Recent developments are reviewed and some new results are presented in the study of time in quantum mechanics and quantum electrodynamics as an observable, canonically conjugate to energy. This paper deals with the maximal Hermitian (but nonself-adjoint) operator for time which appears in nonrelativistic quantum mechanics and in quantum electrodynamics for systems with continuous energy spectra and also, briefly, with the four-momentum and four-position operators, for relativistic spin-zero particles. Two measures of averaging over time and connection between them are analyzed. The results of the study of time as a quantum observable in the cases of the discrete energy spectra are also presented, and in this case the quasi-self-adjoint time operator appears. Then, the general foundations of time analysis of quantum processes (collisions and decays) are developed on the base of time operator with the proper measures of averaging over time. Finally, some applications of time analysis of quantum processes (concretely, tunneling phenomena and nuclear processes) are reviewed.

1. General Introduction

During almost ninety years (e.g., [1, 2]), it is known that time cannot be represented by a self-adjoint operator, with the possible exception of special abstract systems (such as an electrically charged particle in an infinite uniform electric field) and a system with the limited from both below and above energy spectrum (to see later)). (Namely that fact that time cannot be represented by a self-adjoint operator is known to follow from the semiboundedness of the continuous energy spectra, which are bounded from below (usually by the value zero). Only for an electrically charged particle in an infinite uniform electric field, and for other very rare special systems, the continuous energy spectrum is not bounded and extends over the whole energy axis from to .) This fact results to be in contrast with the known sircumstance that time, as well as space, in some cases plays the role just of a parameter, while in some other cases is a physical observable which ought to be represented by an operator. The list of papers devoted to the problem of time in quantum mechanics is extremely large (e.g., [351], and references therein). The same situation had to be faced also in quantum electrodynamics and, more in general, in relativistic quantum field theory (e.g., [1214, 47, 50, 51]).

As to quantum mechanics, the first set of known and cited articles is [321]. The second set of papers on time as an observable in quantum physics [2251] appeared from the end of the eighties and chiefly in the nineties and more recently, stimulated mainly by the need of a self-consistent definition for collision duration and tunneling time. It is noticeable that many of this second set of papers appeared however to ignore the Naimark theorem from [52], which had previously constituted an important basis for the results in [1521]. This Naimark theorem states [52] that the nonorthogonal spectral decomposition 𝐸(𝜆) of a Hermitian operator H is of the Carleman type (which is unique for the maximal Hermitian operator), that is, it can be approximated by a succession of the self-adjoint operators, the spectral functions of which do weakly converge to the spectral function 𝐸(𝜆) of the operator H.

Namely, by exploiting that Naimark theorem, it has been shown by Olkhovsky and Recami [1518, 21] (more details having been added in [2227, 3235, 47, 50, 51]) and, independently, by Holevo [19, 20] that, for systems with continuous energy spectra, time can be introduced as a quantum-mechanical observable, canonically conjugate to energy. More precisely, the time operator resulted to be maximal Hermitian, even if not self-adjoint. Then, in [2325, 3335, 50, 51] it was clarified that time can be introduced also for these systems as a quantum-mechanical observable, canonically conjugate to energy, and the time operator resulted to be quasi-self-adjoint (more precisely, it can be chosen as an almost self-adjoint operator with practically almost any degree of the accuracy).

We intend to justify the association of time with a quantum-physical observable, by exploiting the properties of the maximal Hermitian operators in the case of the continuous energy spectra, and the properties of quasi-self-adjoint operators in the case of the discrete energy spectra.

Then, we analyze the restricted sense of the positive operator value measure (POVM) approach, often used now (see, in particular [2831, 3646, 48, 49]). Finally, we do in a shorten way review our methods of time analysis and joint time-energy analysis which had already proved to be fruitful in tunnelling and nuclear processes.

2. Time as a Quantum Observable and General Definitions of Mean Times and Mean Durations of Quantum Processes

2.1. On Time as an Observable in Nonrelativistic Quantum Mechanics, for Systems with Continuous Energy Spectra

For systems with continuous energy spectra, the following simple operator, canonically conjugate to energy, can be introduced for time which is not self-adjoint, but is Hermitian, and acts on square-integrable space-time wave packets in representation (2.1a), and on their Fourier transforms in representation (2.1b), once the point 𝐸0 is eliminated (i.e., once one deals only with moving packets, i.e., excludes any nonmoving back tails, as well as, of course, the zero flux cases). (Such a condition is enough for operator (2.1a) and (2.1b) to be a “maximal Hermitian” (or “maximal symmetric”) operator [1518, 21] (see also [26, 27, 3335, 52, 53]), according to Akhiezer & Glazman’s terminology. Let us explicitly notice that, anyway, the physically reasonable boundary condition 𝐸=0 can be dispensed with, by having recourse to bilinear operators, as it is simply shown below in the form (2.26) and Appendix A.) It has been shown already in [1518, 21]. The elimination of the point 𝑃 is not restrictive since the “rest” states with the zero velocity, the wave packets with nonmoving rear tails, and the wave packets with zero flux are unobservable.

Operator (2.1b) is defined as acting on the space 𝑓(𝐸) of the continuous, differentiable, square-integrable functions 0||||𝑓(𝐸)2𝑑𝐸<,0|||𝜕𝑓(𝐸)|||𝜕𝐸2𝑑𝐸<,0||||𝑓(𝐸)2𝐸2𝑑𝐸<,(2.2) that satisfy the conditions and the condition which is a space 𝐿2dense in the Hilbert space of 0𝐸< functions defined (only) over the semiaxis (𝑓1,̂𝑡𝑓2̂)=((𝑡𝑓1),𝑓2). Obviously, the operator (2.1a) and (2.1b) is Hermitian, that is, the relation 𝑓(𝐸) holds, only if all square-integrable functions 𝐸=0 in the space on which it is defined vanish for ̂𝑡2.

Also the operator (𝑓1,̂𝑡2𝑓2̂)=((𝑡𝑓1̂),(𝑡𝑓2̂𝑡))=(2𝑓1,𝑓2) is Hermitian, that is, the relation ̂𝑡 holds under the same conditions.

Operator 𝑓0(𝐸) has no Hermitian extension because otherwise one could find at least one function 𝑓0(0)0 which satisfies the condition ̂𝑡 but that is inconsistent with the propriety of being Hermitian. So, according to [53], (0𝑥<) is a maximal Hermitian operator and in accordance with the results of the mathematical theory of operators it is not a self-adjoint operator with equal deficiency indices but it has the deficiency indices (0,1). As a consequence, operator (2.1b) does not allow a unique orthogonal identity resolution.

Essentially because of these reasons, earlier Pauli (e.g., [1, 2]) rejected the use of a time operator; this had the result of practically stopping studies on this subject for about forty years. However, as far back as in [54] von Neumann had claimed that considering in quantum mechanics only self-adjoint operators could be too restrictive. To clarify this issue, let us quote an explanatory example set forth by von Neumann himself [54]: let us consider a particle, free to move in a spatial semiaxis 𝑥=0 bounded by a rigid wall located at ̂𝑝𝑥𝜕=𝑖𝜕𝑥(2.4). Consequently, the operator for the momentum x-component of the particle, which reads is defined as acting on the space of the continuous, differentiable, square-integrable functions 0||||𝑓(𝑥)2𝑑𝑥<,0|||𝜕𝑓(𝑥)|||𝜕𝑥2𝑑𝑥<,0||||𝑓(𝑥)2𝑥2𝑑𝑥<,(2.5) that satisfy the conditions and the condition which is a space dense 𝐿2 in the Hilbert space of 0𝑥< functions defined (only) over the spatial semiaxis ̂𝑝𝑥=𝑖(𝜕/𝜕𝑥). Therefore, operator ̂𝑡 has the same mathematical properties as operator ̂𝑝𝑟=𝑖(𝜕/𝜕𝑟)+(1/𝑟)(0<𝑟<) (2.1a) and (2.1b) and consequently it is not a self-adjoint operator but it is only a maximal Hermitian operator. Nevertheless, it is an observable with an obvious physical meaning. The same properties has also the radial momentum operator (𝐸min<𝐸<𝐸max).

By the way, one can easily demonstrate (e.g., [4, 5]) that in the case of (hypotetical) quantum-mechanical systems with the continuous energy spectra bounded from below and from above 𝜏=/𝑑 the time operator (2.1a) and (2.1b) becomes a really self-adjoint operator and has a discrete time spectrum, with the “the time quantum” 𝑑=|𝐸max𝐸min| where 𝑊(𝑥,𝑡)𝑑𝑡=𝑗(𝑥,𝑡)𝑑𝑡𝑗(𝑥,𝑡)𝑑𝑡,(2.7).

In order to consider time as an observable in quantum mechanics and to define the observable mean times and durations, one needs to introduce not only the time operator, but also, in a self-consistent way, the measure (or weight) of averaging over time. In the simple one-dimensional (1D) and one-directional motion such measure (weight) can be obtained by the the simple quantity: where the probabilistic interpretation of 𝑥 (namely in time) corresponds to the flux probability density of a particle passing through point 𝑡 at time 𝑥 (more precisely, passing through 𝑡 during a unit time interval centered at 𝜌(𝑥,𝑡)), when travelling in the positive x-direction. Such a measure had not been postulated, but is just a direct consequence of the well-known probabilistic (spatial) interpretation of 𝜕𝜌(𝑥,𝑡)𝜕𝑡+div𝑗(𝑥,𝑡)=0(2.8) and of the continuity relation for particle motion in the field of any hamiltonian in the desciption of the 1D Schroedinger equation. (The three-dimensional (3D) case is described in Appendix B.) Quantity 𝜌(𝑥,𝑡) is the probability of finding a moving particle inside a unit space interval, centered at point x, at time t. The probability density 𝑗(𝑥,𝑡) and the flux probability density Ψ(𝑥,𝑡) are related with the wave function 𝜌(𝑥,𝑡)=|Ψ(𝑥,𝑡)|2 by the usual definitions 𝑗(𝑥,𝑡)=Re[Ψ(𝑥,𝑡)(/𝑖𝜇)𝜕Ψ(𝑥,𝑡)/𝜕𝑥] and 𝑗(𝑥,𝑡). The measure (2.7) was firstly investigated in [21, 2327, 3235].

