Time as a Quantum Observable, Canonically Conjugated to Energy, and Foundations of Self-Consistent Time Analysis of Quantum Processes

Abstract

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., [3–51], 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., [12–14, 47, 50, 51]).

As to quantum mechanics, the first set of known and cited articles is [3–21]. The second set of papers on time as an observable in quantum physics [22–51] 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 [15–21]. 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 [15–18, 21] (more details having been added in [22–27, 32–35, 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 [23–25, 33–35, 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 [28–31, 36–46, 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 𝐸≠0 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 [15–18, 21] (see also [26, 27, 33–35, 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 [15–18, 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)𝜕𝐸|||²𝑑𝐸<∞,^∞0||𝑓(𝐸)||²𝐸²𝑑𝐸<∞,(2.2) that satisfy the conditions and the condition which is a space 𝐿²dense in the Hilbert space of 0≤𝐸<∞ functions defined (only) over the semiaxis (𝑓₁,𝑡𝑓̂₂)=((𝑡𝑓̂₁),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 𝑡²̂.

Also the operator (𝑓₁,𝑡²̂𝑓₂)=((𝑡𝑓̂₁),(𝑡𝑓̂₂))=(𝑡²̂𝑓₁,𝑓₂) is Hermitian, that is, the relation 𝑡̂ 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 𝑡̂ but that is inconsistent with the propriety of being Hermitian. So, according to [53], (0≤𝑥<∞) 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 𝑝_𝑥̂=−𝑖ℏ𝜕𝜕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)𝜕𝑥|||²𝑑𝑥<∞,^∞0||𝑓(𝑥)||²𝑥²𝑑𝑥<∞,(2.5) that satisfy the conditions and the condition which is a space dense 𝐿² in the Hilbert space of 0≤𝑥<∞ functions defined (only) over the spatial semiaxis 𝑝_𝑥̂=−𝑖ℏ(𝜕/𝜕𝑥). Therefore, operator 𝑡̂ has the same mathematical properties as operator 𝑝_𝑟̂=−𝑖ℏ(𝜕/𝜕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 (𝐸_min<𝐸<𝐸_max).

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=ℏ/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=|𝐸_max−𝐸_min| where 𝑊(𝑥,𝑡)𝑑𝑡=𝑗(𝑥,𝑡)𝑑𝑡∫^∞−∞𝑗(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)(ℏ/𝑖𝜇)𝜕Ψ(𝑥,𝑡)/𝜕𝑥] and 𝑗(𝑥,𝑡). The measure (2.7) was firstly investigated in [21, 23–27, 32–35].

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_±(𝑥,𝑡)𝑑𝑡=𝑗^±(𝑥,𝑡)𝑑𝑡∫^∞−∞𝑗_±(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, 32–35]. 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^𝜕𝑡^𝑑𝑡^,𝜌_<(𝑥,𝑡)=−^𝑡−∞𝜕𝑗₋𝑥,𝑡^𝜕𝑡^𝑑𝑡^(2.11), one obtains with the initial conditions 𝑁𝑁_>(𝑥,∞;𝑡)≡∫^∞𝑥𝜌_>𝑥^,𝑡𝑑𝑥^=∫^𝑡−∞𝑗₊𝑥,𝑡^𝑑𝑡^>0,_<(−∞,𝑥;𝑡)≡∫^𝑥−∞𝜌_<𝑥^,𝑡𝑑𝑥^=−∫^𝑡−∞𝑗₋𝑥,𝑡^𝑑𝑡^>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)𝑑𝑡=∫^∞0𝑑𝐸(1/2)𝐺^∗(𝑥,𝐸)𝑡𝑣̂𝐺(𝑥,𝐸)+𝑣𝐺^∗(𝑥,𝐸)𝑡𝐺̂(𝑥,𝐸)∫^∞0𝑑𝐸𝑣||𝐺(𝑥,𝐸)||²,(2.15), when travelling in the positive or negative direction, respectively, where Ψ(𝑥,𝑡)=^∞0𝐺(𝑥,𝐸)exp−𝑖𝐸𝑡ℏ𝑑𝐸=^∞0𝑔(𝐸)𝜑(𝑥,𝐸)exp−𝑖𝐸𝑡ℏ𝑑𝐸,(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 𝑥_𝑓>𝑥_𝑖 to (𝑥_𝑖,∞) and 1D particle reflection from the region 𝑥_𝑓≤𝑥_𝑖 into ⟨⟨𝜏𝜏_𝑇𝑅𝑥_𝑖,𝑥_𝑓⟩=⟨𝑡₊𝑥_𝑓⟩−⟨𝑡₊𝑥_𝑖⟩,𝑥_𝑖,𝑥_𝑓⟩=⟨𝑡₋𝑥_𝑓⟩−⟨𝑡₊𝑥_𝑖⟩,(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 ⟨𝑡^𝑛⟩, 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)𝑑𝑡=∫^∞0𝑑𝐸(1/2)𝐺^∗(𝑥,𝐸)𝑓𝑡̂𝑣𝐺(𝑥,𝐸)+𝑣𝐺^∗(𝑥,𝐸)𝑓𝑡̂𝐺(𝑥,𝐸)∫^∞0𝑑𝐸𝑣||𝐺(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=ℏ²𝑘²/2𝜇=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 𝑡̂: this complies with the actual conditions of real experiments, and therefore they does not represent any restriction of generality.

In (2.19), |𝑔(𝐸)|²≈𝐾𝛿(𝐸−𝐸) 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)𝑑𝑡∫^∞−∞𝑗(𝑥,𝑡)𝑑𝑡≈∫^∞−∞𝜌(𝑥,𝑡)𝑓(𝑡)𝑑𝑡∫^∞−∞𝜌(𝑥,𝑡)𝑑𝑡≈∫^∞0𝑑𝐸𝐺^∗(𝑥,𝐸)𝑓𝑡̂𝐺(𝑥,𝐸)∫^∞0𝑑𝐸||𝐺(𝑥,𝐸)||²,(2.22) and (2.19) goes into the more simple relation because of the relations 𝐸=⎧⎪⎨⎪⎩𝐸𝑖ℏ𝜕intheenergy(𝐸)-representation,𝜕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 Δ𝐸Δ𝑡≥ℏ2(2.25) and by utilizing the peculiar mathematical properties of maximal symmetric operators (as in [19–21, 23–25, 33–35, 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=⟨𝑎²⟩−⟨𝑎⟩² 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 [17–21, 23–25, 33–35, 50, 51]). Moreover, relation (2.24) satisfies the Dirac “correspondence principle,” since the classical Poisson brackets 𝑞₀=𝑡, with 𝑝₀=−𝐸 and 𝑡̂, are equal to unity [56]. In [21] (see also [23–25]) 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, 23–25] 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 𝑡̂ 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 𝑡^𝑛̂(𝑛>1), as well as for 𝑡^𝑛̂(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 𝑡^𝑛̂(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 coordinates𝑡̂=−𝑖ℏ2^↔𝜕𝜕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 [28–31, 36–46, 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 [28–31, 36–46, 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 [12–18, 21] (and independently in the papers of Holevo [19, 20], with the same principal idea) much earlier than [28–31, 36–46, 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 [12–14] one more form of the time operator (the so-called bilinear form), where the meaning of the sign (𝑓,𝑡𝑔̂)=(𝑓,(−𝑖ℏ/2)(𝜕/𝜕𝐸)𝑔)+((−𝑖ℏ/2)(𝜕/𝜕𝐸)𝑓,𝑔) is clear from the following definition: 𝐸=0. For this form the direct elimination of the point (𝑓,𝑡𝑓̂) 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 [28–31, 36–46, 48, 49], and it had been published (in [12–14]) 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)exp−𝑖𝐸𝑡ℏ,(2.27): with −∞<𝑘<∞. In such a case the time operator (2.1a), (2.1b), and (2.26) (acting on momentum eigenvector, defined on 𝑑^𝑛𝑔(𝑘)𝑑𝑘^𝑛_𝑘=−∞=𝑑^𝑛𝑔(𝑘)𝑑𝑘^𝑛_𝑘=∞=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)^1/2+_∞0ℏ𝜑𝑥,(2𝑚ℏ𝐸)^1/2exp−𝑖𝐸ℏ𝑡𝑑𝐸(𝐸)^−1/2𝑔−(2𝑚𝐸)^1/2ℏ𝜑𝑥,−(2𝑚ℏ𝐸)^1/2exp−𝑖𝐸ℏ𝑡.(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 𝐸→𝑘 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 ⟨𝜏_dw𝑥_𝑖,𝑥_𝑓⟩=∫^∞−∞𝑑𝑡∫^𝑥𝑓_𝑥𝑖||Ψ(𝑥,𝑡)||²𝑑𝑥∫^∞−∞𝑗_in𝑥_𝑖,𝑡𝑑𝑡(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 ⟨𝜏_dw𝑥_𝑖,𝑥_𝑓⟩=∫^∞−∞𝑡𝑗𝑥_𝑓,𝑡𝑑𝑡−∫^∞−∞𝑡𝑗𝑥_𝑖,𝑡𝑑𝑡∫^∞−∞𝑗_in𝑥_𝑖,𝑡𝑑𝑡,(2.36b) for the initial “dwelling” free 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 𝐴^→^→𝑟,𝑡=_𝑘0𝑑³𝑘𝑘_0→𝜒^→𝑘𝜑^→𝑘,^→𝑟exp−𝑖𝑘₀𝑡,(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 [59–62]. 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={𝑘_𝑥,𝑘_𝑦,𝑘_𝑧} is the electromagnetic vector potential, and 𝑘₀≡𝜔/𝑐=𝜀/ℏ𝑐, 𝑘≡|^→𝑘|=𝑘₀, div𝐴^→=0, 𝑥, where the gauge condition 𝜒^→(^→𝑘)=∑^{𝑧𝑖=𝑦}𝜒_𝑖(^→𝑘)^→𝑒_𝑖(^→𝑘); is assumed.

The axis ^→𝑒_𝑖→𝑒_𝑗=𝛿_𝑖𝑗 has been chosen as the propagation direction, 𝑥_𝑖,𝑥_𝑗≡𝑦,𝑧 with 𝜒𝑖(^→𝑘), ^→𝑘, ^→𝑒_𝑗 is the probability amplitude for the photon to have momentum 𝑥_𝑗 and polarization 𝜑(^→𝑘,^→𝑟)=exp(𝑖𝑘_𝑥𝑥) along 𝜑(^→𝑘,^→𝑟), and it is 𝜌_em(𝑥,𝑡)𝑑𝑥=𝑆⁰𝑑𝑥∫𝑆₀𝑑𝑥,𝑆⁰=𝑠₀𝑑𝑦𝑑𝑧(2.38) in the case of plane waves, while 𝑠₀=[𝐸^→𝐸^∗→+𝐻^→𝐻^∗→]/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), 𝑗_em(𝑥,𝑡)𝑑𝑡=𝑆^𝑥(𝑥,𝑡)𝑑𝑡∫𝑆_𝑥(𝑥,𝑡)𝑑𝑡,𝑆^𝑥(𝑥,𝑡)=𝑠_𝑥𝑑𝑦𝑑𝑧(2.39)) of a photon to be found (localized) in the spatial intervall𝑠_𝑥=𝑐Re[𝐸^→𝐻^∗→]_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)⟩=^∞−∞𝑡𝐽_em,𝑥𝑑𝑡=∫^∞−∞𝑡𝑆_𝑥(𝑥,𝑡)𝑑𝑡∫^∞−∞𝑆_𝑥(𝑥,𝑡)𝑑𝑡,(2.40)) of a photon to pass through the point (plane) x in the time interval 𝜒_𝑖(0)=𝜒_𝑖(∞)=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 [65–67]. 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 𝑠_𝑥, 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 [32–35, 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 [32–35, 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 𝐽_em 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=0𝑔_𝑛𝜑_𝑛(𝑥)exp−𝑖𝜀^𝑛−𝜀₀𝑡ℏ,(2.42) by 𝜑_𝑛(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 [23–25, 50, 51]) to introduce wave packets of the form (where ∑^…𝑛=0|𝑔_𝑛|²=1 are orthogonal and normalized wave functions of system bound states which satisfy exp(−𝑖𝜀₀𝑡/ℏ) 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=2𝜋ℏ/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 𝑡̂ 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 𝑡̂=𝑡−𝑇^∞_𝑛=𝑜Θ𝑡−[2𝑛+1]𝑇2+𝑇^∞_𝑛=0Θ−𝑡−[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)²≥ℏ²⟨𝑁⟩²(2.45) (see Figure 1).

Figure 1 

The periodical saw-tooth function for time operator for the case of (2.42).

