More About Tunnelling Times, the Dwell Time, and the ``Hartman Effect

In a recent review paper [{em Phys. Reports} {bf 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 $<tau_{, rm Pen}>$, $<tau_{, rm Ret}>$, during tunnelling {em inside} a rectangular potential barrier, for various penetration depths $x_{rm f}$; (ii) putting forth and discussing suitable definitions, besides of the mean values, also of the {em variances} (or dispersions) ${rm D} , {tau_{rm T}}$ and ${rm D} , {tau_{, rm R}}$ for the time durations of transmission and reflection processes; (iii) mentioning, moreover, that our definition $<tau_{rm T}>$ for the average transmission time results to constitute an {em improvement} of the ordinary dwell--time ${ove tau}^{rm Dw}$ formula: (iv) commenting, at last, on the basis of our {em new} numerical results, upon some recent criticism by C.R.Leavens. We stress that our numerical evaluations {em confirm} that our approach implied, and implies, the existence of the {em 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.