Traversal-Time Distribution and the Uncertainty Principle in Quantum Tunneling

We derive the exact distribution of times spent in the classically forbidden region by a particle as it tunnels through a square barrier potential. We find that the integrated distribution has a simple interpretation. If one restricts the time spent in the barrier to be {}{τ}, then an effective uncertainty {Δ}V is introduced in the barrier height, such that{Δ}V{τ}{}1. This uncertainty leads to unusual interference effects in the distribution. These interference effects explain why the average speed of a particle traversing a very opaque barrier may exceed the speed of light. {} 1990 The American Physical Society