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 𝐴,𝐵=𝑖ℏ𝑁,(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 Δ𝐸→0 to |𝑔_𝑛|→𝛿_𝑛𝑛′, 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 Δ𝐸≫𝐷) the right part of (2.47) tends to zero since |𝜓(𝑇+𝛾)|²≪𝑇∫^{𝑇/2−𝑇/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 Δ𝑡≪𝑇 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 𝑡̂=𝑖ℏ2_𝑛;>𝑛(−1)^{𝑁𝑛−𝑁𝑛′↔}Δ_𝑛′Δ_𝑛′𝜀_𝑛(2.48) of instants of particle passing through point 𝑁_𝑛=(𝜀_𝑛−𝜀₀)/𝐷, then after a series of bulky transformations he will obtain the following expression: in the energy representation, where 𝐴Δ^∗𝑛↔_𝑛′𝐴_𝑛=⟨𝑡𝐴(𝑥^∗𝑛Δ)⟩=_𝑛′𝐴_𝑛∑−𝐴_{𝑛∞𝑛=0}Δ_𝑛′𝐴^∗𝑛,Δ_𝑛′𝐴_𝑛=𝐴_𝑛′−𝐴_𝑛,𝑔^∗𝑛𝜑^∗𝑛(𝑥)𝑡𝑔̂_𝑛𝜑_𝑛(𝑥)∑^∞𝑛=0||𝑔_𝑛𝜑_𝑛(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 𝑡̂=−𝑖ℏ2^↔ΔΔ𝜀,(2.50)) Operator (2.43) for two levels 𝐷=𝜀₁−𝜀₀→0 acquires the more simple form and when 𝛾_lim, 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 (𝑔_𝑛→𝛿_𝑛𝑛′)) 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., [72–80], apart from [26, 27, 32–35, 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 [12–14, 86, 87] and applied to tunnelling in [88, 89], and finally the Olkhovsky-Recami (O-R) method [26, 27, 32–35, 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, 23–25, 91] for atomic and nuclear collisions. The second group contains methods, utilizing the Feynman path integrals [92–98], the Wigner distribution paths [99–102], and the Bohm approach [103]. The approaches with the Larmor clock [104–107] 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 3.3–3.5), 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 (𝑥_𝑖,𝑥_𝑓)-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)+𝐴^𝑅𝐴𝛼exp(−𝜒𝑥)+𝛽exepx(p−(𝑖𝜒𝑘𝑥𝑥)),,𝑥0≤≤𝑥0≤(re𝑎gi(ornegIi)o,nII),_𝑇exp(𝑖𝑘𝑥),𝑥≥𝑎(regionII),(3.1) and energy 𝑘=(2𝑚𝐸)^1/2/ℏ has the usual form (e.g., [26, 27, 32, 47, 72–81, 90] and a lot of other papers): where 𝐴_𝑅, 𝛼, 𝛽, 𝐴_𝑇, 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, 23–25, 91] and afterwards generalized in [26, 27, 32–35, 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)𝑑𝑡𝐷𝑡_±∫^∞−∞𝑗_±(𝑥,𝑡)𝑑𝑡,(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[(𝑖ℏ/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: 𝐷⟨𝜏𝜏_𝑇𝑇𝑥_𝑖,𝑥_𝑓⟩=⟨𝑡₊𝑥_𝑓⟩−⟨𝑡₊𝑥_𝑖⟩,(3.3)𝑥_𝑖,𝑥_𝑓=𝐷𝑡₊𝑥_𝑓+𝐷𝑡₊𝑥_𝑖,(3.4) (index t is omitted in all expressions for −∞<𝑥_𝑖≤0 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 ⟨𝜏_pen(𝑥_𝑖,𝑥_𝑓)⟩ and 𝐷𝜏_pen(𝑥_𝑖,𝑥_𝑓). For a pure tunnelling process one has Similar expression we have for the penetration (into the barrier region II) temporal quantities 𝑥_𝑓<𝑎 and ⟨𝐷𝜏_𝑅𝜏_𝑅𝑥_𝑖,𝑥_𝑓⟩=⟨𝑡₋𝑥_𝑓⟩−⟨𝑡₊𝑥_𝑖⟩,𝑥_𝑖,𝑥_𝑓)=𝐷𝑡₋𝑥_𝑓+𝐷𝑡₊𝑥_𝑖.(3.6) with |𝑥_𝑖|≫𝑎. For reflections in any point ⟨𝜏_as𝑇⟨𝜏^as_𝑇𝑥_𝑖,𝑥_𝑓⟩=⟨𝑡𝑥_𝑓⟩_𝑇−⟨𝑡𝑥_𝑖⟩_in,(3.7)𝑥_𝑖,𝑥_𝑓⟩=⟨𝜏_𝑇𝑥_𝑖,𝑥_𝑓⟩+⟨𝑡₊𝑥_𝑖⟩−⟨𝑡𝑥_𝑖⟩_in,(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 Ψ_in(𝑥,𝑡)=^∞0𝐺𝑘−𝑘exp𝑖𝑘𝑥−𝑖𝐸𝑡ℏ𝑑𝑘,(3.9) and 𝐸=ℏ²𝑘²/2𝑚,∫^∞0|𝐺(𝑘−𝑘)|²𝑑𝐸=1,𝐺(0)=𝐺(∞)=0,𝑘>0 denote averagings over the fluxes corresponding to ^∞0𝑣^𝑛||𝐺𝐴_𝑇||²𝑑𝐸≅^∞0𝑣^𝑛|𝐺|²𝑑𝐸,𝑛=0,1,𝑣=ℏ𝑘𝑚,(3.10) and ⟨𝜏^as_𝑇𝑥_𝑖,𝑥_𝑓⟩≅⟨𝜏^Ph_𝑇𝑥_𝑖,𝑥_𝑓⟩_𝐸,(3.11), respectively.

For initial wave packets (where 𝐷𝑡₊(𝑥_𝑖)) 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 𝐷𝜏_𝑇(𝑥_𝑖,𝑥_𝑓) into the variance |𝐺|²→𝛿(𝐸−𝐸),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 ⟨𝜏^as_𝑇(𝑥_𝑖,𝑥_𝑓)⟩≅⟨𝜏^Ph_𝑇(𝑥_𝑖,𝑥_𝑗)⟩_𝐸 plus 𝑉₀ which is born by the barrier influence and formally is described by (3.13).

At the quasimonochromatric limit 𝜒𝑎≫1 being 𝜒=[2𝑚(𝑉₀−𝐸)]^1/2/ℏ, we get for 𝑥_𝑖=0,𝑥_𝑓=𝑎 strictly the ordinary phase time, without averaging. For a rectangular barrier with height 𝑎𝑚/ℏ𝑘≫𝐷𝑡₊(𝑥_𝑖) and 𝜏^Phtun=2𝑣𝜒(3.14) (where 𝐷𝜏^Phtun^1/2=𝑣𝑎𝜒𝑘(3.15)), the expressions (3.11) and (3.13), for 𝐺(𝑘−𝑘) 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 [104–107] 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 𝑥_𝑖, 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 ⟨𝜏_𝑇⟨𝑥𝜏_𝑖,𝑥_𝑓tun⟩₊=⟨𝜏^Ph_𝑇𝑥_𝑖,𝑥_𝑓⟩_𝐸,(3.17)(0,𝑎)⟩₊=⟨𝜏^Phtun⟩_𝐸,(3.18) from ⟨𝑡(0)⟩₊=⟨𝑡(0)⟩_in and 𝜏^Phtun_,em=2𝑐𝜒^emfor𝜒_em𝑎≫1.(3.19) from 𝜒_em𝑎>2 because 𝑗(𝑥_𝑓,𝑡)=𝑗_𝑇(𝑥_𝑓,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 𝑗_𝑅 the effective tunnelling velocity is more than c, that is, superluminal. This result agrees with the results of the microwave-tunnelling measurements presented in [111–113] (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 Ψ_in(𝑥_𝑖,𝑡),Ψ_𝑅(𝑥_𝑖,𝑡) and Ψ_𝑇(𝑥_𝑓,𝑡), with 𝜓_in,𝜓_𝑅=𝐴_𝑅exp(−𝑖𝑘𝑥), 𝜓_𝑇 and 𝑗_int(𝑥,𝑡)=Re𝑖ℏ𝑚Ψⁱⁿ(𝑥,𝑡)𝜕Ψ^∗𝑅(𝑥,𝑡)𝜕𝑥+Ψ^𝑅(𝑥,𝑡)𝜕Ψ^∗_in(𝑥,𝑡)∫𝜕𝑥_∞−∞,𝑗_int𝑥_𝑖,𝑡𝑑𝑡=0.(3.21) corresponding to the wave packets ⟨𝜏^Dw𝑥_𝑖,𝑥_𝑓⟩=⟨𝑇⟩_𝐸⟨𝜏_𝑇𝑥_𝑖,𝑥_𝑓⟩+⟨𝑅𝑥_𝑖⟩_𝐸⟨𝜏_𝑅𝑥_𝑖,𝑥_𝑓⟩,(3.22) and ⟨𝑇⟩_𝐸=⟨|𝐴𝑇|²𝑣⟩_𝐸/⟨𝑣⟩_𝐸,⟨𝑅(𝑥_𝑖)⟩_𝐸=⟨𝑅⟩_𝐸+⟨𝑟(𝑥_𝑖)⟩,⟨𝑅⟩_𝐸=⟨|𝐴_𝑅|²𝑣⟩_𝐸/⟨𝑣⟩_𝐸 constructed from the stationary wave functions ⟨𝑇⟩_𝐸+⟨𝑅⟩_𝐸=1 and ⟨𝑟(𝑥)⟩=∫^∞−∞𝐽₊(𝑥,𝑡)−𝐽_in(𝑥,𝑡)𝑑𝑡∫^∞−∞𝐽_in(𝑥,𝑡)𝑑𝑡.(3.23), respectively, and also One obtains from (2.36a) and (2.36b) of Section 2, with −∞, Ψ_in(𝑥_𝑖,𝑡), and One can see that ⟨𝑟(𝑥_𝑖)⟩=0 is negative and tends to 0 when ⟨𝜏^Dw𝑥_𝑖,𝑥_𝑓⟩=⟨𝑇⟩_𝐸⟨𝜏_𝑇𝑥_𝑖,𝑥_𝑓⟩+⟨𝑅⟩_𝐸⟨𝜏_𝑅𝑥_𝑖,𝑥_𝑖⟩(3.24) tends to 𝜒𝑎≫1.

When 𝑥_𝑖=0 and 𝑥_𝑓=𝑎 are sufficiently well separated in time, so that ⟨𝜏^Dw𝑥_𝑖,𝑥_𝑓⟩=ℏ𝑘𝜒𝑉₀_𝐸,(3.25), it follows from (3.22) that the simple weighted average rule is valid. For a rectangular barrier with ⟨𝜏^Dw𝑥_𝑖,𝑥_𝑓⟩=2𝜒𝑣_𝐸(3.26) and quasimonochromatic particles, the expressions (3.22) and (3.24) with ⟨𝑟(𝑥_𝑖)⟩ and 𝐴_𝑅=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 𝜏^La_𝑦,𝑇 and 𝜏^La_𝑧,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 𝜏^La_𝑧,𝑇 are defined by the ratio of the spin-rotation angles around axes |𝐴_𝑇| and 𝜒𝑎≫1 (in turn defined by the developed ⟨𝜏^La_𝑦,tun⟩=⟨𝜏^Dw𝑥_𝑖,𝑥_𝑓⟩=ℏ𝑘𝜒𝑉₀_𝐸⟨𝜏_La𝑧,tun,(3.28)⟩=𝑚𝑎ℏ𝜒_𝐸(3.29)- and |𝐴_𝑇|-spin components, resp.) to the precession (rotation) frequency.

As to ⟨𝜏^La_𝑧,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))/ℏ] 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, 101–103].

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 ⟨𝜏^La_𝑧,tun⟩ for the same reasons as for exp(𝑖𝑆/ℏ).

