Hartman effect and weak measurements that are not really weak

Abstract

We show that in wave packet tunneling, localization of the transmitted particle amounts to a quantum measurement of the delay it experiences in the barrier. With no external degree of freedom involved, the envelope of the wave packet plays the role of the initial pointer state. Under tunneling conditions such "self-measurement" is necessarily weak, and the Hartman effect just reflects the general tendency of weak values to diverge, as postselection in the final state becomes improbable. We also demonstrate that it is a good precision, or a 'not really weak' quantum measurement: no matter how wide the barrier d, it is possible to transmit a wave packet with a width σ small compared to the observed advancement. As is the case with all weak measurements, the probability of transmission rapidly decreases with the ratio σ/d.