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Tunnelling times: An elementary introduction

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 Added by Giovanni Salesi
 Publication date 2004
  fields Physics
and research's language is English




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In this paper we examine critically and in detail some existing definitions for the tunnelling times, namely: the phase-time; the centroid-based times; the Buttiker and Landauer times; the Larmor times; the complex (path-integral and Bohm) times; the dwell time, and finally the generalized (Olkhovsky and Recami) dwell time, by adding also some numerical evaluations. Then, we pass to examine the equivalence between quantum tunnelling and photon tunnelling (evanescent waves propagation), with particular attention to tunnelling with Superluminal group-velocities (Hartman effect). At last, in an Appendix, we add a bird-eye view of all the experimental sectors of physics in which Superluminal motions seem to appear.



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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.
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