When the flux density 𝑊(𝑥,𝑡)𝑑𝑡 changes its sign, the quantity 𝑗(𝑥,𝑡) is no longer positive definite and it acquires a physical meaning of a probability density only during those partial time intervals in which the flux density 𝑊±𝑗(𝑥,𝑡)𝑑𝑡=±(𝑥,𝑡)𝑑𝑡𝑗±(𝑥,𝑡)𝑑𝑡,(2.9) does keep its sign. Therefore, let us introduce the two measures, by separating the positive and the negative flux-direction values (i.e., flux signs): with Θ(𝑧) where 𝑗=𝑗+ is the Heaviside step function. It had been made firstly in [26, 27, 3235]. Actually, one can rewrite the continuity relation (2.8) for those time intervals, for which 𝑗=𝑗 or 𝜕𝜌>(𝑥,𝑡)𝜕𝑡=𝜕𝑗+(𝑥,𝑡),𝜕𝑥𝜕𝜌<(𝑥,𝑡)𝜕𝑡=𝜕𝑗(𝑥,𝑡)𝜕𝑥,(2.10) as follows: respectively. Relations in (2.10) can be considered as formal definitions of 𝜕𝜌</𝜕𝑡 and . Integrating them over time t from 𝑡 to 𝜌>(𝑥,𝑡)=𝑡𝜕𝑗+𝑥,𝑡𝜕𝑡𝑑𝑡,𝜌<(𝑥,𝑡)=𝑡𝜕𝑗𝑥,𝑡𝜕𝑡𝑑𝑡(2.11), one obtains with the initial conditions 𝑁>(𝑥,;𝑡)𝑥𝜌>𝑥,𝑡𝑑𝑥=𝑡𝑗+𝑥,𝑡𝑑𝑡𝑁>0,<(,𝑥;𝑡)𝑥𝜌<𝑥,𝑡𝑑𝑥=𝑡𝑗𝑥,𝑡𝑑𝑡>0,(2.12). Then, it is possible to introduce the quantities which have the meaning of probabilities for the particle wave packet 𝑡 to be located at time (𝑥,) on the semiaxis (,𝑥) and 𝑗+(,𝑡), respectively, as functions of the flux densities 𝑗(𝑥,𝑡) and 𝜌(𝑥,𝑡)𝑑𝑥=1, provided that the normalization condition 𝜌>(𝑥,𝑡) is fulfilled. The right-hand parts of the last couple of equations have been obtained by integrating the rigt-hand parts of the expressions for 𝜌<(𝑥,𝑡) and 𝑗+(,𝑡)=𝑗(,𝑡)=0, and by adopting the boundary conditions 𝑁>(𝑥,;𝑡). Then, by differentiating 𝑁<(,𝑥;𝑡) and 𝜕𝑁>(𝑥,;𝑡)𝜕𝑡=𝑗+(𝑥,𝑡)>0,𝜕𝑁<(,𝑥;𝑡)𝜕𝑡=𝑗(𝑥,𝑡)>0.(2.13) with respect to t, one obtains Finally, from the last four equations one can easily infer that which justify the abovementined probabilistic interpretation of 𝑡(𝑥). Let us stress now that this approach does not assume any new physical postulate in the conventional (Copenhagen-interpretation) nonrelativistic quantum mechanics.

Then, one can eventually define the mean value 𝑡 of the time 𝑡±(𝑥) at which a particle passes through position x (when travelling in only one positive x-direction), and 𝑥 of the time t at which a particle passes through position 𝑡(𝑥)=𝑡𝑗(𝑥,𝑡)𝑑𝑡=𝑗(𝑥,𝑡)𝑑𝑡0𝐺𝑑𝐸(1/2)̂(𝑥,𝐸)𝑡𝑣𝐺(𝑥,𝐸)+𝑣𝐺̂(𝑥,𝐸)𝑡𝐺(𝑥,𝐸)0||||𝑑𝐸𝑣𝐺(𝑥,𝐸)2,(2.15), when travelling in the positive or negative direction, respectively, where Ψ(𝑥,𝑡)=0𝐺(𝑥,𝐸)exp𝑖𝐸𝑡𝑑𝐸=0𝑔(𝐸)𝜑(𝑥,𝐸)exp𝑖𝐸𝑡𝑑𝐸,(2.16) is the Fourier transform of the moving 1D wave packet when going on from the time representation to the energy one, and also the mean durations of particle 1D transmission from 𝑥𝑓>𝑥𝑖 to (𝑥𝑖,) and 1D particle reflection from the region 𝑥𝑓𝑥𝑖 into 𝜏𝑇𝑥𝑖,𝑥𝑓=𝑡+𝑥𝑓𝑡+𝑥𝑖,𝜏𝑅𝑥𝑖,𝑥𝑓=𝑡𝑥𝑓𝑡+𝑥𝑖,(2.18): respectively. (We recall that here we are confining ourselves to systems with continuous spectra only.) Of course, it is possible to pass in (2.17) also to integrals 𝑗±(𝑥,𝑡)=𝑗(𝑥,𝑡)𝜃(±𝑗), similarly to (2.15) by using the unique Fourier (Laplace) transformations and the energy expansion of 𝑡𝑛, but it is evident that they result to be rather bulky.

If one does now generalize the expressions (2.15) and (2.17) for 𝑛=2,3,, with a generic value 𝑓(𝑡) then we will be able to write down for 𝑓(𝑡) with any analytic function of time 𝑓(𝑡)=𝑗(𝑥,𝑡)𝑓(𝑡)𝑑𝑡=𝑗(𝑥,𝑡)𝑑𝑡0𝐺𝑑𝐸(1/2)̂𝑡(𝑥,𝐸)𝑓𝑣𝐺(𝑥,𝐸)+𝑣𝐺̂𝑡(𝑥,𝐸)𝑓𝐺(𝑥,𝐸)0||||𝑑𝐸𝑣𝐺(𝑥,𝐸)2(2.19), the one-to-one relation from the time to the energy representation. For free motion, one has 𝜑(𝑥,𝐸)=exp(𝑖𝑘𝑥), 𝐸=2𝑘2/2𝜇=𝜇𝑣2/2, and 0||||𝐺(𝑥,𝐸)2𝑑𝐸=0||||𝑔(𝐸)2𝑑𝐸=1,(2.20), while the normalization condition is and the boundary conditions are

Conditions (2.21) imply a very rapid decrease till zero of the flux densities near the boundaries 𝐸= and ̂𝑡: this complies with the actual conditions of real experiments, and therefore they does not represent any restriction of generality.

In (2.19), |𝑔(𝐸)|2𝐾𝛿(𝐸𝐸) is defined by relation (2.1b). One should explicitly notice that relation (2.19) does express the complete equivalence of the time and of the energy representations (with their own appropriate averaging weights). This equivalence is a consequence of the existence of the time operator. Actually, for the time and energy operators it holds in quantum mechanics the same formalism as for all other pairs of canonically-conjugate observables.

For quasimonochromatic particles, when 𝑗(𝑥,𝑡), K being a constant, quantity 𝜌(𝑥,𝑡) goes into 𝑓(𝑡)𝑗(𝑥,𝑡)𝑓(𝑡)𝑑𝑡𝑗(𝑥,𝑡)𝑑𝑡𝜌(𝑥,𝑡)𝑓(𝑡)𝑑𝑡𝜌(𝑥,𝑡)𝑑𝑡0𝑑𝐸𝐺̂𝑡(𝑥,𝐸)𝑓𝐺(𝑥,𝐸)0||||𝑑𝐸𝐺(𝑥,𝐸)2,(2.22) and (2.19) goes into the more simple relation because of the relations 𝜕𝐸=𝐸intheenergy(𝐸)-representation,𝑖𝜕𝑡inthetime(𝑡)-representation,(2.23).

Now, one can see that two canonically conjugate operators, the time operator (2.1a), (2.1b), and (2.26) and the energy operator satisfy the typical commutation relation

Although up to now according to the Stone theorem [55] the relation (2.24) has been interpreted as holding only for the pair of the self-adjoint canonically conjugate operators, in both representations, and it was not directly generalized for maximal Hermitian operators, the difficulty of such direct generalization has in fact been by-passed by introducing (<𝑡<) with the help of the single-valued Fourier (Laplace) transformation from the t-axis (0<𝐸<) to the E-semiaxis Δ𝐸Δ𝑡2(2.25) and by utilizing the peculiar mathematical properties of maximal symmetric operators (as in [1921, 2325, 3335, 50, 51]), described in detail, for example, in [52, 53].

Actually, from (2.24) the uncertainty relation (where the standard deviations are D𝑎 quantity D𝑎=𝑎2𝑎2 being the variance 𝑎=𝐸,𝑡; and , while 𝑡 denotes an average over 𝑊(𝑥,𝑡)𝑑𝑡 by the measures 𝑊±(𝑥,𝑡)𝑑𝑡 or 𝑡 in the 𝐸-representation or an average over {𝑞0,𝑝0} similar to the right-hand part of (2.19) in the E-representation) was derived by the simple generalizing of the similar procedures which are standard in the case of self-adjoint canonically conjugate quantities (see [1721, 2325, 3335, 50, 51]). Moreover, relation (2.24) satisfies the Dirac “correspondence principle,” since the classical Poisson brackets 𝑞0=𝑡, with 𝑝0=𝐸 and ̂𝑡, are equal to unity [56]. In [21] (see also [2325]) it was also shown that the differences between the mean times at which a wave packet passes through a pair of points obey the Ehrenfest correspondence principle; in other words, in [21, 2325] the Ehrenfest theorem was suitably generalized.

After what precedes, one can state that, for systems with continuous energy spectra, the mathematical properties of the maximal Hermitian operators (described, in particular, in [49, 53]), like ̂𝑡 in (2.1a), (2.1b), and (2.26) are sufficient for considering them as quantum observables: namely, the uniqueness of the “spectral decomposition” (also called spectral function) for operators ̂𝑡𝑛(𝑛>1), as well as for ̂𝑡𝑛(𝑛>1) guarantees (although such an expansion is not orthogonal) the equivalence of the mean values of any analytic functions of time, evaluated either in the t- or in the E-representations. In other words, the existence of this expansion is equivalent to a completeness relation for the (formal) eigenfunctions of ̂𝑡𝑛(𝑛>1), corresponding with any accuracy to real eigenvalues of the continuous spectrum; such eigenfunctions belonging to the space of the square-integrable functions of the energy E with the boundary conditions (2.2)-(2.3).

From this point of view, there is no practical difference between self-adjoint and maximal Hermitian operators for systems with continuous energy spectra. Let us underline that the mathematica, properties of , are quite enough for considering time asa quantum-mechanical observable (like for energy, momentum, spatial coordinateŝ𝑡=𝑖2𝜕𝜕𝐸(2.26)) without having to introduce any new physical postulates.

Now let us analyse the so-called positive-operator-value-measure (POVM) approach, often used in the second set of papers on time in quantum physics (e.g., in [2831, 3646, 48, 49]). This approach, in general, is well known in the various approaches to the quantum theory of measurements approximately from the sixties and had been applied in the simplest form for the time-operator problem in the case of the free motion already in [57]. Then, in [2831, 3646, 48, 49] (often with certain simplifications and abbreviations) it was affirmed that the generalized decomposition of unity (or POV measures) is reproduced from any self-adjoint extension of the time operator into the space of the extended Hilbert space (usually, with negative values of energy E in the left semiaxis) citing the Naimark’s dilation theorem from [58]. However, it was realized factually only for the simple cases like the particle free motion. As to our approach, it is based on another Naimark's theorem (from [52]), cited above, and without any extension of the physical Hilbert space of usual wave functions (wave packets) with the subsequent return projection to the previous space of wave functions, and, moreover, it had been published in [1218, 21] (and independently in the papers of Holevo [19, 20], with the same principal idea) much earlier than [2831, 3646, 48, 49]. Being based on the earlier published remarkable Naimark theorem [52], it is much more direct, simple and general, and at the same time mathematically not less rigorous than POVM approach.

Let us note that it was introduced by Olkhovsky and Recami in [1214] one more form of the time operator (the so-called bilinear form), where the meaning of the sign ̂(𝑓,𝑡𝑔)=(𝑓,(𝑖/2)(𝜕/𝜕𝐸)𝑔)+((𝑖/2)(𝜕/𝜕𝐸)𝑓,𝑔) is clear from the following definition: 𝐸=0. For this form the direct elimination of the point ̂(𝑓,𝑡𝑓) is not necessary because it is eliminated automatically in 𝑡𝑗(𝑥,𝑡)𝑑𝑡 and in 𝐸=0 by such bilinearity. And such an elimination of the point 𝐸 is not only more simple but also more physical than an elimination made in [2831, 3646, 48, 49], and it had been published (in [1214]) much more earlier.