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 [92–98] 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⁻=𝐽−𝐽⁺ 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 [99–102]. 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 ⟨𝜏^Dw𝑥_𝑖,𝑥_𝑓⟩=⟨𝑇⟩_𝐸⟨𝜏_𝑇𝑥_𝑖,𝑥_𝑓⟩+⟨𝑅_𝑀𝑥_𝑖⟩_𝐸⟨𝜏_𝑅𝑥_𝑖,𝑥_𝑓⟩,(3.31), so that the flux at position x can be separated into positive and negative components: with 𝑅_𝑀(𝑥) and ⟨𝑅⟩_𝐸. 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 𝜏_Dw𝑥_𝑖,𝑥_𝑓=^∞0𝑑𝑡^𝑥2_𝑥1||Ψ(𝑥,𝑡)||²𝑑𝑥=𝑇𝜏_𝑇𝑥_𝑖,𝑥_𝑓+𝑅𝜏_𝑅𝑥_𝑖,𝑥_𝑓,(3.32) instead of our 𝜏𝜏_𝑇𝑥_𝑖,𝑥_𝑓=^∞0𝑑𝑡^𝑥2_𝑥1||Ψ(𝑥,𝑡)||²Θ𝑥−𝑥_𝑐𝑑𝑥/𝑇,_𝑅𝑥_𝑖,𝑥_𝑓=^∞0𝑑𝑡^𝑥2_𝑥1||Ψ(𝑥,𝑡)||²Θ𝑥_𝑐−𝑥𝑑𝑥/𝑅,(3.33). The dwell time decomposition in this approach takes the form with 𝑅. Asymptotically, 𝑥_𝑐=𝑥_𝑐(𝑡) tends to 𝑇=^∞−∞||Ψ(𝑥,𝑡)||²Θ𝑥−𝑥_𝑐𝑑𝑥.(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 𝑥_𝑐 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 𝐽𝐽((𝑥𝑥,,𝑡𝑡))_𝑇𝑅=𝐽(𝑥,𝑡)Θ𝑥−𝑥_𝑐(𝑡),=𝐽(𝑥,𝑡)Θ𝑥_𝑐(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 (⟨𝜏_tun⟩ and 𝐴_𝑇, resp.), sometimes essential, this approach gives one more alternative—in separating the flux by the line 𝐸=ℏ²𝑘²/(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(𝑉₀−𝐸)]^1/2/ℏ,𝑉₀ and 𝜏^Phtun→2/(𝜐𝜒) being the transmission amplitude and the particle kinetic energy, resp.) on the barrier width a for sufficiently large 𝜒𝑎≫1. In particular, for a rectangular potential barrier 𝜐=ℏ𝑘/𝑚⟨𝜏^Dwtun⟩, ⟨𝜏^𝐿𝑦,_tun⟩, Re𝜏^𝐹_tun being the barrier height, and ℏ𝑘/(𝜒𝑉₀) when ⟨𝜏_tun⟩=⟨𝑡₊(𝑎)⟩−⟨𝑡₊(0)⟩=⟨𝜏^Phtun⟩_𝐸−⟨𝑡₊(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)𝑑𝑡∫^∞−∞𝑗±(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 𝜏^𝐿𝑧,_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 𝜏^𝐵−𝐿_tun denotes averaging over the initial wave-packet energy spread and 𝑎𝑚/(ℏ𝜒), 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, 32–35, 47] by numerous calculations for gaussian electron wave packets with narrow momentum spreads (see also Section 3.6).

As to the other Larmor time [𝐷_dyn𝜏_tun]^1/2, from Sections 4.2 and 3.4 it follows that the Bttiker-Landauer time 𝐷_dyn𝜏_tun=𝐷𝜏_tun−𝐷𝑡₊(0) [111–113], and the imaginary part of the complex tunnelling time 𝐷𝜏_tun=⟨𝜏²_tun⟩−⟨𝜏_tun⟩² [92], obtained within the Feynman approach, which are equal to (3.39), they become equal to ⟨𝜏²_tun⟩=⟨[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 ⟨𝜏_tun(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=0.02A⁻¹, 𝐸=5eV𝑘=0.04A⁻¹ (it was shown in [26, 27, 33–35]). 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 0.06A⁻¹ from a are presented for electronic wave packets and rectangular barriers with the same parameters as in [47] (⟨𝜏^Phtun⟩; mean electron energies Δ𝑘 with ⟨𝜏^Phtun⟩ (curves 1a, 2a, and 3a, resp.); energy ⟨𝜏_tun(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 ⟨𝜏_tun(0,𝑎)⟩ from ⟨𝜏^Phtun⟩, 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 𝑉₁≪𝑉₀. Since all 𝑉₁𝑚^1/2𝑣𝜒𝑎/[2(𝑉₀−𝐸)]^3/2≪2 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)exp−𝑖𝐸𝑡ℏ𝑑𝑘,(3.41) axis through a rectangular potential barrier with height 𝑔(𝑘)=𝑔₀(𝑘)exp−𝑖𝑘−𝑘₀𝑥₀=𝐶exp−𝑘−𝑘⁰2Δ𝑘²−𝑖𝑘−𝑘₀𝑥₀,(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/ℏ has the usual form (3.1). The time dependent wave packet Ψ_in(𝑥,𝑡)≅𝐶₀𝑠^−1/2_inexp−𝑥−𝑥⁰+𝜈𝑡²2𝑠_in+𝑖𝑘⁰𝑥−𝑖𝐸₀𝑡ℏ,(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[(𝑖ℏ/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)𝑑𝑡∫^∞0𝑗(𝑎,𝑡)𝑑𝑡,(3.44) and in the absence of barrier crosses the point 𝑥=0 at the instant ⟨𝑡₊(0)⟩=∫^∞0𝑡𝑗₊(0,𝑡)𝑑𝑡∫^∞0𝑗₊(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)𝜓_𝑇(𝑘𝑎)exp−𝑖𝐸𝑡ℏ𝑑𝑘,(3.46) represents the positive values of the flux density 𝜓_𝑇(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 𝜓_𝑇(𝑘,𝑎) is then defined as 𝜓_𝑇(𝑘,𝑎)=⎧⎪⎪⎨⎪⎪⎩2𝑖𝑘𝜒𝐷_<2𝑖𝑖𝑘𝑞,𝐸<𝑉₀,𝐷_>,𝐸>𝑉₀,(3.47).

The flux density 𝑞=[2𝑚(𝐸−𝑉₀)]^1/2/ℏ contains the wave function where 𝑁_𝑎=^∞0𝑗(𝑎,𝑡)𝑑𝑡(3.49) is the value of the stationary wave function (3.1) at point 𝑁_𝑎=ℏ𝑚Re^∞0𝑑𝑘𝑔(𝑘)𝜓_𝑇(𝑘,𝑎)^∞0𝑑𝑘^𝑘^𝑔𝑘^𝜓_𝑇𝑘^,𝑎^∞0exp−𝑖ℏ2𝑚𝑘²−𝑘^2𝑡𝑑𝑡.(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 𝑡≤0, The normalization integral in (3.45), can be evaluated using (3.42), (3.46), and (3.47): If ||𝜓_𝑇(𝑘,𝑎)||²=⎧⎪⎪⎨⎪⎪⎩4𝑘²𝜒²4𝑘²𝜒²+𝑉²⁰sinh²(𝜒𝑎),𝐸<𝑉⁰4𝑘₂,𝑞²4𝑘²𝑞²+𝑉²⁰sin²(𝑞𝑎),𝐸>𝑉⁰.(3.52) is chosen sufficiently far from the left of the barrier, the contribution of 𝐼_𝑎=^∞0𝑡𝑗(𝑎,𝑡)𝑑𝑡.(3.53) in the integral (3.49) from 𝐼_𝑎=ℏ𝑚Re^∞0𝑑𝑘𝑔(𝑘)𝜓_𝑇(𝑘,𝑎)^∞0𝑑𝑘^𝑔𝑘^𝜓_𝑇𝑘^,𝑎^∞0𝑘^𝑡exp−𝑖ℏ2𝑚𝑘²−𝑘^2𝑡𝑑𝑡,(3.54) is negligible small. In this case, the lower integration limit over t in (3.50) can be taken as 𝐼_𝑎=2𝜋𝑚ℏ^∞0𝑔²⁰(𝑘)1𝑘𝜓¹(𝑘,𝑎)𝜓^2(𝑘,𝑎)−𝜓₂(𝑘,𝑎)𝜓^2(𝑘,𝑎)−𝑥₀||𝜓(𝑘,𝑎)||²𝑑𝑘,(3.55). Then the integration over time gives 𝜓₁(𝑘,𝑎)≡Re𝜓_𝑇(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 ⟨𝑡(𝑎)⟩=𝑚ℏ^∞0𝑔²⁰1𝑘𝜓¹𝜓^2−𝜓₂𝜓^1−𝑥₀||𝜓_𝑇||²𝑑𝑘/^∞0𝑔²⁰||𝜓_𝑇||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 𝐴_𝑅(𝑘)=⎧⎪⎪⎨⎪⎪⎩𝑉⁰sinh(𝜒𝑎)𝐷_<𝑉₀,𝐸<𝑉₀,sin(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=0.01Å⁻¹ and 𝑎 in (3.45) can be obtained only numerically.

Figure 5 shows the results of the calculations of the tunnelling time Δ𝑘=0.05Å⁻¹ as a function of the width a of the barrier, for electrons with energy Δ𝑘=0.1Å⁻¹eV through the rectangular barrier of potential with height Δ𝑘=0.1Å⁻¹eV and Δ𝑡≅Δ₀𝑡1+(Δ𝑣)²𝑚2ℏ²𝑡²^1/2,(3.60). We can see the manifestation of the HFE for Δ₀𝑡∼ℏ/Δ𝐸 (curve (a)), with the asymptotic behavior of the tunneling time approaching the constant value with increasing Δ𝑣=ℏΔ𝑘/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 𝜏^ph for large 𝜏^ph) than the decrease of 𝑗₊(0,𝑡) proportional to 𝑗_in(0,𝑡),𝑗_in. For the center of the transmitted wave packet in point Re[(𝑖ℏ/2𝑚)Ψ_in(𝜕Ψ^∗_in/𝜕𝑥)] such delay has to be considerably smaller because the value Δ𝑣 is smaller than the limited value Δ𝑘.𝑡₀,𝑡^0,𝑡^0 (𝑥=0 being the phase tunnelling time) due to the advance caused by the difference between 𝑡_𝑓,𝑡^𝑓 and 𝑡^𝑓 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 ⟨𝜏_tun(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 ⟨𝜏^Phtun⟩ 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.

Figure 2 

Schematic view of particle scattering and tunneling through a potential barrier.

Figure 3 

Orientations of spin amd magnetic field for the case of the “Larmor clock”.

Figure 4 

The dependences of Δ𝑘=0.01Å, Δ𝑘=0.05Å and Δ𝑘=0.1Å from a.

Figure 5 

The tunneling time 𝑡𝑓,𝑡𝑓, as a function of the barrier depth 𝑡𝑓, for different values of 𝑡0,𝑡0,: (a) 𝑡0, (b) 𝜌(𝑥=𝑎,𝑡), and (c) 𝜌(𝑥=0,𝑡).

Figure 6 

Qualitative description of the time shift between peaks 𝑚 and 𝐸=ℏ2𝑘2/2𝑚=𝑚𝑣2/2, and 𝑉0(𝑉0>𝐸) and 𝑎 for the probability density 𝑙 and −ℏ22𝜕𝜕𝑥22𝜓(𝑥)+𝑉(𝑥)𝜓(𝑥)=𝐸𝜓(𝑥),(3.61) of a wave packet.

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., [118–121]) 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 [122–126]). 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(𝑥≤0),II(0≤𝑥≤𝑎),III(𝑎≤𝑥≤𝑙+𝑎),IV(𝑙+𝑎≤𝑥≤𝑙+2𝑎), and width V(𝑥≥𝑙+2𝑎), the quantity 𝜓𝜓_III=𝑒^𝑖𝑘𝑥+𝐴_1R𝑒^{−𝑖𝑘𝑥},𝜓_III=𝛼₁𝑒^−𝜒𝑥+𝛽₁𝑒^𝜒𝑥,𝜓_IV=𝐴_1T𝑒^𝑖𝑘𝑥+𝐴_2R𝑒^{−𝑖𝑘𝑥},𝜓=_V𝐴_1T𝛼₂𝑒^{−𝜒(𝑥−𝑙−𝑎)}+𝛽₂𝑒^{𝜒(𝑥−𝑙−𝑎)},=𝐴_T𝑒^𝑖𝑘𝑥,𝐴_T≡𝐴_1T𝐴_2T,(3.62) being the distance between them see Figure 7. The stationary Schroedinger equation is where 𝐴_1R,𝐴_2R,𝐴_1T,𝐴_2T,𝛼₁,𝛼₂,𝛽₁ outside the barriers and 𝛽₂ inside the potential barriers. In various regions 𝜓_{I,II,III,IV,V} and 𝑑𝜓_{I,II,III,IV,V}/𝑑𝑥, the solutions of (3.61) are the following: where 𝜒𝑎→∞,, and quantities 𝐴_1R⟶𝑖𝑘+𝜒𝑖𝑘−𝜒,𝐴^1T𝛼⟶𝑒^−𝜒𝑎𝑒^{−𝑖𝑘(𝑙+𝑎)}𝐴,₁⟶2𝑖𝑘𝑖𝑘−𝜒,𝛽¹⟶𝑒^−2𝜒𝑎(𝑘−𝑖𝜒)sin𝑘𝑙𝐴𝜒𝐴,_2R⟶𝑒^{2𝑖𝑘(𝑙+𝑎)}𝑖𝑘+𝜒𝑖𝑘−𝜒,𝐴^2T⟶𝑒^{2𝑖𝑘(𝑙+𝑎)}−4𝑖𝑘𝜒(𝑖𝑘−𝜒)²𝛼₂,⟶𝑒^{𝑖𝑘(𝑙+𝑎)}2𝑖𝑘𝑖𝑘−𝜒,𝛽²⟶𝑒^{𝑖𝑘(𝑙+𝑎)−2𝜒𝑎}−2𝑖𝑘(𝑖𝑘+𝜒)(𝑖𝑘−𝜒)²,(3.63), and 𝐴=2𝑘𝜒2𝑘𝜒cos𝑘𝑙+𝜒²−𝑘²sin𝑘𝑙.(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 𝜏^Phtun=ℏ𝜕arg𝐴^𝑇𝑒^{𝑖𝑘(𝑙+2𝑎)}𝜕𝐸⟶𝑣2𝜒,(3.65) and their derivatives 𝐴_𝑇(𝑘)=exp(−2𝑖𝑘𝑎)𝐷(𝑘),(3.66) at points 𝐷(𝑘)=cosh²(𝜒𝑎)+14sinh²+𝑖sinh(𝜒𝑎)𝛿cosh((𝜒𝜒𝑎𝑎))+𝜎²1cos(2𝑘𝑙)−𝛿²4𝜎²sinh(𝜒𝑎)sin(2𝑘𝑙),(3.67). The obtained expressions for them for opaque barriers, when 𝛿=(𝜒²−𝑘²)/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.

