Early times in tunneling

Abstract

Exact analytical solutions of the time-dependent Schr\"odinger equation with the initial condition of an incident cutoff wave are used to investigate the traversal time for tunneling. The probability density starts from a vanishing value along the tunneling and transmitted regions of the potential. At the barrier width it exhibits, at early times, a distribution of traversal times that typically has a peak $\tau_p$ and a width $\Delta \tau$. Numerical results for other tunneling times, as the phase-delay time, fall within $\Delta \tau$. The B\"uttiker traversal time is the closest to $\tau_p$. Our results resemble calculations based on Feynman paths if its noisy behaviour is ignored.