Tunneling times and “ Superluminal ” Tunneling : a Brief Review

In the First Part of this paper [that was submitted for pub. in 1991 and appeared in print in Phys. Reports 214 (1992) 339] we critically review and analyse the main theoretical definitions and calculations of the sub-barrier tunnelling and reflection times. Moreover, we propose a new definition of such durations, on the basis of a recent general formalism of ours (already tested for other types of quantum collisions) within conventional quantum mechanics. At last, we discuss some surprising results regarding the temporal evolution of the tunnelling processes: namely, the fact that QM predicts that tunnelling through opaque barriers takes place with Superluminal group-velocities. Aims of the Second Part [that appeared in print later, in J. de Physique-I 5 (1995) 1351] are: (i) presenting and analysing the results of various numerical calculations (based on our equations) on the penetration and return times < τPen >, < τRet >, during tunnelling inside a rectangular potential barrier, for various penetration depths xf ; (ii) putting forth and discussing suitable definitions, besides of the mean values, also of the variances (or dispersions) D τT and D τR for the time durations of transmission and reflection processes; (iii) mentioning, moreover, that our definition < τT > for the average transmission time results to constitute an improvement of the ordinary dwell–time τ formula. The numerical evaluations confirm that our approach implied, and implies, the existence of Superluminal tunnelling (that we call “Hartman effect”): an effect that in these days (due to the theoretical connections between tunnelling and evanescent–wave propagation) is receiving —at Cologne, Berkeley, Florence and Vienna— indirect, but quite interesting, experimental verifications. Eventually, we briefly analyze some other definitions of tunnelling times. A more detailed review of the same topics (in Italian, however) has been “published” electronically, as LANL Archives # cond-mat/9802126. At 0(∗) Work partially supported by INFN, MURST, CNR (Italy), and by the I.N.R. (Ukrainian Acad. Sc., Kiev).