Figure 7 

Tunneling through two successive potential berriers.

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 cot(𝑘𝑙)=−12𝛿tan(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=cosh²(𝜒𝑎)−14𝛿²𝑣=𝑤𝛿=co1sh(𝜒𝑎)sinshi(n𝜒h𝑎²()𝜒,𝑎),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+2𝑤1+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 𝐷_𝑟=𝐷(𝑘_𝑟) 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 𝐶_𝑟=𝑚ℏ²𝑘𝐷^𝑟,(3.79) (quantity 𝑘 being the resonance value of energy 𝐷(𝑘)=𝐶_𝑟𝐸−𝐸^𝑟+𝐷_𝑟𝐶^∗𝑟||𝐶_𝑟||².(3.80)), we obtain where 𝑢_𝑟, 𝑣_𝑟, and the index prime defines the derivative with respect to 𝐸=𝐸_𝑟. We can rewrite (3.78) in the following form: It follows from (3.70) that at the resonance it is where 𝑤, 𝐷𝑘_𝑟=𝑢^𝑟+𝑖𝑣_𝑟1+𝑤_𝑟.(3.82) and 𝑘 are the values at 𝐷^𝑘_𝑟=𝑢^𝑟−𝑢^𝑟𝑤^𝑟−2𝑙𝑣_𝑟𝑤_𝑟1+𝑤_𝑟+𝑖𝑣^𝑟−𝑣^𝑟𝑤^𝑟−2𝑙𝑢_𝑟𝑤_𝑟1+𝑤_𝑟.(3.83) of the functions 𝑢𝑢^+𝑣𝑣^=(1+𝑤)𝑤^,(3.84), 𝐷_𝑟𝐶^∗𝑟=𝑖𝑚ℏ²𝑘_𝑟𝑢^𝑟𝑣_𝑟−𝑢_𝑟𝑣^𝑟1+𝑤_𝑟+2𝑙𝑤_𝑟,(3.85), and ||𝐶_𝑟||²=𝑚ℏ²𝑘_𝑟²𝑢^𝑟²+𝑣^𝑟²−𝑤^𝑟²+4𝑙𝑢^𝑟𝑣_𝑟−𝑢_𝑟𝑣^𝑟1+𝑤_𝑟𝑤_𝑟+4𝑙²𝑤^2𝑟.(3.86), respectively. On inserting (3.81) into (3.74), we obtain Differentiating (3.74) with respect to ||𝐶_𝑟||²=𝑚ℏ²𝑘_𝑟²𝑢^𝑟𝑣_𝑟−𝑢_𝑟𝑣^𝑟1+𝑤_𝑟+2𝑙𝑤_𝑟.(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=𝐸_𝑟, one can see that where 𝛽=ℏ²𝑘_𝑟𝜒_𝑟𝑚𝜒^𝑟𝑘_𝑟𝑎+2𝜒_𝑟𝑙𝑤_𝑟+𝜎^2𝑟cosh(𝜒_𝑟𝑎)sinh(𝜒_𝑟𝑎)⁻¹,(3.91) and ||𝐶_𝑟||²=1𝛽².(3.92) are the values of ||𝐴_𝑇(𝑘)||²=𝛽²𝐸−𝐸_𝑟²+𝛽².(3.93), and 𝜏^Phtun=ℏ𝑑arg𝐴^𝑇exp𝑖𝑘(2𝑎+𝑙)𝑑𝐸.(3.94) at 𝜏^Phtun=𝑚ℏ𝑘𝑙−𝐷¹𝐷^2−𝐷₂𝐷^1|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 𝜏^Phtun=ℏ𝑑𝑑𝐸arg𝐶1_𝑟exp(𝑖𝑘𝑙)𝐸−𝐸_𝑟+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 𝜒𝑎≫1 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 𝑢^𝑣−𝑢𝑣^=1𝜒(1+𝑤)𝜎²𝑢cosh(𝜒𝑎)sinh(𝜒𝑎)+𝛿𝑘𝑎_,𝑤−𝑢𝑤^=2𝜒𝛿(𝑢+𝑤)+14𝜎²𝑣𝑤_cosh(𝜒𝑎)sinh(𝜒𝑎),−𝑣^𝑤=1𝜒𝑤4−𝛿²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 𝜒𝑎≫1, it holds Differentiating the functions 𝜏^Phtun≈2𝑚ℏ𝑘𝜒+4ℏ𝑚𝑘𝑙exp(−2𝜒𝑎)𝜎4²+1+𝛿4²cos(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 𝜏^Phtun, and far from the resonances, Namely, when 𝜒𝑎 increases, the second term in (3.106) decreases as ||𝐴_𝑇||²≈321𝜎²exp(−4𝜒𝑎)14𝜎²+1−14𝛿²cos(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 [59–62])—also for photons. This can explain the results explaining the experimental fact that |𝐴_𝑇(𝑘)|² has been actually observed to increase (very slowly, almost linearly, and probably with very small oscillations) on exp(−4𝜒𝑎), which was recently found by Longhi et al. [130] and by Nimtz [131].
We would like now to underline that, in quantum experiments, the tunneling-time of a nonrelativistic particle is expected to be practically measurable for large 𝜒𝑎 values only (but not too large, of course, to avoid that the tunneled particles are too few). Therefore, in order to be able to reproduce theoretically any experimental results, it seems to be necessary studying the behavior of the transmission coefficient for large 𝜒𝑎. Indeed, from (3.1) and (3.38) it follows that which shows that, with increasing 𝑙, the transmission coefficient square 𝛼 decreases as 𝑟>𝑅₂, that is, even more quickly than the second term in the right-hand side of (3.106). Consequently, from the experimental point of view, practically no tunneling can take place for those values of 𝑅₁ which make negligible the mentioned second term in (3.106). Moreover, for the values of 𝑅₂ for which the tunneling probability is experimentally significant, the second term in the right-hand side of (3.106), which is slowly exponentially decreasing with 𝑟<𝑅₁, results to depend very weakly on the distance l between the barriers: more precisely, it will depend almost linearly (but very slightly) on 𝑉₁, with even slighter oscillations.

Some Words on Other Papers on Tunneling through A Double Barrier
In addition to [127, 129], in [132, 133] there are studies of the numerical and asymptotic analytical expressions of the wave packets travelled through a double-barrier potential (also not taking into account the multiple internal reflections between two separated barriers). In particular, in [132] the resonant and nonresonant dynamics of a Gaussian quantum wave packet tunneling through a double-barrier system has been analyzed as a function of the initial spectrum characteristics and of the potential parameters. The behavior of the tunneling time shows that there are situations where the Hartman effect occurs, while, when the resonances are dominant, the tunneling time can become very large and the HFE does not take place [133].
The authors of [134] have studied the relativistic quantum mechanical problem of a Dirac relativistic tunneling through two successive barriers, and have shown that in the limit of opaque barriers the generalized HFE also occurs. However, their results for the phase and dwell times show an almost linear increasing with the separation between the barriers and tend to saturate only when the barriers become extremely opaque.

3.8. Particle Tunneling through Three-Dimensional Barriers

Introduction
The particle tunneling through a three-dimensional (3D) barrier has been usually studied in a simplified way (in the framework of the WKB approximation and with using only the elementary time-dependent description) in applications for some concrete tasks such as 𝑟Ψ⁰⁽¹⁾(𝑟,𝑡)=^𝑉1₀𝑑𝐸𝑔(𝐸)𝑒^{−𝑖𝑘𝑟−𝑖𝐸𝑡/ℏ},𝑅₂≤𝑟<∞,(3.108)-decay (see, e.g., [135–140]). Here, we intend to study the nonrelativistic particle tunneling through a 3D potential barrier with a spherical symmetry without WKB approximation, following [141, 142]. We will consider, in the central part of the system, also the presence of a spherical well. We will refer to the various regions in this way: region (I), with 𝑘, represents the external region of null potential, region (II), delimited by 𝐸=ℏ²𝑘²/2𝑚 and 𝑟, is the barrier region, and the internal region (III), with 𝑟Ψ^𝑅(1)(𝑟,𝑡)=^𝑉1₀𝑑𝐸𝑔(𝐸)𝐴^(ex)𝑅𝑒^{−𝑖𝑘𝑟−𝑖𝐸𝑡/ℏ},𝑅₂≤𝑟<∞.(3.109), is the well. We will describe the impact of the particles with this potential as a sequence of two successive processes: in the first stage we think to an ingoing wave packet impinging from outside on the barrier, producing a reflected wave in the external region (I), tunneling through the barrier, and finally penetrating in the well where it is represented by an ingoing mode. In the second phase, we will consider the presence of an outgoing wave from the well (III), which, after a reflection against the internal side of the barrier, tunnels through the barrier and produces, finally, an outgoing mode in the external region (I). The scheme of these processes is sketched in Figure 8.
Multiple reflections inside the barrier can also develop, but, on account of the fact that this phenomenon is more sophisticated and usually present at a low level, we will neglect it in this first approach, considering them only afterwards. The present analysis is a first step for the self-consistent time-dependent study of the emission of protons or alpha-particles from a spherical compound nucleus, or of the photon emission from a glass–air spherical system.

Figure 8 

Schematic description of the impact process.

