. Furthermore, their evolution in time at the entrance and exit of the barrier is obtained. The numerical results show repeated reversal in the current density at the barrier entrance, while being unidirectional at the exit. However, the probability of entering the barrier over an extremely long time tends to equal the corresponding probability of exiting the barrier. Owing to the fact that the wave packet expands on both of its sides, as time goes by, it is possible to have transmission even if the particle’s initial momentum points away from the barrier. The effect, in question, becomes evident in diagrams for the transmitted probability beyond the barrier exit in terms of the particle initial momentum, over a region.