2.2. On the Momentum Representation of the Time Operator

In [19, 20], it had been demonstrated by Holevo that in the continuous spectrum case, instead of the energy (0<𝐸<-) representation, with 𝑘, in (2.1a), (2.1b), (2.26), (2.2), (2.3), and (2.7) one can also use the momentum (<𝑘<-) representation, with the advantage that Ψ(𝑥,𝑡)=𝑑𝑘𝑔(𝑘)𝜑(𝑥,𝑘)exp𝑖𝐸𝑡,(2.27): with <𝑘<. In such a case the time operator (2.1a), (2.1b), and (2.26) (acting on momentum eigenvector, defined on 𝑑𝑛𝑔(𝑘)𝑑𝑘𝑛𝑘==𝑑𝑛𝑔(𝑘)𝑑𝑘𝑛𝑘==0,𝑛=0,1,2,,(2.28), is already formally self-adjoint, with the boundary conditions except for the fact that we have excluded point Ψ(𝑥,𝑡)=0𝑑𝐸(𝐸)1/2𝑔(2𝑚𝐸)1/2𝜑𝑥,(2𝑚𝐸)1/2exp𝑖𝐸𝑡+0𝑑𝐸(𝐸)1/2𝑔(2𝑚𝐸)1/2𝜑𝑥,(2𝑚𝐸)1/2exp𝑖𝐸𝑡.(2.29); an exclusion which has now only the physical meaning of nonobserving the rest (motionless) state), being inessential mathematically (this had been considered in [19, 20, 50, 51]). In fact, it is one more argument in favor of that time is an observable in the same degree as any other quantity to which a self-adjoint operator corresponds.

Let us now compare choice (2.27) with choice (2.16); namely let us rewrite (2.27) as follows: If we now introduce the weight as a “two-dimensional” vector, then the norm being

If the wave packet (2.27) is one directional and 0𝑑𝑘, then the integral 𝐸𝑘 goes on to the integral ||||𝑑𝑃(𝑥,𝑡)𝑍(𝑥,𝑡)𝑑𝑥=Ψ(𝑥,𝑡)2𝑑𝑥||||Ψ(𝑥,𝑡)2𝑑𝑥(2.33) and the two-dimensional vector goes on to a scalar quantity. In such a case, the boundary conditions (2.2) and (2.3) can be replaced by relations of the same form, provided that the replacement (𝑥,𝑥+𝑑𝑥) is performed.

2.3. The Second Measure of Time Averaging (in the Cases of Particle Dwelling in Spatial Regions)

One can easily see that the weight can be considered as the meaning of the probability for a particle to be localized, or to sojourn, or to dwell in the spatial region (𝑥1,𝑥2) at the moment t , independently from the motion processes. As a consequence, the quantity will have the meaning of the probability of particle dwelling in the spatial range 𝑗(𝑥,𝑡)𝑑𝑡=||||Ψ(𝑥,𝑡)2𝑑𝑥,(2.35) at the instant 𝜏dw𝑥𝑖,𝑥𝑓=𝑑𝑡𝑥𝑓𝑥𝑖||||Ψ(𝑥,𝑡)2𝑑𝑥𝑗in𝑥𝑖,𝑡𝑑𝑡(2.36a). Taking into account the equality which evidently follows from the 1D continuity relation (2.8), the mean dwell time can be presented in the following form: with the flux density 𝜏dw𝑥𝑖,𝑥𝑓=𝑥𝑡𝑗𝑓,𝑡𝑑𝑡𝑥𝑡𝑗𝑖,𝑡𝑑𝑡𝑗in𝑥𝑖,𝑡𝑑𝑡,(2.36b) for the initial “dwellingfree motion through point 𝑗(𝑥,𝑡). The expression (2.36a) can be rewritten in the following equivalent form taking into account the continuity relation (2.8) for the total flux density 𝑡 in the interval 𝐴𝑟=,𝑡𝑘0𝑑3𝑘𝑘0𝜒𝑘𝜑𝑘,𝑟exp𝑖𝑘0𝑡,(2.37) at time 𝐴(𝑟,𝑡) (the details of the derivation one can see in [22, 47]).

Thus, in correspondence with two measures above, (2.7), (2.9), (2.36a), and (2.36b), when integrating on time, we get different two kinds of time distributions (mean values, variances, etc.) being with different physical meanings (referring to the particle moving, passing, transferring, traversing, transmitting, etc. in the case of the measures (2.7) and (2.9) and of particle staying, dwelling, living, sojourning, etc. in the case of the measure (2.36a) and (2.36b), resp.).

2.4. Extension of the Notion of Time as a Quantum-Physical Observable Quantity to Quantum Electrodynamics

The formal mathematical analogy between the stationary and time-dependent Schroedinger equation for nonrelativistic particles and the stationary and time-dependent Helmholtz equation for electromagnetic wave propagation was studied in [5962]. In the time-dependent case, these equations are no longer mathematically equivalent, since the former is first-order in the time derivative whereas the latter is second order. However, here we will deal with the comparison of their solutions, considering not only the formal mathematical analogy between them but also such similarity of the probabilistic interpretation of the wave function for a particle and of an electromagnetic wave packet (being according to [63, 64] the “wave function for a single photon”) which is sufficient for the identical definition of mean time instants and durations (and distribution variances, etc.) of propagation, collision, tunnelling, processes for particles and photons.

In the first quantization for the 1D case, the single-photon wave function can be probabilistically described by the wave packet (see, e.g., [63, 64]) where, as usual, 𝑘={𝑘𝑥,𝑘𝑦,𝑘𝑧} is the electromagnetic vector potential, and 𝑘0𝜔/𝑐=𝜀/𝑐, 𝑘|𝑘|=𝑘0, div𝐴=0, 𝑥, where the gauge condition 𝜒(𝑘)=𝑧𝑖=𝑦𝜒𝑖(𝑘)𝑒𝑖(𝑘); is assumed.

The axis 𝑒𝑖𝑒𝑗=𝛿𝑖𝑗 has been chosen as the propagation direction, 𝑥𝑖,𝑥𝑗𝑦,𝑧 with 𝜒𝑖(𝑘), 𝑘, 𝑒𝑗 is the probability amplitude for the photon to have momentum 𝑥𝑗 and polarization 𝜑(𝑘,𝑟)=exp(𝑖𝑘𝑥𝑥) along 𝜑(𝑘,𝑟), and it is 𝜌em𝑆(𝑥,𝑡)𝑑𝑥=0𝑑𝑥𝑆0𝑑𝑥,𝑆0=𝑠0𝑑𝑦𝑑𝑧(2.38) in the case of plane waves, while 𝑠0=[𝐸𝐸+𝐻𝐻]/4𝜋 is a linear combination of evanescent (decreasing) and antievanescent (increasing) waves in the case of “photon barriers” (various band-gap filters, or even undersized segments of waveguides for microwaves, frustrated-total-internal-reflection regions for light, etc.). Although it is not possible to localize a photon in the direction of its polarization, nevertheless for 1D propagations, it is possible to use the space-time probabilistic interpretation of (2.37) and define the following probability density: (𝐸=(1/𝑐)𝜕𝐴/𝜕𝑡 being the energy density, the electromagnetic field being (𝑥,𝑥+𝑑𝑥), 𝑗em𝑆(𝑥,𝑡)𝑑𝑡=𝑥(𝑥,𝑡)𝑑𝑡𝑆𝑥(𝑥,𝑡)𝑑𝑡,𝑆𝑥𝑠(𝑥,𝑡)=𝑥𝑑𝑦𝑑𝑧(2.39)) of a photon to be found (localized) in the spatial intervall𝑠𝑥=𝑐Re[𝐸𝐻]𝑥/8𝜋 along axis x at the moment t, and the flux probability density (with (𝑡,𝑡+𝑑𝑡) being the energy flux density, 𝑡(𝑥)=𝑡𝐽em,𝑥𝑑𝑡=𝑡𝑆𝑥(𝑥,𝑡)𝑑𝑡𝑆𝑥(𝑥,𝑡)𝑑𝑡,(2.40)) of a photon to pass through the point (plane) x in the time interval 𝜒𝑖(0)=𝜒𝑖()=0, quite similarly to the probabilistic quantities for particles. The justification and convenience of such definitions is evident, every time that there is a coincidence of the wave packet group velocity and the velocity of the energy transport which was established for electromagnetic waves [6567]. Hence, (1) in a certain sense, for the time analysis along the motion direction, the wave packet (2.24) is quite similar to a wave packet for nonrelativistic particles and (2) similarly to the conventional nonrelativistic quantum mechanics, one can define the mean time of photon (electromagnetic wave packet) passing through point x : where the form (2.1b) of time operator is valid also for photons with natural boundary conditions 𝑠0 in the energy representation 𝑠𝑥, quite similarly to (2.1b)–(2.3) for nonrelativistic particles in the energy representation.

The energy density 𝜕𝑠0+𝜕𝑡𝜕𝑠𝑥𝜕𝑥=0,(2.41) and energy flux density 𝑡(𝑥),D𝑡(𝑥) satisfy the relevant continuity equation which is Lorentz-invariant for the spatially 1D propagation [3235, 47, 50, 51]. As a consequence, it is self-evident that also in this case of photons we can use the same energy representation of the time operator as for particles in nonrelativistic quantum mechanics, and hence verify the equivalence of calculations of 𝐽 and so on, in the both time and energy representations. Then, the same interpretation one can use for the propagation of electromagnetic wave packets (photons) in media and waveguides when collisions, reflections, and tunnelling can take place. Then, one can introduce the same form of the time operator as for particles in nonrelativistic quantum mechanics and hence verify the equivalence of calculations of mean values, variances, and so on, for time durations of photon motions, interactions and so on, with the measure (2.7)–(2.9), in the both time and energy representations [3235, 47, 50, 51]. It is also possible to introduce the second measure in time averaging, quite similarly to (2.36a) and (2.36b). In other words, in the cases of 1D photon propagations time is a quantum-physical observable also in quantum electrodynamics.

In the case of fluxes which change their signs with time we introduce quantities 𝐽em with the same physical meaning as for particles. Therefore, expressions for mean values and variances of distributions of propagation, tunnelling, transmission, penetration, and reflection durations can be obtained in the same way as in the case of nonrelativistic quantum mechanics for particles (with the substitution of 𝜓(𝑥,𝑡)=𝑛=0𝑔𝑛𝜑𝑛𝑖𝜀(𝑥)exp𝑛𝜀0𝑡,(2.42) by 𝜑𝑛(𝑥)).

2.5. Time as an Observable and Time-Energy Uncertainty Relation for Quantum-Mechanical Systems with Discrete Energy Spectra

For systems with discrete energy spectra it is natural (following [2325, 50, 51]) to introduce wave packets of the form (where 𝑛=0|𝑔𝑛|2=1 are orthogonal and normalized wave functions of system bound states which satisfy exp(𝑖𝜀0𝑡/) being the system Hamiltonian, 𝑛=0, here we factually omitted a nonsignificant phase factor 𝑡=0 as being general for all terms of the sum 𝑇=2𝜋/𝐷,𝐷) for describing the evolution of systems in the regions of the purely discrete spectrum. Without limiting the generality, we choose moment 𝐻 as an initial time instant.

Firstly, we will consider those systems, whose energy levels are spaced with distances for which the maximal common divisor is factually existing. Examples of such systems are harmonic oscillator, particle in a rigid box, and spherical spinning top. For these systems the wave packet (2.42) is a periodic function of time with the period (Poincaré cycle time) 𝑡𝜓(𝑡) being the maximal common divisor of distances between system energy level.