3.8.1. Model Picture

(i) Impact from Outside
We will start by considering an initial wave packet, defined in the outer region (I) by means of a superposition of ingoing spherical waves, and moving from outside towards the barrier region (II) where the potential has value 𝑟Ψ⁽¹⁾_II(𝑟,𝑡)=^𝑉1₀𝑑𝐸𝑔(𝐸)𝛼₁𝑒^−𝜒𝑟+𝛽₁𝑒^𝜒𝑟𝑒^{−𝑖𝐸𝑡/ℏ},𝑅₁≤𝑟<𝑅₂,(3.110): where 𝑟Ψ^𝑇(1)(𝑟,𝑡)=^𝑉1₀𝑑𝐸𝑔(𝐸)𝐴^(in)𝑇𝑒^{−𝑖𝐾𝑟−𝑖𝐸𝑡/ℏ},0<𝑟<𝑅₁,(3.111) and 𝐾=2𝑚(𝐸+𝑉₀)/ℏ² are the wave number and the kinetic energy, respectively.
A section of the potential along the 𝑟Ψ^𝐽(1)(𝑟,𝑡)=^𝑉1₀𝑑𝐸𝑔(𝐸)Φ^𝐽(1)(𝑘,𝑟)𝑒^{−𝑖𝐸𝑡/ℏ},(3.112)-axis is shown in Figure 9.
When the wave encounters the barrier, it is partially reflected with wave function: At the same time, the wave packet tunnels through the barrier in region (II), where it is represented by the following function: where 𝑅. After the tunnelling through the barrier, the wave penetrates inside the well and in region (III) we can write where ΦΦ⁽¹⁾_I=𝑒^{−𝑖𝑘𝑟}+𝐴^(ex)𝑅𝑒^𝑖𝑘𝑟,𝑅₂≤𝑟<∞,_(1)IIΦ_(1)III=𝛼₁𝑒^−𝜒𝑟+𝛽₁𝑒^𝜒𝑟,𝑅₁≤𝑟≤𝑅₂,=𝐴^(in)𝑇𝑒^{−𝑖𝐾𝑟},0<𝑟≤𝑅₂.(3.113) is the wave number inside the well.
In general, we can say that the wave packet is described by the expression where the index ^𝑉1₀𝑑𝐸||𝑔(𝐸)||²=𝑁,𝑁<∞.(3.114) is I, II, III, 0, 4𝜋^∞𝑅₂𝑟²𝑑𝑟||Ψ⁰⁽¹⁾(𝑟,𝑡)||²=1,(3.115) or 𝑉₁ depending on the particular mode considered, 𝐸<𝑉₁ being the stationary wave functions: Furthermore, 𝑙>0 is a normalized amplitude weight factor such as We normalize (3.1) by the following condition: which is strictly possible only in the limit 𝐴^ex_𝑅,𝛼₁,𝛽₁, going to infinity. For the sake of simplicity we neglect the contribution of the energy above the barrier, considering only cases where 𝐴^(in)𝑇.
All the previous formulas are written in the case of orbital momentum 𝑟=𝑅₂, and the extension of this theory to the general case with 𝑟=𝑅₁ can be obtained by replacing the terms 𝐴^ex_𝑅=−𝑒^{−2𝑖𝑘𝑅2}𝑎^ex_𝑅,𝑎^ex_𝑅=(𝜒+𝑖𝑘)(𝑖𝐾+𝜒)+𝑒^{−2𝜒(𝑅2−𝑅1)}(𝜒−𝑖𝑘)(𝑖𝐾−𝜒)(𝜒−𝑖𝑘)(𝑖𝐾−𝜒)+𝑒^{−2𝜒(𝑅2−𝑅1)}(𝜒+𝑖𝑘)(𝑖𝐾+𝜒),(3.116) and 𝜒(𝑅₂−𝑅₁) with the spherical Hanckel function of the first and second kinds.
In addition, we have that −𝑒^{−2𝑖𝑘𝑅2}((𝜒+𝑖𝑘)/(𝜒−𝑖𝑘)) and 𝐴ⁱⁿ_𝑇=𝑒^{−𝑖𝑘𝑅2+𝑖𝜅𝑅1}𝑎ⁱⁿ_𝑇,𝑎ⁱⁿ_𝑇=4𝑖𝑘𝜒𝑒^{−𝜒(𝑅2−𝑅1)}(𝜒−𝑖𝑘)(𝑖𝐾−𝜒)+𝑒^{−2𝜒(𝑅2−𝑅1)}(𝜒+𝑖𝑘)(𝑖𝐾+𝜒)(3.117) are, respectively, the external reflection amplitude factor, the evanescent and antievanescent wave amplitude factors during the first tunnelling and the internal transmission amplitude factor. Their analytical expression can be found by imposing the continuity condition for both the stationary wave functions and their first derivatives at the points 𝑗⁽¹⁾_I,𝑗⁽¹⁾_II and 𝑗⁽¹⁾_III, finding for the external reflection coefficient the expression: which, in the limit ^∞0𝑗^𝑛(1)(𝑟,𝑡)𝑑𝑡≈^∞−∞𝑗^𝑛(1)(𝑟,𝑡)𝑑𝑡.(3.119) tending to infinity, becomes 𝐸=⟨𝐸⟩,.
For the internal transmission coefficient we get that, in the same limit as before, tends to 0.
The calculation of the probability fluxes can be used for controlling if the quantities (3.116) and (3.117) satisfy the conservation law.
The fluxes 𝛿(𝐸−⟨𝐸⟩),, and ⟨𝐸⟩ in the three regions I, II, and III are in general defined as One can use the approximation For quasimonochromatic wave packet centred around the value ^∞−∞𝑗⁰⁽¹⁾(𝑟,𝑡)𝑑𝑡=^∞−∞𝑗^𝑅(1)(𝑟,𝑡)𝑑𝑡+^∞−∞𝑗^𝑇(1)(𝑟,𝑡)𝑑𝑡(3.121) that is, when the factor ||𝐴^(ex)𝑅⟨𝐸⟩||²+𝐾⟨𝐸⟩𝑘⟨𝐸⟩||𝐴^(in)𝑇⟨𝐸⟩||²=1,(3.122) is a delta function 𝑉₀=0 with ||𝐴^(ex)𝑅⟨𝐸⟩||²+||𝐴^(in)𝑇⟨𝐸⟩||²=1.(3.123) in the open interval 𝜏^ph(in)𝑇 one obtains Hence, from the conservation law for the probability fluxes one obtains that for 𝜏^ph(ex)𝑅 becomes The transmission probability through the barrier from outwards is represented by the last of expressions (3.120).
The phase times 𝜒(𝑅₂−𝑅₁) and 2/𝑣𝜒 can be defined as the evident generalization of the following 1D definitions: where 𝑅₂. The quantities 𝑟Ψ⁽ⁱⁿ⁾(𝑟,𝑡)=^𝑉1₀𝑑𝐸𝐺(𝐸)𝑒^{𝑖𝐾𝑟−𝑖𝐸𝑡/ℏ},0≤𝑟≤𝑅₁(3.125) and 𝑟, in the limit 𝑟Ψ^𝑅(2)(𝑟,𝑡)=^𝑉1₀𝑑𝐸𝐺(𝐸)𝐴^(in)𝑅𝑒^{−𝑖𝐾𝑟−𝑖𝐸𝑡/ℏ},0≤𝑟≤𝑅₁.(3.126) approaching infinity, go to 𝑟Ψ⁽²⁾_II(𝑟,𝑡)=^𝑉1₀𝑑𝐸𝐺(𝐸)𝛼₂𝑒^−𝜒𝑟+𝛽₂𝑒^𝜒𝑟𝑒^{−𝑖𝐸𝑡/ℏ},𝑅₁≤𝑟≤𝑅₂,(3.127) and 0, respectively.
So, we can see from the previous expressions, for analogy with the same 1D quantities, the manifestation of the HFE effect and also the absence of dependence on the geometrical characteristics of the barriers 𝑟Ψ⁽²⁾_I(𝑟,𝑡)=^𝑉1₀𝑑𝐸𝐺(𝐸)𝐴^(ex)𝑇𝑒^{𝑖𝑘𝑟−𝑖𝐸𝑡/ℏ},𝑅₂≤𝑟<∞.(3.128) and 𝐺(𝐸).

Figure 9 

Schematic description of the impact of the wave with the barrier from outside.

(ii) Emission from The Barrier
Now we will study the evolution of a wave coming out from the central core of the system, with a wave function given by constructed from the overlapping stationary solutions propagating in the positive 𝐴^(ex)𝑇-direction from the well region. In Figure 10, the scheme of the various waves is presented.
When the wave impinges the barrier from inside, a reflected wave is formed, whose wave function is Afterwards, in the region (II) of the barrier a system of evanescent and antievanescent waves develops while in the outer region a propagating wave will exist As in the previous case we have that 𝑣=ℏ𝑘/𝑚 is a normalized amplitude weight factor, and 𝜏^ph(ex)𝑇 and 𝜏^ph(in)𝑅 are the internal reflection, the evanescent and antievanescent and the external transmission amplitude factors, respectively. With calculations similar to that of the first part we obtain the explicit expressions of the amplitude factors as follows: Also in this case we can demonstrate the conservation of the current fluxes and we can introduce the phase times: with ΨΨ_III(𝑘,𝑟)=𝑒^{−𝑖𝑘𝑟}−𝑆𝑒^𝑖𝑘𝑟,Ψ_III(𝑘,𝑟)=𝛼𝑒^−𝜒𝑟+𝛽𝑒^𝜒𝑟,(𝑘,𝑟)=𝐴𝑒^{−𝑖𝐾𝑟}−𝑒^𝑖𝐿𝑟.(3.133). The quantities 𝑆 and 𝛼, in the limit 𝛽 approaching infinity, go to 𝐴 (with 𝑟=𝑅₁) and 0, respectively.

Figure 10 

Schematic view of the second phase of the scattering process, the emission from inside.

(iii) Scattering Matrix
Finally, we will connect the two mechanisms of scattering described above in one single scattering event (see Figure 11), introducing the matrix of scattering 𝑟=𝑅₂ and considering the multiple reflections inside the potential well.
For this purpose we describe the stationary wave functions in the various regions as follows: We can now find |𝑆|=1, 𝐴=𝐴^(in)𝑇1+𝐴ⁱⁿ_𝑅𝑆=−𝐴_(ex)𝑅,(3.136)−𝑖𝐴𝐴^(ex)𝑇=−𝐴^(ex)𝑅+𝐴^(ex)𝑇𝐴^(in)𝑇1+𝐴^(in)𝑅.(3.137), 1/(1+𝐴ⁱⁿ_𝑅), and 𝐴^(in)𝑇1−𝐴^(in)𝑅+𝐴^(in)𝑅²−𝐴^(in)𝑅³+⋯𝑒^{−𝑖𝑘𝑟}=𝐴^(in)𝑇1+𝐴^(in)𝑇𝐴_(in)𝑇𝑒^{−𝑖𝑘𝑟},1−𝐴^(in)𝑅+𝐴^(in)𝑅²−𝐴^(in)𝑅³+⋯𝑒^𝑖𝑘𝑟=𝐴^(in)𝑇1+𝐴^(in)𝑇𝑒^𝑖𝑘𝑟(3.138) by connecting the various expression of the wave function and its derivative in 𝜏^phsc=ℏ(𝜕arg𝑆𝑒^{𝑖𝑘𝑅2}/𝜕𝐸)−(−𝑅₂/𝑣) and 𝜒(𝑅₂−𝑅₁).
Imposing the continuity conditions we have One can easily see that 𝑆, and that The physical meaning if the term 𝐴₁=𝐵₁+𝑒^{−2𝜒(𝑅2−𝑅1)}𝐵₂ is directly connected to the presence of an infinite sequence of multiple internal reflections that can be described by the stationary wave functions for the ingoing and the outgoing waves, respectively.
The scattering phase time 𝐵₁=𝐾cos(𝐾𝑅₁)+𝜒sin(𝐾𝑅₁) that in the limit 𝐵₂=𝐾cos(𝐾𝑅₁)−𝜒sin(𝐾𝑅₁) approaching infinity goes to 𝑆. So in this limit, the scattering phase time coincides with (𝐸−𝐸_𝑟),𝐸_𝑟.

Figure 11 

Schematic view of the whole scattering process.

Resonances
In all the previous analysis we have disregarded the possibility that resonances develop during tunneling and scattering. To take into account the insurgence of resonances we rewrite the scattering matrix 𝐴₁(𝐸_𝑟)=0 in the following form: where, from the comparison with (3.134), we set: 𝑆=𝛼𝐸−𝐸^𝑟−𝑖Γ/2𝐸−𝐸_𝑟+𝑖Γ/2,(3.141), Γ2=−𝑘𝐸^𝑟𝐴₂𝐸_𝑟𝜒𝐸_𝑟𝜕𝐴₁/𝜕𝐸_{𝐸=𝐸𝑟}=2𝑘𝐸^𝑟𝑒^{−2𝜒(𝐸𝑟)(𝑅2−𝑅1)}𝐵₂𝐸_𝑟𝜒𝐸_𝑟𝜕𝐴₁/𝜕𝐸_{𝐸=𝐸𝑟}.(3.142), 𝜒(𝑅₂−𝑅₁), Γ2=2𝑘𝐸^𝑟𝑒^{−2𝜒(𝐸𝑟)(𝑅2−𝑅1)}𝐹𝐸_𝑟,(3.143), 𝐹𝐸_𝑟=ℏ𝐾𝐸^𝑟𝜒𝐸_𝑟√𝑚/21/√𝑉₀+𝐸_𝑟1+𝜒𝐸_𝑟𝑅₁+1/√𝑉₁−𝐸_𝑟1+𝐾𝐸_𝑟𝑅₁.(3.144).
In the region of a resonance, we can develop 𝑉=𝑍¹𝑍₂𝑒²𝑟,(3.145) into a series of powers of 𝑍₁𝑒 being the eigenvalue of the energy for the resonance, considered as a solution of the transcendental equation 𝑍₂𝑒, and then we obtain the following: that shows better its resonant character if written in the form with If 𝜂=𝑍₁𝑍₂𝑒²𝑚/ℏ²𝑘 is very large, we can neglect all the terms containing the negative exponential factor and write finally with

