From the paradoxes of the standard wave-packet analysis to the definition of tunneling times for particles

摘要

We develop a new variant of the wave-packet analysis and solve the tunnelingtime problem for one particle. Our approach suggests an individual asymptoticdescription of the quantum subensembles of transmitted and reflected particlesboth at the final and initial stage of tunneling. We find the initial states ofboth subensembles, which are non-orthogonal. The latter reflects ultimately thefact that at the initial stage of tunneling it is impossible to predict whethera particle will be transmitted through or reflected off the barrier. At thesame time, in this case, one can say about the to-be-transmitted andto-be-reflected subensembles of particles. We show that before the interactionof the incident packet with the barrier both the number of particles and theexpectation value of the particle's momentum are constant for each subensemble.Besides, these asymptotic quantities coincide with those of the correspondingscattered subensemble. On the basis of this formalism we define individualdelay times for both scattering channels. Taking into account the spreading ofwave packets, we define the low bound of the scattering time to describe thewhole quantum ensemble of particles. All three characteristic times are derivedin terms of the expectation values of the position and momentum operators. Thecondition which must be fulfilled for completed scattering events is derived.We propose also the scheme of a {\it gedanken} experiment that allows one toverify our approach.