In the t-representation the relevant energy operator ̂𝑡 is a self-adjoint operator acting in the space of periodical functions whereas the function 𝑡 does not belong to the same space. In the space of periodical functions the time operator 𝑡, even in the eigen representation, has to be also a periodical function of time ̂𝑡=𝑡𝑇𝑛=𝑜Θ𝑡[2𝑛+1]𝑇2+𝑇𝑛=0Θ𝑡[2𝑛+1]𝑇2,(2.43). This situation is quite similar to the case of angular momentum (e.g., [68, 69]). Utilizing the example and result from [54], let us choose, instead of 𝑡, a periodical function which is the so-called saw-function of (Δ𝐴)2(Δ𝐵)22𝑁2(2.45) (see Figure 1).

This choice is convenient because the periodical function of time operator (2.43) is linear function (one-directional) within each Poincaré interval, that is, time conserves its flowing and its usual meaning of an order parameter for the system evolution.

The commutation relation of the self-adjoint energy and time operators acquires in this case (discrete energies and periodical functions) the form Let us recall (see, e.g., [70, 71]) that a generalized form of uncertainty relation holds for two self-adjoint operators 𝐵𝐴,=𝑖𝑁,(2.46) and 𝑁, canonically conjugate each to other by the commutator 𝛾 being a third self-adjoint operator. One can easily obtain where the parameter +𝑇/2 (with an arbitrary value between 𝑑𝑡 and 𝑇/2 ) is introduced for the univocality of calculating the integral on right part of (2.47) over +𝑇/2 in the limits from Δ𝐸0 to |𝑔𝑛|𝛿𝑛𝑛, just similarly to the procedure introduced in [68] (see also [70, 71]).

From (2.47) it follows that when |𝜓(𝑡)|2 (i.e., when Δ𝐸𝐷) the right part of (2.47) tends to zero since |𝜓(𝑇+𝛾)|2𝑇𝑇/2𝑇/2|𝜓(𝑡)|2𝑑𝑡 tends to a constant. In this case, the distribution of time instants of wave packet passing through point x in the limits of one Poincaré cycle becomes uniform. When Δ𝑡𝑇 and 𝑡(𝑥), the periodicity condition may be inessential for 𝑥, that is, (2.47) passes to uncertainty relation (2.11), which is just the same one as for systems with continuous spectra.

In principle, one can obtain the expression for the time operator (2.43) also in energy representation. If one will calculate the mean value ̂𝑡=𝑖2𝑛;>𝑛(1)𝑁𝑛𝑁𝑛Δ𝑛Δ𝑛𝜀𝑛(2.48) of instants of particle passing through point 𝑁𝑛=(𝜀𝑛𝜀0)/𝐷, then after a series of bulky transformations he will obtain the following expression: in the energy representation, where 𝐴𝑛Δ𝑛𝐴𝑛=𝐴𝑛Δ𝑛𝐴𝑛𝐴𝑛Δ𝑛𝐴𝑛,Δ𝑛𝐴𝑛=𝐴𝑛𝐴𝑛,𝑡(𝑥)=𝑛=0𝑔𝑛𝜑𝑛̂(𝑥)𝑡𝑔𝑛𝜑𝑛(𝑥)𝑛=0||𝑔𝑛𝜑𝑛||(𝑥)2.(2.49); the bilinear operation, denominated by |Ψ(𝑥,𝑡)|2., signifies (Of course, one has to average over the flux density, but for the simplicity in this case it is possible to make averaging over ̂𝑡=𝑖2ΔΔ𝜀,(2.50)) Operator (2.43) for two levels 𝐷=𝜀1𝜀00 acquires the more simple form and when 𝛾lim, the expression (2.50) passes to the differential form which coincides with (A.1) from Appendix A, that is, it is equivalemt to operator (2.1b) for the continuous energy spectra.

In general cases, for excited states of nuclei, atoms, and molecules, level distances in discrete spectra have not strictly defined the maximal common divisor and hence, they have not the strictly defined time of the Poincaré cycle. Also there is no strictly defined passage from the discrete part of the spectrum to the continuous part. Nevertheless, even for those systems one can introduce an approximate description (and with any desired degree of the accuracy within the chosen maximal limit of the level width, let us say, 𝑛=0) by quasicycles with quasiperiodical evolution and for sufficiently long intervals of time the motion inside such systems (however, less than (𝑔𝑛𝛿𝑛𝑛)) one can consider as a periodical motion also with any desired accuracy. For them one can choose (define) a time of the Poincare’ cycle with any desired accuracy, including in one cycle as many quasicycles as it is necessary for demanded accuracy. Then, with the same accuracy the quasi-self-adjoint time operator (2.43) or (2.48) can be introduced and all time characteristics can be defined.

In the degenerate case when at-the-state (2.42) the sum 𝛼 contains only one term 𝑥, the evolution is absent and the time of the Poincare' cycle is equal formally to infinity.

If a system has both (continuous and discrete) regions of the energy spectrum, one can easily use the forms (2.1a), (2.1b), and (2.26) for the continuous energy spectrum and the forms (2.43) and (2.48) for the discrete energy spectrum.

3. Applications for Tunneling Phenomena

3.1. Introduction

The developments of the study of tunneling processes in nuclear physics ((0,𝑎);-radioactivity, nuclear subbarrier fission, fusion, proton radioactivity and so on), then in various other fields of physics and especially the advent of high-speed electronic (and now microwave and optical) devices, based on tunnelling processes, generated an interest in the tunnelling time analysis and stimulated the publication of not only a lot of theoretical studies but already a lot of theoretical reviews on tunneling times (e.g., [7280], apart from [26, 27, 3235, 47]). And during many years, there had not been not only the consensus in the theoretical definition of the tunneling time for particles, but also there had been some declarations about the incompatibility of some approaches both quantitavely and in the physical interpretation. Among the reasons of such situation there had been the following ones:

(i)the problem of defining the tunneling time is closely connected with general fundamental problems of time as a quantum-physical observable and the general definition of quantum-collision durations, and the acquaintance with the principal solution of these problems had not got a wide prevalence yet till 2000–2004 (e.g., [47, 81]);(ii)the motion of particles inside a potential barrier is a quantum phenomenon without any direct classical limit (namely for particles);(iii)there are essential physical and mathematical differences in initial, boundary, and external conditions of various definition schemes.

Following [47, 80], we arrange the majority of approaches into several groups which are based on (1) the time-dependent wave packet description; (2) averaging over an introduced set of kinematic paths, distribution of which is supposed to describe the particle motion inside a barrier; (3) introducing a new degree of freedom, constituting a physical clock for measurements of tunnelling times. Separately, by one’s self, the dwell time stands. The last has ab initio the presumptive meaning of the time that the incident flux has to be turned on, to provide the accumulated particle storage in the barrier [22, 80].

The first group contains the so-called phase times, firstly mentioned in [82, 83] and applied to tunnelling in [84, 85], the times of the motion of wave packet spatial centroids, earlier considered for general quantum collisions in [1214, 86, 87] and applied to tunnelling in [88, 89], and finally the Olkhovsky-Recami (O-R) method [26, 27, 3235, 47, 90] of averaging over unidirectional fluxes, basing on the representation of time as a quantum-mechanical observable and on the generalization of the definitions, introduced in [21, 2325, 91] for atomic and nuclear collisions. The second group contains methods, utilizing the Feynman path integrals [9298], the Wigner distribution paths [99102], and the Bohm approach [103]. The approaches with the Larmor clock [104107] and the oscillatory barrier [108, 109] pertain to the third group.

Certainly, the basic self-consistent definition of tunnelling durations (mean values, variances of distributions, etc.) has to be elaborated quite similarly to the definitions of other physical quantities (distances, energies, momenta, etc.) on the base of utilizing all necessary properties of time as a quantum-physical observable (time operator, canonically conjugated to energy operator; the equivalency of the averaged quantities in time and energy representations with adequate measures, or weights, of averaging). For such definition, the description of solutions of the time-dependent Schroedinger equation by moving wave packets, which are typical in quantum collision theory (e.g., [110]), is quite natural for utilizing. Then one can expect that in the framework of the conventional quantum mechanics every known definition of tunnelling times can be shown, after appropriate analysis, to be (at least in the asymptotic region, used for typical boundary conditions in quantum collision theory) either a particular case of the general definition or an equivalent one or the definition which is valid not for tunnelling but for some accompanying process, different from tunnelling.

Here such a definition with the necessary formalism is presented (Section 3.2) and a brief comparison with various approaches is given (Sections 3.33.5), basing on the O-R formalism. In Section 3.6 the Hartman and Fletcher effect, with its generalization and its violations, is described. The tunneling through a double barrier is described in Section 3.7. The particle tunneling through three-dimensional barriers is presented in Section 3.8. The quaternion description of tunneling phenomena is mentioned in Section 3.9.

3.2. The O-R Formalism of Defining Tunnelling Durations, Based on Utilizing Properties of Time as a Quantum-Mechanical Observable

We confine ourselves to the simplest case of particles moving only along the (𝑥𝑖,𝑥𝑓)-direction, and consider a time-independent barrier in the interval 𝑉0—see Figure 1, in which a larger interval 𝑚, containing the barrier region, is also indicated.

As it is well known, in the case of a rectangular potential barrier of the height 𝐸<𝑉0, the stationary wave function for a particle with mass 𝜓(𝑘,𝑥)=exp(𝑖𝑘𝑥)+𝐴𝑅𝐴exp(𝑖𝑘𝑥),𝑥0(regionI),𝛼exp(𝜒𝑥)+𝛽exp(𝜒𝑥),0𝑥𝑎(regionII),𝑇exp(𝑖𝑘𝑥),𝑥𝑎(regionII),(3.1) and energy 𝑘=(2𝑚𝐸)1/2/ has the usual form (e.g., [26, 27, 32, 47, 7281, 90] and a lot of other papers): where 𝐴𝑅, 𝛼, 𝛽, 𝐴𝑇, 𝑖𝜒 and 𝑥 are the amplitudes of the reflected, evanescent, antievanescent and transmitted waves, respectively.

Inside a barrier here we have not usual propagating waves but a superposition of an evanescent (decreasing) and antievanescent (growing) waves with an imaginary wave number 𝑥. Just for this reason, for particle tunnelling (with subbarrier energies) through a barrier any direct classical limit does not really exist. However, one can see the direct classical limit for waves (more strictly, for time-dependent wave packet tunnelling, considered later). And we can remind real evanescent and antievanescent waves inside the layers with lesser refraction numbers between the layers with larger refraction numbers in the cases of the frustrated total internal reflection, well known in classical optics and in classical acoustics.

Following the definition of collision durations, put forth firstly in [21, 2325, 91] and afterwards generalized in [26, 27, 3235, 47] (see also [81]), we can eventually define the mean values of the time at which a particle passes through position 𝑡±(𝑥)=𝑡𝑗±(𝑥,𝑡)𝑑𝑡𝑗±,(𝑥,𝑡)𝑑𝑡𝐷𝑡±(𝑥)=𝑡2𝑗±(𝑥,𝑡)𝑑𝑡,(3.2), travelling in the positive or negative direction of the 𝑗±(𝑥,𝑡)-axis, and the variances of the distributions of these times, respectively, as Re[(𝑖/𝑚)Ψ(𝑥,𝑡)𝜕Ψ(𝑥,𝑡)/𝜕𝑥] being the positive or negative values, respectively, of the probability flux density Ψ(𝑥,𝑡)𝑡=𝐸 for an evolving time-dependent normalized wave packet 𝑡. We recall here the equivalence of canonically conjugated time and energy representations, with appropriate measures of averaging, in the following sense: 𝜏𝑇𝑥𝑖,𝑥𝑓=𝑡+𝑥𝑓𝑡+𝑥𝑖,(3.3)𝐷𝜏𝑇𝑥𝑖,𝑥𝑓=𝐷𝑡+𝑥𝑓+𝐷𝑡+𝑥𝑖,(3.4) (index t is omitted in all expressions for <𝑥𝑖0 for the sake of the simplicity). This equivalence is a consequence of the unique time-operator existence.