The Case of A Rectangular Well with A Coulomb Barrier
In this case we can initiate from a simple Coulomb barrier (|𝑆|=1 and 𝐴=2𝑖𝑒^{𝑖𝑘𝑅2}𝑘𝐺^0𝑘,𝜂,𝑅₁+𝑖𝐹^0𝑘,𝜂,𝑅₁𝑘1−𝑒^{2𝑖𝐾𝑅1}+𝐺₀𝑘,𝜂,𝑅₁+𝑖𝐹₀𝑘,𝜂,𝑅₁𝑖𝐾1+𝑒^{2𝑖𝐾𝑅1}(3.148) are the charges of the daughter nucleus and the emitted particle, resp.) instead of a rectangular potential barrier. Instead of the functions 𝐹^0 we have to use the coulomb functions 𝐺^0 in the field of the coulomb barrier with (where 𝐺₀ is the Sommerfeld parameter, 𝑘𝑅₁).
In this case, taking also 𝐴^ex_𝑅=−𝐺⁰𝑘,𝜂,𝑅₁+𝑖𝐹₀𝑘,𝜂,𝑅₁𝑖𝐾−𝐺^0𝑘,𝜂,𝑅₁+𝑖𝐹^0𝑘,𝜂,𝑅₁𝑘𝐴𝐺₀𝑘,𝜂,𝑅₁+𝑖𝐹₀𝑘,𝜂,𝑅₁𝑖𝐾+𝐺^0𝑘,𝜂,𝑅₁+𝑖𝐹^0𝑘,𝜂,𝑅₁𝑘(3.149)_in𝑇=2𝑖𝑘𝑒^{𝑖𝐾𝑅1}𝐺₀𝑘,𝜂,𝑅₁+𝑖𝐹₀𝑘,𝜂,𝑅₁𝑖𝐾+𝐺^0𝑘,𝜂,𝑅₁+𝑖𝐹^0𝑘,𝜂,𝑅₁𝑘,(3.150), we obtain (one can easily see that here 𝐹₀𝐺^0−𝐺₀𝐹^0=1 too) for the scattering (see Figure 12). Here, 𝑘 and 𝑘→0 signify the derivatives of 2𝜂≫𝑘𝑅₁ and 𝐺₀⟶2𝜌𝜋^1/2𝐺𝐼₀2(2𝜋𝜌)^1/2exp(𝜋𝜂),with𝐼₀2(2𝜋𝜌)^1/2⟶1,_0⟶−22𝜂𝜋^1/2𝐾₀2(2𝜋𝜌)^1/2exp(𝜋𝜂),with𝐾₀2(2𝜋𝜌)^1/2⟶ln1𝐹₀𝛾(2𝜋𝜌)^1/2,𝐹_0⟶(𝜋𝜌)^1/2𝐼₁2(2𝜋𝜌)^1/2exp(−𝜋𝜂),with𝐼₁2(2𝜋𝜌)^1/2⟶(2𝜋𝜌)^1/2,⟶(2𝜋𝜂)^1/2𝐼₀2(2𝜋𝜌)^1/2exp(−𝜋𝜂),(3.153) with respect to 𝛾≈1.781…, respectively.
Similarly, in this case we also obtain for the impact from outside (see Figure 13), and for the emission from the barrier (see Figure 14). In the derivation of (3.147), (3.148), (3.150), and (3.152) we had used the known relation 𝑅₁ for the wronskian.
It is easy to be convinced that relations (3.122), (3.131), (3.136), and (3.137) are also valid for a coulomb barrier. We can also repeat the same reasonings on the physical meaning of the term → which has been made in connection with the formula (3.137).
For very small 𝛼→𝛽 when 𝑓_𝛼𝛽(𝐸) (more precisely, when 𝑓_𝛼𝛽(𝐸)=𝐶^𝛼𝛽𝐸−𝐸_𝑟+𝑖Γ/2,(4.1)), 𝐸 being the Euler constant, and if (𝐸_𝑟−Γ/2,𝐸_𝑟+Γ/2),𝐸_𝑟, the transmission (penetration) probability from outside through the Coulomb barrier into the internal rectangular potential well Γ becomes which contains the same exponential factor 𝑔(𝐸) as the known quasiclassical (WKB) approximation, containing nevertheless the other pre-exponential factor. It is quite understandable because the quasiclassical approximation is not applicable near the point Δ𝐸 where there is an abrupt potential change and one cannot use the approximate notion of the turning point.

Figure 12 

Schematic stationary description of a scattering in the presence of a coulomb barrier (l = 0).

Figure 13 

Schematic description of the impact of a wave packet with a Coulomb barrier from outside.

Figure 14 

Schematic view of the emission from inside through a coulomb barrier.

3.9. An Odd Description of Tunneling Phenomena

For the sake of the completeness, it is possible to mention the quaternion description of tunneling phenomena. In [143–145], the new quaternion description of tunneling phenomena has initiated (the quaternion description of quantum mechanics one can see, e.g., in [146]). The authors show a noteful difference between the complex and quaternionic formulations of tunneling phenomena which could be matter of further theoretical discussions and could represent the starting point for a possible experimental investigation.

4. Applications for Nuclear Reactions and Decays

4.1. Narrow Resonance Δ𝐸≪𝐸_𝑟),𝑚_𝛽 Exponential Decay

Let us firstly explain how a typical isolated Lorentzian (Breit-Wigner) resonance, in the cross-section of a quantum collision or nuclear reaction 𝑟_𝛽, is connected with an exponential law of the decay function of the correspondent compound or radioactive nucleus (somewhat generalizing the similar derivation from [147] and mainly following [148]). We represent the reaction amplitude 𝑘=√2𝑚_𝛽𝐸/ℏ as where 𝑇_𝛼𝛽(𝐸)=𝑁_𝛽(𝐸)𝑓_𝛼𝛽(𝐸),𝑇_𝛼𝛽(𝐸) is a constant or a smooth function of the final-particle kinetic energy 𝑇 in the region 𝑆 and 𝑇_𝛼𝛽=𝛿_𝛼𝛽−𝑆_𝛼𝛽,𝑁_𝛼𝛽(𝐸) being the resonance energy and width, respectively.

The final-particle wave packet in the 1D radial asymptotic limit is described by where 𝑧_𝛽≥𝑅_𝛽,𝑅_𝛽 is a smooth weight amplitude with an energy spread 𝑧_𝛽=𝑅_𝛽 (usually Ψ_𝛽𝑧_𝛽,𝑡=^∞0𝑑𝐸𝑔(𝐸)𝑇_𝛼𝛽(𝐸)exp−𝑖𝐸𝑡ℏ,(4.4) and 𝑇_𝛼𝛽(𝐸)=𝑇_𝛼𝛽exp(𝑖𝑘𝑅_𝛽) are the final-particle mass and radial coordinate, respectively, and 𝐸. For short-ranged interactions (including also screened Coulomb potentials), it can be rewritten as with 𝑇_𝛼𝛽(𝐸) is the exp(−𝑖𝑘𝑅_𝛽)-matrix elements connected with the exp(𝑖𝑘𝑅_𝛽)-matrix elements by known relation 𝑇_𝛼𝛽(𝐸) is an unessential smooth function of Γ≪Δ𝐸≪𝐸_𝑟,(4.5) is the axis along the direction of the final-particle emission imposed by the registration geometry, Ψ_𝛽𝑅_𝛽,𝑡=𝐴^∞0𝑑𝐸exp[−𝑖𝐸𝑡/ℏ]𝐸−𝐸_𝑟+𝑖Γ/2,(4.6) is the interaction radius in the final channel. In the simplest case one can fix 𝐴 and where Ψ_𝛽𝑅_𝛽,𝑡=⎧⎪⎨⎪⎩𝐵exp−𝑖𝐸^𝑟𝑡0,ℏ−(Γ/2ℏ)𝑡,ffoorr𝑡𝑡><00,,(4.7) is a smooth function of −∞: in accordance with the analytical 𝐵-matrix theory, 𝑡−𝑡_in contains the factor 𝑡_in=ℏ(𝜕arg𝑔/𝜕𝐸) and consequently this factor is being cancelled by 𝑡 in Γ∼𝐸^1/2. For condition one can rewrite (4.4) in the following simplified form: where 𝛽 is a constant. In the approximation (4.5) and 𝑧_𝛽 constant one obtain (moving the lower integration limit in (4.15) from 0 to 𝑗_𝛽(𝑧_𝛽,𝑡)=Re[Ψ_𝛽(𝑧_𝛽,𝑡)(𝑖ℏ/2𝑚)𝜕Ψ^∗𝛽(𝑧_𝛽,𝑡)/𝜕𝑧_𝛽] and utilizing the residue theorem). Here ∫^∞−∞𝑗_𝛽(𝑧_𝛽,𝑡)𝑑𝑡=1 is a constant and more precisely there must be 𝑧_𝛽=𝑅_𝛽 (with 𝐼(𝑡)=𝑗^𝛽𝑅_𝛽,𝑡∫^∞−∞𝑑𝑡𝑗_𝛽𝑅_𝛽,𝑡(4.8)) instead of 𝐼(𝑡)=Γℏexp−Γℏ𝑡.(4.9). The form (4.7) is valid also with slight modifications in the cases when lim_{𝑧𝛽→𝑅𝛽}−𝑚𝑖ℏ_𝛽𝑇^𝛼𝛽𝜕exp(𝑖𝑘𝑧_𝛽)𝜕𝑧_𝛽=𝑣𝑇_𝛼𝛽(4.10) or (𝑣=ℏ𝑘/𝑚_𝛽). is a linear function of Ψ_𝛽(𝑅_𝛽,𝑡) (see, e.g., [149]).

The evolution of the particle Ψ_𝛽 passing through position ^∞0𝑑𝑡Ψ_𝛽𝑅_𝛽,𝑡exp−𝑖𝐸𝑡ℏ=𝐵^∞0𝑑𝑡exp−𝑖𝐸−𝐸_𝑟𝑡/ℏ−Γ𝑡/2ℏ=𝑖𝐵𝐸−𝐸_𝑟+𝑖Γ/2,(4.11) during the unitary time interval, centered at 𝑧_𝛽>𝑅_𝛽, is described by the probability flux density Ψ_𝛽𝑧_𝛽,𝑡=^∞0𝑑𝑘𝐺(𝑘)𝐷(𝑘)𝑘−𝑘₀𝑘+𝑘₀exp𝑖𝑘𝑧^𝛽−𝑖𝐸𝑡ℏ,(4.12) with the adequate normalization 𝐺(𝑘)=𝑔ℏ²𝑘²𝐷(𝑘)=2𝑚2_𝛽𝑚_𝛽𝑑𝑑𝐸𝑘,ℏ²𝑁^𝛽𝑘₀(𝐸)𝐶_𝛼𝛽,=1√2𝑘𝑘^4𝑟_𝑟+𝛾²+𝑘^2𝑟−𝑖𝑘^4𝑟+𝛾²−𝑘^2𝑟,=√2𝑚_𝛽𝐸_𝑟𝛾=Γ𝑚ℏ_𝛽,ℏ².(4.13). The emission probability (per a time unit) in the vicinity of the compound nucleus (near 𝐺(𝑘)) is equal to (On the presence of the violations of (4.9) for very small t and for very large t (see, in particular, in [57]).) In obtaining (4.9) we took into account that 𝑦=𝑖ℏ𝑡2𝑚_𝛽𝑘−𝑚^𝛽𝑧_𝛽ℏ𝑡,(4.15) If Ψ_𝛽𝑅_𝛽,𝑡=⎧⎪⎨⎪⎩0,for𝑧^𝛽const⋅exp𝑖𝑘_𝑟>𝑣_𝑟𝑡,𝑧_𝛽−𝑖𝐸^𝑟𝑡ℏ−2Γℏ𝑡−𝑣_𝑟𝑧^𝛽,for𝑧^𝛽≤𝑣_𝑟𝑡,(4.16) has a form (4.7), the Fourier transform of 𝑣_𝑟=ℏ𝑘/𝑚_𝛽 is equal to which is proportional to the amplitude from (4.1).

For 𝛽 one can rewrite (4.3) in a following way: with Since Δ𝐸≪Γ.(4.17) and Δ𝐸 are smooth functions of 𝑔(𝐸). under the condition (4.5), then, introducing in (4.14) a new variable and performing the transformations quite similar to those which were used in [30–32], it is possible to obtain with Δ𝐸. The wave function (4.16) can be applied for macroscopic distances Ψ_𝛽𝑅_𝛽,𝑡=⎧⎪⎪⎨⎪⎪⎩0,for𝑧^𝛽const>𝑣_𝑟𝑡−𝑡⁰_in;𝐸−𝐸_𝑟+𝑖Γ/2⋅exp𝑖𝑘^𝑟𝑧_𝛽−𝑖𝐸^𝑟𝑡ℏ−Δ2ℏ𝐸𝑣𝑡_𝑟−−𝑧𝑡^𝛽0_in,for𝑧^𝛽≤𝑣_𝑟𝑡−𝑡⁰_in(4.19), near a detector which registers particles 𝑣=ℏ𝑘/𝑚_𝛽.

