J. Phys. I France
Volume 5, Numéro 10, October 1995
Page(s) 1351 - 1365
DOI: 10.1051/jp1:1995202
J. Phys. I France 5 (1995) 1351-1365

More about Tunnelling Times, the Dwell Time and the "Hartman Effect"

Vladislav S. Olkhovsky1, Erasmo Recami2, Fabio Raciti3 and Aleksandr K. Zaichenko1

1  Institute for Nuclear Research, Ukrainian Academy of Sciences, Kiev, Ukraine an I.N.F.N., Sezione di Catania, 57 Corsitalia, Catania, Italy
2  Facoltà di Ingegneria, Università Statale di Bergamo, Dalmine (BG), Italy; I.N.F.N., Sezione di Milano, Milan, Italy; Dept. of Applied Mathematics, State University at Campinas, Campinas, S.P., Brazil
3  Dipartim. di Fisica, Università di Catania, Catania, Italy

(Received 18 April 1995, received in final form 29 May 1995, accepted 20 June 1995)

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 $<\tau_{\rm Pen}>$, $<\tau_{\rm Ret}>$, during tunnelling 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 variances (or dispersions) $D\tau_{\rm T}$ and $D\tau_{\rm
R}$ 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 improvement of the ordinary dwell-time $\bar{\tau}^{D{\rm w}}$ 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.

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