For transmissions from region I to region III we have with 𝜏pen(𝑥𝑖,𝑥𝑓) and 𝐷𝜏pen(𝑥𝑖,𝑥𝑓). For a pure tunnelling process one has Similar expression we have for the penetration (into the barrier region II) temporal quantities 𝑥𝑓<𝑎 and 𝜏𝑅𝑥𝑖,𝑥𝑓=𝑡𝑥𝑓𝑡+𝑥𝑖,𝐷𝜏𝑅𝑥𝑖,𝑥𝑓)=𝐷𝑡𝑥𝑓+𝐷𝑡+𝑥𝑖.(3.6) with |𝑥𝑖|𝑎. For reflections in any point 𝜏as𝑇𝑥𝑖,𝑥𝑓𝑥=𝑡𝑓𝑇𝑥𝑡𝑖in,(3.7)𝜏as𝑇𝑥𝑖,𝑥𝑓=𝜏𝑇𝑥𝑖,𝑥𝑓+𝑡+𝑥𝑖𝑥𝑡𝑖in,(3.8) one has

We stress that these definitions hold within the framework of conventional quantum mechanics, without introducing any new physical postulate.

In the asymptotic cases, when in, where Ψin(𝑥,𝑡)=0𝐺𝑘𝑘exp𝑖𝑘𝑥𝑖𝐸𝑡𝑑𝑘,(3.9) and 𝐸=2𝑘2/2𝑚,0|𝐺(𝑘𝑘)|2𝑑𝐸=1,𝐺(0)=𝐺()=0,𝑘>0 denote averagings over the fluxes corresponding to 0𝑣𝑛||𝐺𝐴𝑇||2𝑑𝐸0𝑣𝑛|𝐺|2𝑑𝐸,𝑛=0,1,𝑣=𝑘𝑚,(3.10) and 𝜏as𝑇𝑥𝑖,𝑥𝑓𝜏Ph𝑇𝑥𝑖,𝑥𝑓𝐸,(3.11), respectively.

For initial wave packets (where 𝐷𝑡+(𝑥𝑖)) with sufficiently small energy (momentum) spreads when we get where are the phase transmission time obtained by the stationary-phase approximation. At the same approximation and with a small contribution of 𝐷𝜏𝑇(𝑥𝑖,𝑥𝑓) into the variance |𝐺|2𝛿(𝐸𝐸),𝐸 (that can be realized for sufficiently large energy spreads, i.e., short wave packets) we get For the opposite case of very small energy spreads (quasimonochromatic particles) it follows that, instead of the expression (3.13), the general expression (3.4) becomes just the item of 𝜏as𝑇(𝑥𝑖,𝑥𝑓)𝜏Ph𝑇(𝑥𝑖,𝑥𝑗)𝐸 plus 𝑉0 which is born by the barrier influence and formally is described by (3.13).

At the quasimonochromatric limit 𝜒𝑎1 being 𝜒=[2𝑚(𝑉0𝐸)]1/2/, we get for 𝑥𝑖=0,𝑥𝑓=𝑎 strictly the ordinary phase time, without averaging. For a rectangular barrier with height 𝑎𝑚/𝑘𝐷𝑡+(𝑥𝑖) and 𝜏Phtun=2𝑣𝜒(3.14) (where 𝐷𝜏Phtun1/2=𝑎𝑘𝑣𝜒(3.15)), the expressions (3.11) and (3.13), for 𝐺(𝑘𝑘) and 𝑡(0)in=0, pass into the known expressions (coincident with the phase time [26, 27, 47, 83]), and (coincident with one of the Larmor times [104107] and the Buettiker-Landauer time [108] and also with the imaginary part of the complex time in the Feynman path-integration approach: see later Section 3.5), respectively.

For real weight amplitude 𝑥𝑖, when (,0), from (3.8) we obtain

By the way, if the measurement conditions are such that only the positive-momentum components of wave packets are registrated, that is, (𝑎,) being the projector onto positive-momentum states, then for any 𝜏𝑇𝑥𝑖,𝑥𝑓+=𝜏Ph𝑇𝑥𝑖,𝑥𝑓𝐸,(3.17)𝜏tun(0,𝑎)+=𝜏Phtun𝐸,(3.18) from 𝑡(0)+=𝑡(0)in and 𝜏Phtun,em=2𝑐𝜒emfor𝜒em𝑎1.(3.19) from 𝜒em𝑎>2 because 𝑗(𝑥𝑓,𝑡)=𝑗𝑇(𝑥𝑓,𝑡).

In the particular case of quasimonochromatic electromagnetic wave packets, using the stationary-phase method under the same boundary and measurememt conditions as considered for particles, we obtain the identical expression for the phase tunnelling time

From (3.19), we can see that when 𝑗𝑅 the effective tunnelling velocity is more than c, that is, superluminal. This result agrees with the results of the microwave-tunnelling measurements presented in [111113] (see also [114] where moreover, the effective tunnelling velocity was identified with the group velocity of the final wave packet corresponding to a single photon).

3.3. Analysis of the Mean Dwell Time in the Light of the Olkhovsky-Recami Formalism

In Section 2, it was analyzed the meaning of two forms of the expression for the mean dwell time (2.36a) and (2.36b) from Section 2. Taking into account that the total flux Ψin(𝑥𝑖,𝑡),Ψ𝑅(𝑥𝑖,𝑡) and Ψ𝑇(𝑥𝑓,𝑡), with 𝜓in,𝜓𝑅=𝐴𝑅exp(𝑖𝑘𝑥), 𝜓𝑇 and 𝑗int(𝑥,𝑡)=Re𝑖𝑚Ψin(𝑥,𝑡)𝜕Ψ𝑅(𝑥,𝑡)+Ψ𝜕𝑥𝑅(𝑥,𝑡)𝜕Ψin(𝑥,𝑡),𝜕𝑥𝑗int𝑥𝑖,𝑡𝑑𝑡=0.(3.21) corresponding to the wave packets 𝜏Dw𝑥𝑖,𝑥𝑓=𝑇𝐸𝜏𝑇𝑥𝑖,𝑥𝑓𝑥+𝑅𝑖𝐸𝜏𝑅𝑥𝑖,𝑥𝑓,(3.22) and 𝑇𝐸=|𝐴𝑇|2𝑣𝐸/𝑣𝐸,𝑅(𝑥𝑖)𝐸=𝑅𝐸+𝑟(𝑥𝑖),𝑅𝐸=|𝐴𝑅|2𝑣𝐸/𝑣𝐸 constructed from the stationary wave functions 𝑇𝐸+𝑅𝐸=1 and 𝑟(𝑥)=𝐽+(𝑥,𝑡)𝐽in(𝑥,𝑡)𝑑𝑡𝐽in(𝑥,𝑡)𝑑𝑡.(3.23), respectively, and also One obtains from (2.36a) and (2.36b) of Section 2, with , Ψin(𝑥𝑖,𝑡), and One can see that 𝑟(𝑥𝑖)=0 is negative and tends to 0 when 𝜏Dw𝑥𝑖,𝑥𝑓=𝑇𝐸𝜏𝑇𝑥𝑖,𝑥𝑓+𝑅𝐸𝜏𝑅𝑥𝑖,𝑥𝑖(3.24) tends to 𝜒𝑎1.

When 𝑥𝑖=0 and 𝑥𝑓=𝑎 are sufficiently well separated in time, so that 𝜏Dw𝑥𝑖,𝑥𝑓=𝑘𝜒𝑉0𝐸,(3.25), it follows from (3.22) that the simple weighted average rule is valid. For a rectangular barrier with 𝜏Dw𝑥𝑖,𝑥𝑓2=𝜒𝑣𝐸(3.26) and quasimonochromatic particles, the expressions (3.22) and (3.24) with 𝑟(𝑥𝑖) and 𝐴𝑅=0 pass to the known expressions (taking account of the interference term 𝑥1) (when the interference term 𝑥 is equal to 0).

When 𝑦, that is, a barrier is transparent, the mean dwell time (3.22) is automatically equal to

It is not clear how to define directly the variance of the dwell-time distribution. The approach, proposed in [115], is rather sophisticated, withan artificial abrupt switching on the initial wave packet. It is possible to define the variance of the dwell-time distribution indirectly, in particular, by means of relation (3.22), with the help of the variances of the transmission-time and reflection-time distributions, or by means of relation (2.36a) from Section 2, with the help of the variances of the positions 𝜏La𝑦,𝑇 and 𝜏La𝑧,𝑇.

3.4. Analysis of the Larmor and Buettiker-Landauer Clocks

One can often realize that the introducing of additional degrees of freedom as “clocks” does in a certain degree distort the true values of the tunnelling time. The Larmor clock uses the phenomenon of changing the spin orientation (the Larmor precession or spin-flip) in a weak homogeneous magnetic field covered the barrier region. If initially the particle spin is polarized in the 𝑧 direction, after tunnelling the spin develops small 𝑦 and 𝑦 components (see Figure 3).

The Larmor times 𝑧 and 𝜏La𝑧,𝑇 are defined by the ratio of the spin-rotation angles around axes |𝐴𝑇| and 𝜒𝑎1 (in turn defined by the developed 𝜏La𝑦,tun=𝜏Dw𝑥𝑖,𝑥𝑓=𝑘𝜒𝑉0𝐸,(3.28)𝜏La𝑧,tun=𝑚𝑎𝜒𝐸(3.29)- and |𝐴𝑇|-spin components, resp.) to the precession (rotation) frequency.

As to 𝜏La𝑧,tun, in the reality it is not a precession but a jump to position “spin-up” or “spin-down” (spin-flip) accompanied by the Zeeman energy-level splitting [79, 104, 105]. Due to the Zeeman splitting, the component of the spin, that is parallel to the magnetic field, corresponds to a higher tunnelling energy and hence tunnels preferentially, and namely therefore one can realize that this time is connected with the energy dependence of exp[𝑖𝑆(𝑥(𝑡))/] and coincides with the expression (3.15) (of course, at the same approximations when (3.15) is valid).

For an opaque rectangular barrier with 𝑆 the expressions (for mean Larmor times) had been obtained in [79, 101103].

In [26, 27, 32, 116], it was noted that, if the magnetic field region is infinite, the expression (3.28) passes into the expression (3.14) for the phase tunnelling time, after averaging over the small energy spread of the wave packet.

The work of the Buttiker-Landauer clock is connected withthe modulation cycle (absorption or emission of modulation quanta) caused by the oscillating part of a barrier, during tunneling. Also in this case one can realize that the coincidence of the Buttiker-Landauer time with (3.15) is connected withthe energy dependence of 𝜏La𝑧,tun for the same reasons as for exp(𝑖𝑆/).

