REAL TIME QUANTUM TUNNELING

The time evolution of the probability and current densities stemming from an initial state of a particle in the form of a wave packet on the left hand side of a square barrier is studied. In particular, the space distribution at a given time of the above densities is given. Furthermore, the evolution in time at the entrance and exit of the barrier is given. The numerical results show repeated reversal in the current density at the barrier entrance, while being unidirectional at the exit. However, the probability entering the barrier over a long time equals the corresponding probability 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 a diagram for the transmission coefficient in terms of the particle initial momentum, over a relatively small region.