As was shown in quant-ph/0405028, the state of a tunneling particle can be uniquely presented as a coherent superposition of two states to describe alternative sub-processes, transmission and reflection. In this paper, on the basis of the stationary wave functions for these sub-processes, we give new definitions of the dwell times for transmission and reflection. In the case of rectangular potential barriers the dwell times are obtained explicitly. In contrast with the well-known Büttiker's dwell-time, our dwell time for transmission increases exponentially, for the under-barrier tunneling, with increasing the barrier's width. By our approach the well-known Hartman effect is rather an artifact resulted from an improper interpretation of the wave-packet tunneling and experimental data, but not a real physical effect accompanying the tunneling phenomenon.
As was shown in quant-ph/0405028, the state of a tunneling particle can be uniquely presented as a coherent superposition of two states to describe alternative sub-processes, transmission and reflection. In this paper, on the basis of the stationary wave functions for these sub-processes, we give new definitions of the dwell times for transmission and reflection. In the case of rectangular potential barriers the dwell times are obtained explicitly. In contrast with the well-known Büttiker's dwell-time, our dwell time for transmission increases exponentially, for the under-barrier tunneling, with increasing the barrier's width. By our approach the well-known Hartman effect is rather an artifact resulted from an improper interpretation of the wave-packet tunneling and experimental data, but not a real physical effect accompanying the tunneling phenomenon.
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 ,, during tunnelling {\em inside} a rectangular potential barrier, for various penetration depths xf; \ (ii) putting forth and discussing suitable definitions, besides of the mean values, also of the {\em variances} (or dispersions) DτT and DτR for the time durations of transmission and reflection processes; \ (iii) mentioning, moreover, that our definition $$ 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.
We develop a new quantum-mechanical approach to scattering a particle on a one-dimensional (1D) system of two identical rectangular potential barriers, which implies modelling the dynamics of its subprocesses -- transmission and reflection -- at all stages of scattering. On its basis we define, for each subprocess, the dwell time as well as the local (exact) and asymptotic (extrapolated) group times. Our concept of the asymptotic transmission group time confirms the validity of the Wigner phase time in the opaque limit, as well as the existence of the usual and generalized Hartman effects predicted on its basis. On the energy scale, this concept is valid everywhere in the high energy region as well as in the low energy region, excepting resonance points and their neighborhoods. On the contrary, the Buttiker dwell time is valid, as the transmission time, just only at the resonance points. Our concept of the transmission dwell time predicts monotonous growth of the tunneling time when the distance between the opaque barriers increases. By our approach only this time scale yields the true time spent, on average, by transmitted particles in the region occupied by the system. We explain why the asymptotic and local transmission group times cannot play this role and why the concept of {\it transmission group} velocity lies beyond the scope of special relativity. And else, all the transmission times admit only indirect measurements. Hence the unambiguous interpretation of all tunneling-time experiments is impossible when the transmission dynamics at all stages of scattering is unknown.
We develop a new quantum-mechanical approach to scattering a particle on a one-dimensional (1D) system of two identical rectangular potential barriers, which implies modelling the dynamics of its subprocesses -- transmission and reflection -- at all stages of scattering. On its basis we define, for each subprocess, the dwell time as well as the local (exact) and asymptotic (extrapolated) group times. Our concept of the asymptotic transmission group time confirms the validity of the Wigner phase time in the opaque limit, as well as the existence of the usual and generalized Hartman effects predicted on its basis. On the energy scale, this concept is valid everywhere in the high energy region as well as in the low energy region, excepting resonance points and their neighborhoods. On the contrary, the Buttiker dwell time is valid, as the transmission time, just only at the resonance points. Our concept of the transmission dwell time predicts monotonous growth of the tunneling time when the distance between the opaque barriers increases. By our approach only this time scale yields the true time spent, on average, by transmitted particles in the region occupied by the system. We explain why the asymptotic and local transmission group times cannot play this role and why the concept of {\it transmission group} velocity lies beyond the scope of special relativity. And else, all the transmission times admit only indirect measurements. Hence the unambiguous interpretation of all tunneling-time experiments is impossible when the transmission dynamics at all stages of scattering is unknown.
In a recent review paper [Phys. Rpeorts 214 (1992) 339] we proposed, within conventional quantum mechanics, new definitions fot the subarrier 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 $\\left \\langle tau_P_e_n \\right \\rangle $ , $\\left \\langle tau_R_e_t\\right \\rangle $ during tunnelling inside a rectangular potential barrier, for various penetration depths xf ; (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 tima durations of trasmission and reflection processes; (iii) mentioning that our definition $\\left \\langle tau_T \\right \\rangle $ for the average transmission time results to constituite an improvemente of the ordinary dwell-time $ \\bar tau^D^W\\ $ formula: (iv) commenting upon some recent criticism by C.R. Leavens, on the basis of our new numerical results. 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 Wien - indirect, but quite interesting, experimental verifications. At last we briefly analyze some other definitions of tunnelling times.
A {\it completed} scattering of a particle on a static one-dimensional (1D) potential barrier is a combined quantum process to consist from two elementary sub-processes (transmission and reflection) evolved coherently at all stages of scattering and macroscopically distinct at the final stage. The existing model of the process is clearly inadequate to its nature: all one-particle "observables" and "tunneling times", introduced as quantities to be common for the sub-processes, cannot be experimentally measured and, consequently, have no physical meaning; on the contrary, quantities introduced for either sub-process have no basis, for the time evolution of either sub-process is unknown in this model. We show that the wave function to describe a completed scattering can be uniquely presented as the sum of two solutions to the Schrödinger equation, which describe separately the sub-processes at all stages of scattering. For symmetric potential barriers such solutions are found explicitly. For either sub-process we define the time spent, on the average, by a particle in the barrier region. We define it as the Larmor time. As it turned out, this time is just Buttiker's dwell time averaged over the corresponding localized state. Thus, firstly, we justify the known definition of the local dwell time introduced by Hauge and co-workers as well by Leavens and Aers, for now this time can be measured; secondly, we confirm that namely Buttiker's dwell time gives the energy-distribution for the tunneling time; thirdly, we state that all the definitions are valid only if they are based on the wave functions for transmission and reflection found in our paper. Besides, we define the exact and asymptotic group times to be auxiliary in timing the scattering process.
The proper basis for the calculation of transmission and reflection times for wave packets scattered off arbitrary tunneling structures in one dimension is considered. With packets narrow in wave-number space, we demonstrate that the classic phase times are indeed correct to lowest order. Explicit, general expressions for the leading correction terms for finite wave packets are given. The physics associated with these corrections is discussed. We also consider the dwell time, as it is currently defined, and derive a general relation between this dwell time and the phase times. This relation shows when the dwell time can and cannot be used. Finally, we discuss wave packets transmitted from narrow resonances, and derive an explicit, exact formula for the tunneling time with resonance transmission from a symmetric double barrier. Comparison with earlier approximate results is made.