3.5. Analysis of the Mean Tunnelling Times, Defined by Averaging over Kinematic Paths

The Feynman path-integral approach to quantum mechanics was applied in [9298] to study and calculate the mean tunnelling time averaged over all possible paths, that have the same beginning and end, with the complex weight factor 𝑓(𝑥,𝑝), where 𝐽(𝑥)=𝐽+(𝑥)+𝐽(𝑥),(3.30) is the action associated with the path 𝐽+(𝑥)=0(𝑝/𝑚)𝑓(𝑥,𝑝)𝑑𝑝. Namely, such weighting of tunnelling times implies their distribution with a real and an imaginary component [79]. In [92], the real and imaginary parts of the obtained complex tunnelling time were found to be equal to 𝐽=𝐽𝐽+ and 𝐽±, respectively.

An interesting development of this approach, the instanton version, is presented in [97, 98]. The instanton-bounce path is a stationary point of the Euclidean action. The latter is obtained by the analytic continuation to imaginary time in the Feynman-path integrals containing the factor 𝐽±. This path obeys a classical equation of motion in the potential barrier with the sign reversed. In [97, 98], the instanton bounces were considered as real physical processes. The bounce duration was calculated in real time and was found to be in good agreement with the one evaluated by the phase-time method. The temporal density of bounces was estimated in imaginary time and the obtained result coincided with (3.13) for the square root of the distribution variance at the limit of the phase-time approximation. Here one can see a manifestation of the virtual equivalence of the Schroedinger representation and the Feynman path-integral approach to quantum mechanics.

Another definition of the tunnelling time is connected with the Wigner distribution paths [99102]. The basic idea of this approach, finally formulated by Muga, Brouard, and Sala, is that the distribution of the tunnelling times in the dynamical evolution of wave packets through barriers can be well approximated by a classical ensemble of particles with a certain distribution function, namely the Wigner function 𝜏Dw𝑥𝑖,𝑥𝑓=𝑇𝐸𝜏𝑇𝑥𝑖,𝑥𝑓+𝑅𝑀𝑥𝑖𝐸𝜏𝑅𝑥𝑖,𝑥𝑓,(3.31), so that the flux at position x can be separated into positive and negative components: with 𝑅𝑀(𝑥) and 𝑅𝐸. Then formally the same expressions (3.3), (3.5), and (3.6) for the transmission, tunneling, and penetration durations and so on, as in the O-R formalism, were obtained with the substitution of 𝜏Dw𝑥𝑖,𝑥𝑓=0𝑑𝑡𝑥2𝑥1||||Ψ(𝑥,𝑡)2𝑑𝑥=𝑇𝜏𝑇𝑥𝑖,𝑥𝑓+𝑅𝜏𝑅𝑥𝑖,𝑥𝑓,(3.32) instead of our 𝜏𝑇𝑥𝑖,𝑥𝑓=0𝑑𝑡𝑥2𝑥1||||Ψ(𝑥,𝑡)2Θ𝑥𝑥𝑐𝜏𝑑𝑥/𝑇,𝑅𝑥𝑖,𝑥𝑓=0𝑑𝑡𝑥2𝑥1||||Ψ(𝑥,𝑡)2Θ𝑥𝑐𝑥𝑑𝑥/𝑅,(3.33). The dwell time decomposition in this approach takes the form with 𝑅. Asymptotically, 𝑥𝑐=𝑥𝑐(𝑡) tends to 𝑇=||||Ψ(𝑥,𝑡)2Θ𝑥𝑥𝑐𝑑𝑥.(3.34) and (3.31) takes formally the known form (3.24).

One more alternative is the stochastic method for wave packets [108]. It also leads to real times but its numerical implementation is not trivial [109].

In [110], the Bohm approach to quantum mechanics was used to choose a set of classical paths which do not cross. The Bohm formulation can provide, on the one hand, a strict equivalent to the Schroedinger equation, and on the other hand, a base for the nonstandard interpretation of quantum mechanics [79]. The obtained in [110] expression for the mean dwell time is not only positive definite but gives the unambiguous distinction between particles that are transmitted or reflected: with where 𝑥𝑐 and 𝐽(𝑥,𝑡)=𝐽(𝑥,𝑡)𝑇+[𝐽(𝑥,𝑡)]𝑅,(3.35) are here the mean transmission and reflection probability, respectively, the bifurcation line 𝐽(𝑥,𝑡)𝑇=𝐽(𝑥,𝑡)Θ𝑥𝑥𝑐,(𝑡)𝐽(𝑥,𝑡)𝑅𝑥=𝐽(𝑥,𝑡)Θ𝑐.(𝑡)𝑥(3.36), separating transmitted and reflected trajectories, is defined by relation Factually, in addition to the difference in the temporal integration in this and our formalisms (𝜏tun and 𝐴𝑇, resp.), sometimes essential, this approach gives one more alternative—in separating the flux by the line 𝐸=2𝑘2/(2𝑚): with

3.6. On the Hartman and Fletcher Effect, Its Generalization and Its Violations

Firstly the Hartman and Fletcher effect (HFE) was revealed and studied in [84, 85] within the stationary-phase method for a 1D motion of quasimonochromatic nonrelativistic particles tunnelling through potential barriers. It consists in the absence of the dependence of the phase tunnelling time (which is the mean tunnelling time 𝐷±=1±exp(2𝜒𝑎) within the stationary-phase method when it is possible to neglect the interference between incident and reflected waves out of a barrier [26, 27, 32], 𝜒=[2𝑚(𝑉0𝐸)]1/2/,𝑉0 and 𝜏Phtun2/(𝜐𝜒) being the transmission amplitude and the particle kinetic energy, resp.) on the barrier width a for sufficiently large 𝜒𝑎1. In particular, for a rectangular potential barrier 𝜐=𝑘/𝑚𝜏Dwtun, 𝜏𝐿𝑦,tun, Re𝜏𝐹tun being the barrier height, and 𝑘/(𝜒𝑉0) when 𝜏tun=𝑡+(𝑎)𝑡+(0)=𝜏Phtun𝐸𝑡+(0),(3.38) (𝐸 is the particle velocity before entering into a barrier).

Now we will test the validity of HFE for all other known theoretical expressions for mean tunnelling times. If we firstly take from the mean dwell time 𝑡±(𝑥)=𝑡𝑗±(𝑥,𝑡)𝑑𝑡𝑗±(𝑥,𝑡)𝑑𝑡with𝑗±(𝑥,𝑡)=Θ(±𝑗)𝑗(𝑥,𝑡),(3.39), the mean Larmor time 𝑗(𝑥,𝑡) and the real part of the complex tunnelling time obtained by averaging over the Feynman paths (0,𝑎), which are equal 𝜏𝐿𝑧,tun for quasimonochromatic particles and opaque rectangular barriers, we immediately easily see that also in these cases there is no dependence on the barrier width and consequently HFE is valid.

The validity of HFE for the mean tunnelling time within the O-R approach, is directly seen from the expression (where 𝜏𝐵𝐿tun denotes averaging over the initial wave-packet energy spread and 𝑎𝑚/(𝜒), being the probability flux density for a wave packet moving along axis x through a barrier located in the interval 𝑎 and it was confirmed in [26, 27, 3235, 47] by numerous calculations for gaussian electron wave packets with narrow momentum spreads (see also Section 3.6).

As to the other Larmor time [𝐷dyn𝜏tun]1/2, from Sections 4.2 and 3.4 it follows that the Bttiker-Landauer time 𝐷dyn𝜏tun=𝐷𝜏tun𝐷𝑡+(0) [111113], and the imaginary part of the complex tunnelling time 𝐷𝜏tun=𝜏2tun𝜏tun2 [92], obtained within the Feynman approach, which are equal to (3.39), they become equal to 𝜏2tun=[𝑡+(𝑎)𝑡+(0)]2 that is, proportional to the barrier width +𝐷𝑡+(0), in the opaque rectangular barrier limit (as one can see from (3.15) and (3.29)). These times are not mean times but mean-square fluctuations in the tunnelling-time distribution because they are equal to 𝜏tun(0,𝑎) where 𝑉0=10eV is the dynamical tunneling-time variance caused by the barrier influence only and defined by the equation 𝐸=2.5,5,7.5eV with Δ𝑘=0.02A1, 𝐸=5eV𝑘=0.04A1 (it was shown in [26, 27, 3335]). Hence, they are not connected with the peak (or group) velocities of tunnelling particles but with the relevant tunneling velocity distribution over the barrier region.

In Figure 4 the dependences of the values of 0.06A1 from a are presented for electronic wave packets and rectangular barriers with the same parameters as in [47] (𝜏Phtun; mean electron energies Δ𝑘 with 𝜏Phtun (curves 1a, 2a, and 3a, resp.); energy 𝜏tun(0,𝑎) with 𝑎 and 𝑡+(0) (curves 4a and 5a, resp.)). The curves 𝑡+(0) corresponding to different energies and 𝑎 merge practically into one curve 6. Since the dependence of 𝑡+(0) from a is very weak, the dependence of 𝜏tun(0,𝑎) from 𝜏Phtun, is defined mainly by the dependence of 𝑖𝑉1(𝑉1>0) from a (curves 1b–5b, correspondent to 1a–5a, resp.).

All these calculations manifest the negative values of 𝑉0. Such “acausal” advance can be interpreted as a result of the superposition and interference of incoming and reflected waves. The reflected-wave packet extinguishes the back edge of the incoming-wave packet, and the larger is the barrier width, the larger is the part of the back edge of the incoming-wave packet which is extinguished by the superimposing reflected-wave packet, -up to the saturation when the contribution of the reflected wave packet becomes almost constant and independent from 𝑉1𝑉0. Since all 𝑉1𝑚1/2𝑣𝜒𝑎/[2(𝑉0𝐸)]3/22 are negative, the values of 𝑥=𝑥0 are always positive and, moreover, larger than |𝑥0|𝑎 in accordance with (3.16).

All presented here results are obtained for transparent media (without absorption and dissipation). As it was theoretically demonstrated in [114] in nonrelativistic quantum mechanics, HFE vanishes for barriers with absorption. As it follows from [114], if one describes the absorption by adding the imaginary term 𝑡=0 to 𝑥, then for small absorptions, when 𝑉0 and [0,𝑎], HFE does not vanish and remains practically valid. This was confirmed experimentally for electromagnetic (microwave) tunnelling in [115].

Now we will consider wave packets with large momentum spreads and with the initial condition of the wave-packet center motion from the distant point 𝑚 (with 𝐸<𝑉0) at the instant 𝜓(𝑥,𝑡) in order to analyze the influence of rather strong wave-packet time spreading before the entering into the barrier [117].

First, we will formulate explicitly initial conditions which take into account the irreversibility of the wave packet spreading. Further, we will propose a particularly convenient form of the O-R tunnelling time which allows the control of the accuracy in numerical calculations. Finally, we will present and explain the strong decrease shown by the tunneling time as the momentum spread increases, in such condition that we can say that the the Hartman and Fletcher effect is violated. We analyze the case of the 1D tunnelling of particles along the Ψ(𝑥,𝑡)=0𝑔0(𝑘)𝜓(𝑘,𝑥)exp𝑖𝐸𝑡𝑑𝑘,(3.41) axis through a rectangular potential barrier with height 𝑔(𝑘)=𝑔0(𝑘)exp𝑖𝑘𝑘0𝑥0=𝐶exp𝑘𝑘02Δ𝑘2𝑖𝑘𝑘0𝑥0,(3.42), localized in the interval 𝐶=[(2𝜋)3/2Δ𝑘]1/2. The chosen here stationary wave function for a particle with mass 𝑘 and energy (2𝑚𝑉0)1/2/ has the usual form (3.1). The time dependent wave packet Ψin(𝑥,𝑡)𝐶0𝑠1/2in𝑥exp𝑥0+𝜈𝑡22𝑠in+𝑖𝑘0𝑥𝑖𝐸0𝑡,(3.43) is formed with wave functions (3.1): with the weight amplitude 𝑥 being the normalization coefficient. Here, since in calculations the only subbarrier part of the wave packet was considered, integrating over 𝑣 in (3.42) had to be made at the limits from 0 to 𝑥=0 [117].

