‘Superluminal paradox’ in wave packet propagation and its quantum mechanical resolution

Abstract

We analyse in detail the reshaping mechanism leading to apparently ‘superluminal’ advancement of a wave packet traversing a classically forbidden region. In the coordinate representation, a barrier is shown to act as an effective beamsplitter, recombining envelopes of the freely propagating pulse with various spacial shifts. Causality ensures that none of the constituent envelopes are advanced with respect to free propagation, yet the resulting pulse is advanced due to a peculiar interference effect, similar to the one responsible for ‘anomalous’ values which occur in Aharonov’s ‘weak measurements’. In the momentum space, the effect is understood as a bandwidth phenomenon, where the incident pulse probes local, rather than global, analytical properties of the transmission amplitude T(p). The advancement is achieved when T(p) mimics locally an exponential behaviour, similar to the one occurring in Berry’s ‘superoscillations’. Seen in a broader quantum mechanical context, the ‘paradox’ is but a consequence of an attempt to obtain ‘which way?’ information without destroying the interference between the pathways of interest. This explains, to a large extent, the failure to adequately describe tunnelling in terms of a single ‘tunnelling time’.