More about tunnelling, the dwell time and the Hartman effect

In a recent review paper [Phys. Reports 214 (1992) 339] we proposed, within conventional quantum mechanics, new definitions for the sub-barrier tunnelling and reflection times. Aims of the present paper are: i) presenting and analysing the results of various numerical calculations (based on our equations) on the penetration and return times < tau(Pen) >, < tau(Ret) >, during tunnelling inside a rectangular potential barrier, for various penetration depths x(f); ii) putting forth and discussing suitable definitions, besides of the mean values, also of the variances (or dispersions) D tau(T). and D tau(R) for the time durations of transmission and reflection processes; nl) mentioning, moreover, that our definition < tau(T) > for the average transmission time results to constitute an improvement of the ordinary dwell-time tau(DW) formula: iv) commenting, at last, on the basis of our new numerical results, upon some recent criticism by C.R. Leavens. We stress that our numerical evaluations confirm that our approach implied, and implies, the existence of the 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. Eventully, we briefly analyze some other definitions of tunnelling times.