The initial wave function turns out to be whose center transits from point 𝑗(𝑥,𝑡)=Re[(𝑖/2𝑚)Ψ(𝑥,𝑡)𝜕Ψ(𝑥,𝑡)/𝜕𝑥] at the instant 𝑥, moves along axis 𝑥=𝑎 from the left to the right with the velocity 𝑡(𝑎)=0𝑡𝑗(𝑎,𝑡)𝑑𝑡0𝑗(𝑎,𝑡)𝑑𝑡,(3.44) and in the absence of barrier crosses the point 𝑥=0 at the instant 𝑡+(0)=0𝑡𝑗+(0,𝑡)𝑑𝑡0𝑗+(0,𝑡)𝑑𝑡,(3.45).

The flux density 𝑗+(0,𝑡), as usually in the O-R method, has been considered as a function of the arrival times at point 𝑗(0,𝑡), so that the mean time instant for particles passing through point 𝑥=0 (the mean time of the particle exit from the barrier), has been chosen as while the mean time instant for particle passing through point 𝜏=𝑡(𝑎)𝑡+(0) (the mean time of the particle entrance into the barrier) is defined by relation where Ψ(𝑎,𝑡)=0𝑔(𝑘)𝜓𝑇(𝑘𝑎)exp𝑖𝐸𝑡𝑑𝑘,(3.46) represents the positive values of the flux density 𝜓𝑇(𝑘,𝑎), corresponding, therefore, to particles moving through point 𝑥=𝑎 from left to right (entering the barrier), and the integrals have been limited to positive times only, due to initial condition (3.43).

The tunnelling time 𝜓𝑇(𝑘,𝑎) is then defined as 𝜓𝑇(𝑘,𝑎)=2𝑖𝑘𝜒𝐷<,𝐸<𝑉0,2𝑖𝑖𝑘𝑞𝐷>,𝐸>𝑉0,(3.47).

The flux density 𝑞=[2𝑚(𝐸𝑉0)]1/2/ contains the wave function where 𝑁𝑎=0𝑗(𝑎,𝑡)𝑑𝑡(3.49) is the value of the stationary wave function (3.1) at point 𝑁𝑎=𝑚Re0𝑑𝑘𝑔(𝑘)𝜓𝑇(𝑘,𝑎)0𝑑𝑘𝑘𝑔𝑘𝜓𝑇𝑘,𝑎0exp𝑖𝑘2𝑚2𝑘2𝑡𝑑𝑡.(3.50), corresponding to the transmitted wave. Using the usual expression for the amplitude of the transmitted wave, we can 𝑥0 represent in the form where 𝑡0, The normalization integral in (3.45), can be evaluated using (3.42), (3.46), and (3.47): If ||𝜓𝑇||(𝑘,𝑎)2=4𝑘2𝜒24𝑘2𝜒2+𝑉20sinh2(𝜒𝑎),𝐸<𝑉0,4𝑘2𝑞24𝑘2𝑞2+𝑉20sin2(𝑞𝑎),𝐸>𝑉0.(3.52) is chosen sufficiently far from the left of the barrier, the contribution of 𝐼𝑎=0𝑡𝑗(𝑎,𝑡)𝑑𝑡.(3.53) in the integral (3.49) from 𝐼𝑎=𝑚Re0𝑑𝑘𝑔(𝑘)𝜓𝑇(𝑘,𝑎)0𝑑𝑘𝑔𝑘𝜓𝑇𝑘,𝑎0𝑘𝑡exp𝑖𝑘2𝑚2𝑘2𝑡𝑑𝑡,(3.54) is negligible small. In this case, the lower integration limit over t in (3.50) can be taken as 𝐼𝑎𝑚=2𝜋0𝑔201(𝑘)𝑘𝜓1(𝑘,𝑎)𝜓2(𝑘,𝑎)𝜓2(𝑘,𝑎)𝜓2(𝑘,𝑎)𝑥0||||𝜓(𝑘,𝑎)2𝑑𝑘,(3.55). Then the integration over time gives 𝜓1(𝑘,𝑎)Re𝜓𝑇(𝑘,𝑎) and the integral (3.50) can be cast in the form where Now we evaluate the integral

Using (3.50) and (3.51), we can write which can be set in the form where 𝑚𝑡(𝑎)=0𝑔201𝑘𝜓1𝜓2𝜓2𝜓1𝑥0||𝜓𝑇||2𝑑𝑘/0𝑔20||𝜓𝑇||2𝑑𝑘.(3.57), 𝜏0. After differentiating (3.47) over 𝑗(0,𝑡), one can see the following: From (3.51) and (3.56) it follows that, with an appropriate choice of 𝐴𝑅𝑉(𝑘)=0sinh(𝜒𝑎)𝐷<,𝐸<𝑉0,𝑉0sin(𝑞𝑎)𝐷>,𝐸>𝑉0.(3.59), the mean instant 𝑡+(0) can be defined by the simple relation As regards the calculation of 𝐼+=𝑡𝑗+(0,𝑡)𝑑𝑡, the wave function entering the flux density 𝑁+=𝑗+(0,𝑡)𝑑𝑡 is where Since for calculating 10 we consider only the positive values of 𝑥0=6/Δ𝑘, the integrals Δ𝑘=0.01Å1 and 𝑎 in (3.45) can be obtained only numerically.

Figure 5 shows the results of the calculations of the tunnelling time Δ𝑘=0.05Å1 as a function of the width a of the barrier, for electrons with energy Δ𝑘=0.1Å1eV through the rectangular barrier of potential with height Δ𝑘=0.1Å1eV and Δ𝑡Δ0𝑡1+(Δ𝑣)2𝑚22𝑡21/2,(3.60). We can see the manifestation of the HFE for Δ0𝑡/Δ𝐸 (curve (a)), with the asymptotic behavior of the tunneling time approaching the constant value with increasing Δ𝑣=Δ𝑘/𝑚. Curves (b) and (c) show, on the contrary, the strong decrease presented by the tunneling times when the wave packets are characterized by larger momentum spread Δ𝑡 and 𝑡. For Δ𝑣 the tunneling time is even negative. The violation of the HFE is strongly evident.

This effect can be explained with the following reasons [117]: the time spread of the Gaussian wave packet (3.43) is described by the relation where Δ𝑡, 𝑡. The value of Δ𝑘 strongly increases in time 𝑡 for large velocity spread |𝑥0|,. So, the center of the initial wave packet during wave-packet approaching the barrier and increasing time 1/Δ𝑘 will appear even farther due to the essentially stronger increase of 𝑥=𝑎 with time 𝜏 (proportional to 𝜏ph for large 𝜏ph) than the decrease of 𝑗+(0,𝑡) proportional to 𝑗in(0,𝑡),𝑗in. For the center of the transmitted wave packet in point Re[(𝑖/2𝑚)Ψin(𝜕Ψin/𝜕𝑥)] such delay has to be considerably smaller because the value Δ𝑣 is smaller than the limited value Δ𝑘.𝑡0,𝑡0,𝑡0 (𝑥=0 being the phase tunnelling time) due to the advance caused by the difference between 𝑡𝑓,𝑡𝑓 and 𝑡𝑓 being 𝑥=𝑎 (see, in particular, [26, 27]). So, one can expect that the time interval between the transit of the center of the wave packet through the entrance and the exit of the barrier has to be smaller than in the case of the validity of the HFE, that is, for sufficiently large velocity spreads 𝜏tun(0,𝑎) becoming even negative.

Figure 2 shows qualitatively the behavior of the time shift of the outgoing wave packet with respect to the ingoing wave packet with different momentum spread 𝜏Phtun represent the time instants when the wave packet peak passes through the initial point 𝑡+(0) of the barrier. 𝜏, and 𝑎 represent the transit time through the final point Δ𝑘 of the barrier.

In any case a point of the wave packet preparation has to be located at some finite distance from the barrier and therefore the wave packet center arrives to the barrier during a finite time interval. A wave packet is spreading during its motion and its width does always remain a finite one.

In conclusion, the O-R definition with strictly formulated initial conditions can be especially useful for investigations of the particle tunnelling accompanied by the quantum dissipation. A strong decrease of tunneling times, much more strong than in the case of the validity of the HFE, for wave packets with the large momentum spread can be easily explained by the sufficiently rapid spreading of such wave packets during the initial motion before the entrance into the barrier and also inside the barrier.

Some authors (see, e.g., [118121]) have extended the study of particle tunneling phenomena to a completely relativistic case, using the Dirac equation. All these papers indicate the apparent superluminal tunneling through opaque barriers, showing a behavior similar to that of the HFE.

However, the complete review of the Dirac relativistic tunneling has to include the so-called Klein paradox when the reflection coefficient is greater than 1 and when the transmission coefficient has nonvanishing values (see, e.g., the papers [122126]). Its origin is now usually described as the electropositron pair production for large potential step but it is not possible to develop a simple relationship between the time-dependent pair production process with a finite lifetime and the time-independent transmission coefficient in general. The problem of the Klein paradox and the Klein tunneling has to be studied in a self-consistent way and reviewed separately in another paper.

3.7. Tunneling through a Double Barrier

In this section, we confine ourselves by the approximation of not taking into account the multiple internal reflections between two separated barriers. Some words on such considering will be said at the end of Appendix F (before the subdivision, the case of photon tunneling).