Let us remember that an exponential law (4.9) and also the asymptotic (4.16) are valid only under conditions (4.5), that is, when all energies (or continuum states) around 𝑡⁰_in=ℏ(𝜕arg𝑔₀/𝜕𝐸) are completely populated in the large region with the width 𝜎_𝛼𝛽. If, on the contrary, The emission probability is nonexponential and does essentially depend on 𝑡_min=𝑧_𝛽/𝑣+𝑡⁰_in,(𝑡_min,∞) and the form of 𝜎_𝛼𝛽=||𝑓_𝛼𝛽||²=const𝐸−𝐸_𝑟²+Γ²/4.(4.21) If one will take the Lorentzian form also for 𝐹(𝐸,𝜃), that is, with 𝑓(𝐸,𝜃)=𝑓_coul(𝐸,𝜃)+(2𝑖𝑘)⁻¹_𝜆_≠𝑙𝑓(2𝜆+1)𝑃_𝜆(cos𝜃)exp2𝑖𝜂_𝜆exp(2𝑖𝛿^𝑏𝜆−1_𝑙,res,(𝐸,𝜃)=(2𝑖𝑘)⁻¹(2𝜆+1)𝑃_𝜆(cos𝜃)exp2𝑖𝜂_𝜆exp2𝑖𝛿^𝑏𝑙𝐸−𝐸^res−𝑖Γ/2𝐸−𝐸_res+𝑖Γ/2−1(4.23) const or smooth inside 𝛿^𝑏𝑙 under conditions (4.5) and (4.17), instead of (4.7) the expression (with 𝑙 and 𝑙) will be obtained. The cross-section 𝜃, which is proportional to where 𝐵𝐴==𝑓𝑓((𝐸𝐸,,𝜃𝜃))++((𝑖𝑘𝑘))⁻¹₋₁(2𝑙+1)𝑃_𝑙(cos𝜃)exp2𝑖𝜂_𝑙+𝑖𝛿^𝑏𝑙sin𝛿^𝑏𝑙,(2𝑙+1)𝑃_𝑙(cos𝜃)exp2𝑖𝜂_𝑙+𝑖𝛿^𝑏𝑙cos𝛿^𝑏𝑙,(4.25) being the operative registration time interval of detector, after integrating in (4.20) acquires the Breit-Wigner form

4.2. The Real Possibility of the Phenomenon of the Delay-Advance in Proton Scattering by Nuclei

Near an isolated resonance the proton-nucleus (or nucleus-nucleus) scattering amplitude 𝜏(𝐸,𝜃)=2𝑅𝑣+ℏ𝜕𝜕a𝐸rg𝐹≡2𝑣𝑅+Δ𝜏(𝐸,𝜃),(4.26) is with is the Coulomb scattering amplitude, Δ𝜏(𝐸,𝜃)=ℏΓ2𝐸−𝐸^res²+Γ²4⁻¹−ℏRe𝛼2𝐸−𝐸^res−Im𝛼2²+(Re𝛼)²4⁻¹(4.27) and 𝛼=Γ𝐵/𝐴 being the background nuclear 0<Re(𝐵/𝐴)<1-scattering phase-shift and the Coulomb |Im𝛼/2|≪𝐸_res-scattering phase shift, respectively, k is the wave number, 𝐸=𝐸_res is the scattering angle in the c.m.s. Rewriting (4.22) in the form where we obtain (as it was firstly made in [150]) the following expression for the quasimonochromatic total scattering duration 𝐸: where Δ𝜏 is the projectile velocity and ¹²C is the interaction radius, is the time delay, Δ𝐸≫Γ,𝐷. When Δ𝐸,Γ and 𝐷, near 𝑓_𝛼𝛽(𝐸) the time delay 𝑓_𝛼𝛽(𝐸)=𝐶^𝑛𝛼𝛽exp−𝐸𝜏^𝑛2ℏ+𝑖𝐸ℏ𝑡^𝑛,(4.28) can become negative, that is, an advance can appear instead of a delay.

Because of the existing Coulomb barrier, there is a certain indefiniteness in the choice of the exact value of the physical interaction radius 𝜏_𝑛; it lies between the minimal value, determined by the equality of the energy 𝑡_𝑛 and the Coulomb barrier at the point of the exit of the final charged particle from the barrier, and the maximal value, determined by the practical vanishing of the external tail of the (screened) Coulomb barrier or by the evident causality condition 𝜏_𝑛 in the case of negative values of 𝑡_𝑛.

The time analysis of proton scattering by nuclei 𝐶^𝑛𝛼𝛽 and Δ𝐸 at the range of isolated resonances distorted by the nonresonant background, with the help of the experimental study of the interference in the accompanied bremsstrahlung, had resulted in [151, 152] the revealing of the real possibility of such delay-advance phenomenon.

4.3. The Phenomenon of Time Resonances (Explosions)

Sometimes, in the cases of the dense and strongly overlapper resonances in high-energy nuclear reactions, when Δ𝐸≪2ℏ𝜏_𝑛(4.29) (𝜎_𝛼𝛽=||𝑓_𝛼𝛽||²=const⋅exp−𝐸𝜏^𝑛𝐼(𝑡)=𝜏_𝑛ℏ,(4.30)2𝜋1𝑡−𝑡_𝑛²+𝜏^4𝑛/4,(4.31), and 𝐿^𝑐(𝑡)=1−^𝑡(0)𝑑𝑡𝐼(𝑡).(4.32) being the experimental energy resolution, mean reasonance width and mean distance between resonances, resp., for any contributed resonant spin, parity and total moment quantum numbers), it is possible to approximate the reaction amplitude 𝐿^𝑐(𝑡) by the form where 𝐿^𝑐(𝑡) and 𝐼(𝑡) are constants (with the time dimension), 𝑓_𝛼𝛽 and 𝑓_𝛼𝛽(𝐸)=^𝑣_𝑛=1𝐶^𝑛𝛼𝛽exp−𝐸𝜏^𝑛2ℏ+𝑖𝐸ℏ𝑡^𝑛,(4.33) determine the exponential dependence on energy for the correspondent cross-section and the linear dependence on energy for the amplitude phase, respectively, (𝑣=2,3,…), is a constant or a very smooth function (inside 𝜎_𝛼𝛽=|𝑓_𝛼𝛽|²) of the final-particle energy E, then the correspondent cross-section and the emission probability under the condition will be respectively, (the detailed physical and mathematical justification of the form like (4.28) see, e.g., in [148]). The evolution of compound-nucleus surviving (at instant t during the life and decay after the formation) can be described by the following function: From (4.31)-(4.32), one can deduce the strongly nonexponential form of Δ𝐸 and Δ𝐸𝑡_𝑛≪𝐸𝜏_𝑛andΔ𝐸𝜏_𝑛≪𝐸𝑡_𝑛 like depicted in Figure 15.

Figure 15 

𝜎inc,𝑘𝐸𝑘=2𝑛=1||||𝐶𝑛exp𝑖𝑡𝑛−𝜏𝑛2𝐸ℏ𝑘||||2=2𝑛=1||𝐶𝑛||2exp−𝐸𝑘𝜏𝑛ℏ+2Re𝐶∗1𝐶2exp𝑖𝑡2−𝑡1−𝜏1+𝜏22𝐸ℏ𝑘.(4.35) (curve 1) and 𝜎inc,𝑘(𝐸𝑘) (curve 2).

When p+¹²C→⁷Be has a more general form like with several terms 𝐶₁=0.35,𝐶₂=0.05(𝜃=160^∘) the cross-section 𝑡̂=−𝑖ℏ2^↔𝜕𝜕𝐸(A.1) contains not only exponentially decreasing terms but also oscillating terms with factors (𝑓,𝑡𝑔̂)≡(𝑓,(−𝑖ℏ/2)(𝜕/𝜕𝐸)𝑔)+((−𝑖ℏ/2)(𝜕/𝜕𝐸)𝑓,𝑔) and 𝐸=0. In the case of two terms (𝑓,𝑡𝑓̂) in (4.33) formula (4.30) can be rewritten as where the terms with 𝜕𝜌^→𝑟,𝑡𝜕𝑡+div𝑗^→𝑟,𝑡=0(B.1) are neglected if the conditions ^→𝑟 are supposed.

In particular, for inclusive energy spectra of the kth final fragment we will use the following expression: In Figure 16, an example of the calculated inclusive energy spectra ^→𝑟={𝑟,𝜃,𝜑}, in arbitrary units and in semilogarithmic scale, is presented in comparison with the experimental data (see from [148]).

(a) 𝑦,𝑧

(b) 𝑟

Figure 16 

Inclusive process 𝑥 (2.1 GeV protons).

Appendices

A. On the Bilinear Time Operator

One can rewrite the linear operator (2.1b) into the form of the bilinear operator with div𝑗(^→𝑟,𝑡) and without changing the form (2.1a). Such bilinear form of the time operator had been firstly introduced in [12–14]. By adopting expression (A.1) for the time operator, the point 𝜕𝑗_𝑥(^→𝑟,𝑡)/𝜕𝑥 happens to be automatically emiminated both in (1/𝑟²)𝜕[𝑟²𝑗_𝑟(^→𝑟,𝑡)]/𝜕𝑟 and in the transformation of the integral 𝑊_𝑥^→𝑟,𝑡𝑑𝑡=𝑗^𝑥^→𝑟,𝑡𝑑𝑡∫^∞−∞𝑗_𝑥^→𝑟,𝑡𝑑𝑡(B.2) over time into the integral over energy.

B. The Measure of Averaging over Time in the 3D Case

In the 3D case of the Schroedinger equation, the continuity equation (2.8) in Section 2 has to be rewritten as (see, e.g., [153, 154]), 𝑗_𝑥(^→𝑟,𝑡)being the radius vector of a particle moving in the extential potential or the radius vector for the relative motion of two interacting particles, 𝑗_𝑟(^→𝑟,𝑡) in Cartesian coordinates or 𝑗(𝑥,𝑡) in spherical coordinates. Within the framework of the quantum collision theory the motion direction is often described by any Cartesian axis (let us say, 𝑗_𝑥(^→𝑟,𝑡), with fixed values of 𝑗_𝑟(^→𝑟,𝑡)) or by variable radial coordinate 𝑊_𝑥(^→𝑟,𝑡), with fixed values of angular coordinates 𝑊_𝑟(^→𝑟,𝑡). The term 𝑗_𝑥(^→𝑟,𝑡) acquires the form 𝑗_𝑟(^→𝑟,𝑡) or 𝑊_±(𝑥,𝑡), respectively.

Then, (2.7) from Section 2 has to be substituted by or where the probabilistic interpretations of 𝑗_𝑟,±(𝑥,𝑡)=𝑗_𝑟(𝑥,𝑡)𝜃(±𝑗_𝑟), and 𝑡̂ in time are the same as the probabilistic interpretation, contained in Section 2 for the 1D flux 𝑡²̂, as (one can be convinced in this by the evident generalization).

When the flux density 𝑡𝑡̂𝜑̂^{𝑡𝛿,𝜂}(𝐸)≈𝑡𝜑^{𝑡𝛿,𝜂}(𝐸),₂𝜙^{𝑡𝛿,𝜂}(𝐸)≈𝑡²𝜙^{𝑡𝛿,𝜂}(𝐸),(C.1) (or 𝜑^{𝑡𝛿,𝜂}(𝐸)=𝐶exp𝑖𝐸𝑡ℏ𝑓^𝛿(𝐸)𝑔_𝜂(𝐸),(C.2)) changes its sign, the quantity 𝑡̂ (or 𝑓_𝛿(𝐸)=2sin𝛿𝐸/ℏ𝐸=⎧⎪⎨⎪⎩3𝐸𝐸/ℏ,𝜂²−2𝐸𝜂³1,ffoorr0𝜂≤≤𝐸𝐸,≤𝜂,(C.3)) 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 𝛿 (or exp(𝑖𝐸𝑡/ℏ);) does keep its sign. Therefore, it is possible, quite similarly as 𝑔_𝜂(𝐸) in Section 2, to introduce the two measures, by separating the positive and the negative flux-direction values (i.e., flux signs): with 𝐸>0 and 𝐸≤0. respectively.

C. Approximate Eigenvalues and Approximate Orthonormalized Eigenfunctions of the Time Operator

Following [4, 5] (see also [21, 23–25, 50, 51]), one can specify the following approximate eigenvalues and eigenfunctions of the operator 𝜑^{𝑡𝛿,𝜂}(𝐸) in (2.1a), (2.1b), and (2.26) and simultaneously of the operator 𝑡̂: where C is an arbitrary constant, t is the continuous real eigenvalue of the operator 𝛿𝑡≪𝛿ℏ𝜂^1/2𝛿≪1,𝑡²≪𝛿𝜂ℏ^3/2(C.4), 𝜂→0 is a positive parameter that describes the width of the wave packet formed from the functions 𝐷𝑡=⟨𝜑^{𝑡𝛿,𝜂}(𝐸)||𝑡²̂||𝜑^{𝑡𝛿,𝜂}(𝐸)⟩−||⟨𝜑^{𝑡𝛿,𝜂}(𝐸)|𝑡|̂𝜑^{𝑡𝛿,𝜂}(𝐸)⟩||²(C.5) the sequence of functions 𝜑^{𝑡𝛿,𝜂}(𝐸) has the limit as 𝐶 which is equal to the generalized function ‖𝜑^{𝑡𝛿,𝜂}(𝐸)‖ is equal to 1 for 𝜑^𝛿𝑡(𝐸)=𝐶exp𝑖𝐸𝑡ℏ𝑓^𝛿(𝐸),(C.6) and to 0 for 𝑔_𝜂(𝐸)→Θ(𝐸). As is readily seen by direct calculation of the left-hand side of (C.1), the functions 𝑖ℏ(𝜕/𝜕𝑡) approximate the eigenfunctions of the operartors 𝐻(𝑝_𝑥̂,𝑥̂,…) and 𝐻Ψ=𝑖ℏ(𝜕Ψ/𝜕𝑡) the more accurately, and are the more nearly orthogonal for different 𝐸𝑡̂⟶⟶𝑇𝐻𝑝𝑝_𝑥̂_𝑥̂,𝑥̂,…,,𝑥̂,…,(D.1), the better the relations hold as 𝑇(𝑝_𝑥̂,𝑥̂,…) and 𝐻(𝑝_𝑥̂,𝑥̂,…) and under the fulfilment of the condition (B.4), the variance in the state 𝑇(𝑝_𝑥̂,𝑥̂,…) tends to 0. The constant 𝐻(𝑝_𝑥̂,𝑥̂,…) can be chosen to make the norm 𝑇(𝑝_𝑥̂,𝑥̂,…) equal to 1.

