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

Abstract

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 $<{\tau }_{\mathrm{Pen}}>$, $<{\tau }_{\mathrm{Ret}}>$, during tunnelling {\em inside} a rectangular potential barrier, for various penetration depths $x_{\mathrm{f}}$; \ (ii) putting forth and discussing suitable definitions, besides of the mean values, also of the {\em variances} (or dispersions) $\mathrm{D}{\tau }_{\mathrm{T}}$ and $\mathrm{D}{\tau }_{\mathrm{R}}$ for the time durations of transmission and reflection processes; \ (iii) mentioning, moreover, that our definition $<{\tau }_{\mathrm{T}}>$ for the average transmission time results to constitute an {\em improvement} of the ordinary dwell--time ${\ove τ}^{\rm Dw}$ 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.

Accepted for publication on "Journal de Physique I"

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