Phase Time of Nonresonant Tunneling through Two Opaque Barriers
Now let us consider the stationary solution for 1D tunneling of a particle with mass 𝑉(𝑥)=0 and kinetic energy 𝑉(𝑥)=𝑉0, through two equal rectangular barriers with height I(𝑥0),II(0𝑥𝑎),III(𝑎𝑥𝑙+𝑎),IV(𝑙+𝑎𝑥𝑙+2𝑎), and width V(𝑥𝑙+2𝑎), the quantity 𝜓I=𝑒𝑖𝑘𝑥+𝐴1R𝑒𝑖𝑘𝑥,𝜓II=𝛼1𝑒𝜒𝑥+𝛽1𝑒𝜒𝑥,𝜓III=𝐴1T𝑒𝑖𝑘𝑥+𝐴2R𝑒𝑖𝑘𝑥,𝜓IV=𝐴1T𝛼2𝑒𝜒(𝑥𝑙𝑎)+𝛽2𝑒𝜒(𝑥𝑙𝑎),𝜓V=𝐴T𝑒𝑖𝑘𝑥,𝐴T𝐴1T𝐴2T,(3.62) being the distance between them see Figure 7. The stationary Schroedinger equation is where 𝐴1R,𝐴2R,𝐴1T,𝐴2T,𝛼1,𝛼2,𝛽1 outside the barriers and 𝛽2 inside the potential barriers. In various regions 𝜓I,II,III,IV,V and 𝑑𝜓I,II,III,IV,V/𝑑𝑥, the solutions of (3.61) are the following: where 𝜒𝑎,, and quantities 𝐴1R𝑖𝑘+𝜒𝑖𝑘𝜒,𝐴1T𝑒𝜒𝑎𝑒𝑖𝑘(𝑙+𝑎)𝛼𝐴,12𝑖𝑘𝑖𝑘𝜒,𝛽1𝑒2𝜒𝑎(𝑘𝑖𝜒)sin𝑘𝑙𝜒𝐴𝐴,2R𝑒2𝑖𝑘(𝑙+𝑎)𝑖𝑘+𝜒𝑖𝑘𝜒,𝐴2T𝑒2𝑖𝑘(𝑙+𝑎)4𝑖𝑘𝜒(𝑖𝑘𝜒)2,𝛼2𝑒𝑖𝑘(𝑙+𝑎)2𝑖𝑘𝑖𝑘𝜒,𝛽2𝑒𝑖𝑘(𝑙+𝑎)2𝜒𝑎2𝑖𝑘(𝑖𝑘+𝜒)(𝑖𝑘𝜒)2,(3.63), and 𝐴=2𝑘𝜒𝜒2𝑘𝜒cos𝑘𝑙+2𝑘2sin𝑘𝑙.(3.64) are the reflection amplitudes, the transmission amplitudes, and the coefficients of the “evanescent” (decreasing) and “antievanescent” (increasing) waves for barriers 1 and 2, respectively. These 8 quantities can be easily obtained from 8 matching (continuity) conditions for the functions 𝜏Phtun𝐴=𝜕arg𝑇𝑒𝑖𝑘(𝑙+2𝑎)2𝜕𝐸𝑣𝜒,(3.65) and their derivatives 𝐴𝑇(𝑘)=exp(2𝑖𝑘𝑎)𝐷(𝑘),(3.66) at points 𝐷(𝑘)=cosh21(𝜒𝑎)+4sinh2𝜎(𝜒𝑎)2cos(2𝑘𝑙)𝛿21+𝑖sinh(𝜒𝑎)𝛿cosh(𝜒𝑎)+4𝜎2,sinh(𝜒𝑎)sin(2𝑘𝑙)(3.67). The obtained expressions for them for opaque barriers, when 𝛿=(𝜒2𝑘2)/𝑘𝜒 are where
From (3.63) one can derive that the phase tunneling time is which is precisely the same as for one barrier and does not depend not only on the width a of the opaque barrier, but also on the distance l between two opaque barriers. This result is a striking generalization of the HFE, firstly obtained in [127]. It is important to stress that this result holds, however, for nonresonant tunneling, that is, for energies far from the resonances.
Esposito in [128] has generalized this result for the tunneling through an arbitrary number of finite rectangular opaque barriers and it has been shown that the total tunneling phase time depends neither on the barrier thickness nor on the interbarrier separation. It has been also shown the independence of the phase transit time (for nonresonant tunneling).
Now, we will consider the cases of the resonances (between two barriers) and the influence of not very far resonances.

The Resonant Tunneling
If one takes two arbitrary (not necessarily opaque) barriers (cf. Figure 7), the amplitude of the transmitted wave in this case is defined by the formula [88, 129] where 𝑘, and 1cot(𝑘𝑙)=2𝛿tan(𝜒𝑎).(3.70). The dimensionless constants 𝑎 and 𝑚 are connected by the relation
A resonance is characterized by the condition [88, 120] that is, the double barrier becomes totally transparent (without any reflections).
It is easy to see (cf. (3.67), and also [88, 129]) that the values of the wave number 𝑢=cosh21(𝜒𝑎)4𝛿2sinh21(𝜒𝑎),𝑣=𝛿cosh(𝜒𝑎)sinh(𝜒𝑎),𝑤=4𝜎2sinh2(𝜒𝑎)(3.72) for which condition (3.69) is satisfied can be found from the equation We can show that from (3.70) it is possible to find out also the values of parameters 𝐷=𝑢+𝑤cos(2𝑘𝑙)+𝑖[𝑣+𝑤sin(2𝑘𝑙)].(3.74), |𝐷|2=1+2𝑤1+𝑤+𝑢cos(2𝑘𝑙)+𝑣sin(2𝑘𝑙),(3.75), and 1+𝑤+𝑢cos(2𝑘𝑙)+𝑣sin(2𝑘𝑙)=0.(3.76), at the resonance. Indeed, from (3.67) and (3.70) it follows that at a resonance it is On introducing the functions and using (3.68), one can infer that these functions are connected by the relation Then, by using functions (3.72), the denominator of (3.67) can be written in the following form: It follows from (3.74) and (3.73) that and condition (3.71) gets transformed into Taking (3.68) into account, we can write the term 𝐷𝑟=𝐷(𝑘𝑟) of (3.76) in the following form: and this (last) equation can be easily transformed, afterwards, into (3.70). There are no other solutions to (3.76). Hence, (3.70), as well as (3.69) or (3.71) is a general resonance condition. In the region of a resonance, if we limit ourselves to the first two terms of the expansion of (3.67) into a series of powers of 𝐶𝑟=𝑚2𝑘𝐷𝑟,(3.79) (quantity 𝑘 being the resonance value of energy 𝐷(𝑘)=𝐶𝑟𝐸𝐸𝑟+𝐷𝑟𝐶𝑟||𝐶𝑟||2.(3.80)), we obtain where 𝑢𝑟, 𝑣𝑟, and the index prime defines the derivative with respect to 𝐸=𝐸𝑟. We can rewrite (3.78) in the following form: It follows from (3.70) that at the resonance it is where 𝑤, 𝐷𝑘𝑟=𝑢𝑟+𝑖𝑣𝑟1+𝑤𝑟.(3.82) and 𝑘 are the values at 𝐷𝑘𝑟=𝑢𝑟𝑢𝑟𝑤𝑟2𝑙𝑣𝑟𝑤𝑟1+𝑤𝑟𝑣+𝑖𝑟𝑣𝑟𝑤𝑟2𝑙𝑢𝑟𝑤𝑟1+𝑤𝑟.(3.83) of the functions 𝑢𝑢+𝑣𝑣=(1+𝑤)𝑤,(3.84), 𝐷𝑟𝐶𝑟𝑚=𝑖2𝑘𝑟𝑢𝑟𝑣𝑟𝑢𝑟𝑣𝑟1+𝑤𝑟+2𝑙𝑤𝑟,(3.85), and ||𝐶𝑟||2=𝑚2𝑘𝑟2𝑢𝑟2+𝑣𝑟2𝑤𝑟2𝑢+4𝑙𝑟𝑣𝑟𝑢𝑟𝑣𝑟1+𝑤𝑟𝑤𝑟+4𝑙2𝑤2𝑟.(3.86), respectively. On inserting (3.81) into (3.74), we obtain Differentiating (3.74) with respect to ||𝐶𝑟||2=𝑚2𝑘𝑟2𝑢𝑟𝑣𝑟𝑢𝑟𝑣𝑟1+𝑤𝑟+2𝑙𝑤𝑟.(3.88), and inserting the result into (3.81), we get Then, on using (3.73), (3.79), (3.82), (3.83), and relation which follows from (3.73), one finds that at the resonance where (by having recourse to (3.73), (3.79), (3.83), and (3.84)) From (3.73) and (3.84) one gets while, from (3.86) and (3.87), one obtains By differentiating the functions (3.72) with respect to 𝐸=𝐸𝑟, one can see that where 𝛽=2𝑘𝑟𝜒𝑟𝑚𝜒𝑟𝑘𝑟𝑎+2𝜒𝑟𝑙𝑤𝑟+𝜎2𝑟cosh(𝜒𝑟𝑎)sinh(𝜒𝑟𝑎)1,(3.91) and ||𝐶𝑟||2=1𝛽2.(3.92) are the values of ||𝐴𝑇||(𝑘)2=𝛽2𝐸𝐸𝑟2+𝛽2.(3.93), and 𝜏Phtun=𝐴𝑑arg𝑇exp𝑖𝑘(2𝑎+𝑙)𝑑𝐸.(3.94) at 𝜏Phtun=𝑚𝐷𝑘𝑙1𝐷2𝐷2𝐷1|𝐷|2,(3.95). On inserting (3.89) into (3.85) and (3.88), we can then rewrite (3.80) in the noteworthy form where so that (3.80), (3.85), (3.88), and (3.90) yield Finally, by inserting (3.90) into (3.66) and taking account of (3.92), we obtain that near a resonance it holds in general which corresponds to nothing but a Breit and Wigner formula. In other words, one verifies that the Breit-Wigner’s formula has a general validity for our (1D) tunneling near a resonance.
Let us start from the first two parts of relation (3.65), that is, concretely from the By using (3.66) and (3.76), we can rewrite it in the following form: where (the index prime denoting again the derivative with respect to 𝜏Phtun𝑑=1𝑑𝐸arg𝐶𝑟exp(𝑖𝑘𝑙)𝐸𝐸𝑟+𝑖𝛽,(3.101)). On inserting (3.96) into (3.95) and using (3.73), we get in general for the total tunneling phase time the remarkable formula: where It should be noticed that (3.97) holds in general for any (resonant and/or nonresonant) tunneling time through two barriers. From (3.97), (3.98), (3.71), and (3.85), it follows that that the tunneling phase time at a resonance 𝜒𝑎1 is given by the following expression: It follows from (3.65) and (3.66) that Inserting (3.90) into (3.100), we get so that, near a resonance, the behavior of the tunneling phase time in terms of the energy is represented by the interesting following formula: holding for any resonant tunneling through the two barriers. The first term represents the time associated with the particle free flight over the distance 𝑢𝑣𝑢𝑣=1𝜒𝜎(1+𝑤)2,𝑢cosh(𝜒𝑎)sinh(𝜒𝑎)+𝛿𝑘𝑎𝑤𝑢𝑤=2𝜒1𝛿(𝑢+𝑤)+4𝜎2,cosh(𝜒𝑎)sinh(𝜒𝑎)𝑣𝑤𝑣1𝑤=𝜒𝑤4𝛿2.cosh(𝜒𝑎)sinh(𝜒𝑎)𝛿𝑘𝑎(3.104) between the two barriers; while the second term is the time delay caused by the quasibound state assumed by the particle in such an intermediate region.
Esposito in [128] has shown that for the arbitrary number of finite rectangular opaque barriers the resonant energy does not depend on the number of barriers.

The Dependence of The Tunneling Phase Time on The Width of The Setup, Far from The Resonances
When releasing the above condition, we have found in this paper a more complicate expression, given by formulae (3.65), (3.66), and (3.75) above. Anyway, from (3.75) and (3.72) it follows that, for opaque barriers, when 𝜒𝑎1, it holds Differentiating the functions 𝜏Phtun2𝑚𝑚𝑘𝜒+4𝜎𝑘𝑙exp(2𝜒𝑎)24+𝛿1+24cos(2𝑘𝑙)+𝛿sin(2𝑘𝑙)1,(3.106) and 𝜒𝑎 with respect to exp(2𝜒𝑎), one can see that Therefore, from (3.72), (3.98), (3.103), and (3.104), we get that, still for 𝜏Phtun, and far from the resonances, Namely, when 𝜒𝑎 increases, the second term in (3.106) decreases as ||𝐴𝑇||2132𝜎21exp(4𝜒𝑎)4𝜎2+114𝛿2cos(2𝑘𝑙)+𝛿sin(2𝑘𝑙)1,(3.107); while, at the limit when 𝜒𝑎, (3.106) goes into the right-hand side of (3.65).
Of course, our result (3.106) does not hold only for particles but—as well-known (see Section 2.4 and [5962])—also for photons. This can explain the results explaining the experimental fact that |𝐴𝑇(𝑘)|2 has bee