It is curious that the function (C.2) differs from simple wave packets of the form which are typical for “eigendifferentials” in the continuous spectrum of linear self-adjoint operators [54], only for the presence of the factor −𝑖ℏ(𝜕/𝜕𝐸)

D. The Duality of Time and Energy Operators

As it is known, in quantum theory there is a correspondence between energy E and two operators—operator, 𝐸=𝑝^2𝑥/2𝜇 in the t-representation and Hamiltonian operator exp(𝑖𝑘𝑥). The duality of these operators is well seen from the Schroedinger equation 𝑇exp(𝑖𝑘𝑥)=𝑥𝑣exp(𝑖𝑘𝑥),(D.5). The similar duality has to take place for time in quantum theory: besides the general form (2.1a), (2.1b), and (2.26) which is valid for any physical system (in the region of continuous energy spectrum) one can express the time operator in terms of the coordinate and momentum operators using the commutation relation between Hamiltonian operator and time one. So, if one will make the substitution then he will obtain which is similar to (2.24). The commutation relation (D.2) can be used for finding 𝑥̂=𝑖∇_𝑝≡𝑖2^↔∇_𝑝+𝑖2^↔∇^𝑝(+),(E.1) for any concrete physical system with known Ψ∇^∗↔_𝑝Φ≡Ψ^∗∇𝑝Φ−Φ∇𝑝Ψ^∗ (see, e.g., [11]). Choosing the coordinate representation or momentum one for Ψ∇^{∗↔𝑝(+)}Φ≡Ψ^∗∇_𝑝Φ+Φ∇_𝑝Ψ^∗ and 𝑥̂_ℎ≡𝑖2^↔∇_𝑝,(E.2), one does not change the formal expression for Hamiltonian Ψ,𝑥̂Φ=𝑖𝑑³𝑝𝑝₀Ψ^∗(𝑝)∇_𝑝Φ(𝑝),(E.3) but does change the sign in the formal expression for time operator ReΦ,𝑥̂Φ=𝑖𝑑³𝑝𝑝₀Φ^∗(𝑝)∇^↔_𝑝Φ(𝑝),(E.4). It can be easily seen for a free particle: (in the symmetrized form). By the way, formula (D.4b) is equivalent to ⟨Ψ||𝑥̂||Ψ⟩=^→𝛼+Δ^→𝛼+𝑖^→𝛽+Δ^→𝛽,(E.7) (since Δ^→𝛼) and therefore is also a maximal Hermitian operator.

For plane-wave state Δ^→𝛽 in both representations (D.4a) and (D.4b) we obtain the same result: 2𝑖^↔𝜕𝜕𝑝_𝑖,𝑖2^↔𝜕₍₊₎𝜕𝑝_𝑗=2𝑝𝑖²⁰𝛿^𝑖𝑗−2𝑝_𝑖𝑝_𝑗𝑝²⁰,(E.8) being the time of a free motion with the velocity v over the distance x.

E. On Four-Position Operators in Quantum Field Theory

E.1. The Klein-Gordon Case: Three-Position Operators

The usual position operators, being Hermitian and, moreover, self-adjoint, are known to possess real eigen-values: that is, they yield a point-like localization. It is possible [49] to split operator Δ𝛼_𝑖⋅Δ𝛽_𝑗≥14𝛿^𝑖𝑗−2𝑝_𝑖𝑝_𝑗/𝑝²⁰𝑝²⁰_Ψ.(E.9) into two bilinear parts as follows: where 𝑥̂^𝜇ℎ≡−𝑖2^↔𝜕𝜕𝑝_𝜇,(E.10) and 𝑝^𝜇ℎ̂≡+𝑖2^↔𝜕𝜕𝑥_𝜇.(E.11), and where we always referred to a suitable space of wave packets (see, e.g., [12–14, 155–161]). Its Hermitian part [12–14, 155–161] which was expected to yield an (ordinary) point-like localization, was derived also by writing explicitly and imposing hermiticity, that is, the reality of the diagonal elements. The calculation yielded just suggesting to adopt the Lorentz-invariant quantity (E.2) as a bilinear Hermitian position operator. Then, integrating by parts (and due to the vanishing of the surface integral), we verified that (E.2) is equivalent to the ordinary Newton-Wigner operator

We were left with the bilinear anti-Hermitian part whose average values over the considered state (wave packet) were regarding as yielding [66] the sizes of an ellipsoidal localization region.

In general, the extended-type position operator (0,𝑎) will give where 𝑥>𝑎 and 𝜓(𝑘,𝑥) are the mean-errors encountered when measuring the point-like position and the sizes of the localization region, respectively. It is to evaluate the commutators 𝜓≡⎧⎪⎪⎨⎪⎪⎩𝜓^I𝜓=𝜓_in+𝜓_𝑅,𝑥<0,_II𝜓,0<𝑥<𝑎,_III=𝜓_𝑇,𝑥>𝑎,(F.1): where from the noticeable “uncertainty correlations” follow:

E.2. Four-Position Operators

It is tempting to propose as four-position operator the quantity 𝐸, whose Hermitian (Lorentz-covariant) part can be written as to be associated with its corresponding “operator” in four-momentum space

Let us firstly recall the proportionality between the 4-momentum operator and the 4-current density operator in the chronotopical space

Then, let us recall the canonical correspondence (in the 4-position and 4-momentum spaces, resp.) and introduce the 4-position operators in the 4-momentum space (cf. the previous section):

Now, recalling the properties of the time operator as a maximal Hermitian operator in the nonrelativistic case, considered in Section 2.1, we can be easily convinced that the relativistic time operator (E.13a) (for the Klein-Gordon case) has to be also self-adjoint bilinear operator for continuous energy spectra and a maximal Hermitian linear operator due to the boundedness by zero from below for the kinetic energy (or by ∞ from below for the total energy ) for the free particle.

Finally, comparing (E.12) and (E.13a) and (E.13b), we can conclude that the four-position “operator” (E.13a) and (E.13b) can be regarded as a 4-current density operator in the energy-momentum space [12–14].

Of course, similar considerations and conclusions can be carried on for the anti-Hermitian parts also [12–14].

F. On Multiple Internal Reflections during Tunnelling through a Barrier

Introduction
The analysis of multiple internal reflections inside potentials has been considered since a long (see, e.g., [162–164]). The problem is rather trivial for attractive potentials and for over-barrier energies in potential barriers, but things change drastically for under-barrier energies, namely for tunneling. Indeed, in this case decreasing (evanescent) and increasing (antievanescent) waves separately correspond to zero—both stationary and (time-averaged) nonstationary—fluxes. Nonzero fluxes correspond only to linear combinations of both decreasing and increasing waves. As a consequence, evanescent and antievanescent waves cannot be regarded as physical propagating waves. This circumstance was overlooked anywhere up to now. It implies that a physical treatment of the problem requires a description of the tunneling in terms of wave packets.
Analyzing multiple internal reflections in particle and photon tunneling, we will follow [165, 166].

Evolution of Particle Tunneling through A 1d Rectangular Potential Barrier
We confine ourselves to the simplest case of particles moving along the 𝑔(𝐸−𝐄)-direction and tunneling through a rectangular potential barrier of height Ψ(𝑥,𝑡)=𝑑𝐸𝑔(𝐸−𝐄)𝜓(𝑘,𝑥)exp−𝑖𝐸𝑡ℏ,(F.2) in the interval ∫𝑑𝐸|𝑔(𝐸−𝐄)|²=1 (see Figure 1) where by I, II, and III we label the regions 𝐄 and (𝜓_in), respectively. As usually [26, 27, 32, 47], we will use the following expression for the stationary wave function (𝜓_𝑅) with where 𝐴_𝑅, and 𝐴_𝑇 and 𝑔𝐸−𝐄⟶𝐺𝐸−𝐄≡𝑔𝐸−𝐄Θ𝐸−𝑉₀,(F.3) are the particle kinetic energy and mass, respectively. The coefficients (amplitudes) Θ(𝑧) and 𝜓(𝑘,𝑥) can be analytically calculated and are well known.
The tunneling evolution has to be described by the nonstationary description of actually moving wave packets. These are built up in terms of the solutions 𝑥=0 of the stationary Schroedinger equation by using the resolution of the evolution operator, namely, by integrating 𝑥=𝑎 over E from 0 to 𝐴_𝑅 with a weight amplitude 𝐴_𝑇, where we assume the normalization condition 𝛽, quantity 𝜓(𝑘,𝑥) being the average kinetic energy.
By inserting in the integral (F.2) the initial Ψ(𝑥,𝑡),, reflected 𝑥=0 and transmitted (Ψ_in(𝑥,𝑡)=𝑑𝐸𝐺𝐸−𝐄𝜓_in(𝑘,𝑥)exp−𝑖𝐸𝑡ℏ,𝑥<0,(F.4)) wave, instead of the total wave Ψ^ex_𝑅(𝑥,𝑡)=𝑑𝐸𝐺𝐸−𝐄𝜓^ex_𝑅(𝑘,𝑥)exp−𝑖𝐸𝑡ℏ,𝑥<0.(F.5), we obtain the initial, final reflected and transmitted wave packets, respectively, carrying a time delay during the motion or due to the interaction. Of course, there is always a certain distortion in the wave packet form due to the energy dependence of Ψ¹_pen(𝑥,𝑡)=𝑑𝐸𝐺𝐸−𝐄𝛼₀exp(−𝜒𝑥)exp−𝑖𝐸𝑡ℏ,0<𝑥<𝑎,(F.6) and 𝛼₀; but for a wide class of weight amplitudes such a distortion is negligible [6]. Moreover, we can get rid of the wave components with above-barrier energies by introducing the additional transformation (where 𝐴^0𝑅,𝛼₀ is the Heaviside step function) in order to avoid distortions of the under-barrier penetration (tunneling) due to the over-barrier transitions.
In the conventional approach, one requires that the stationary wave function exp(𝑖𝑖𝑘𝑘𝑥e)x||p(_𝑥=0𝑖+𝑘𝑥𝐴)^0𝑅−e𝐴x_0𝑅p(−𝑖𝑘𝑥)||_𝑥=0=𝛼₀exp(𝜒𝑥)||_𝑥=0,exp(−𝑖𝑘𝑥)||_𝑥=0=−𝜒𝛼₀,(F.7) and its first derivative be continuous across both potential discontinuities at 𝐴^0𝑅 and 𝛼₀exp(−𝜒𝑥). This results in four equations in the four unknowns Ψ¹_pen(𝑥,𝑡),, 𝑡, 𝑥=𝑎, and 𝑥=𝑎. If one does not take explicitly into account multiple internal reflections inside the barrier, the general tunneling evolution is quite simply described by passing from the stationary solution 𝜒𝛼−₀𝛼ex₀p(−𝜒𝑥)||_𝑥=𝑎+𝛽₀exp(𝜒𝑥)||_𝑥=𝑎=𝐴^0𝑇exp(𝑖𝑘𝑥)||_𝑥=𝑎,exp(−𝜒𝑥)+𝛽₀exp(𝜒𝑥)||_𝑥=𝑎=−𝑖𝑘𝐴^0𝑇exp(𝑖𝑘𝑥)||_𝑥=𝑎,(F.8) to the nonstationary wave packet 𝛽₀ defined by (F.2)-(F.3), but in a more detailed description including multiple internal reflections one has to take a different course.
Our model of the tunneling process of a nonrelativistic particle is as follows. We are going to solve the problem of a wave packet incident on the first (initial) potential wall;
(1)without taking into account the second (final) potential barrier wall, because the wave packet has not yet reached it in consequence of its finite propagation speed;(2)being careful of not breaking the requirement of the finiteness of the wave function (packet) for infinitely wide barriers (when increasing waves have to be absent); (3)constructing the waves in successive steps of multiple internal reflections from both barrier walls, in such a way that they are analytical continuations of the related expressions, corresponding to current waves in the simpler case of above-barrier energies.
We can therefore distinguish three subsequent steps in the evolution of the tunneling process.

Step 1 (the particle starts tunneling the barrier by crossing the first wall at 𝐴^0𝑇). At this initial step, in region I we have the initial time-dependent wave packet plus the externally reflected (from the initial potential wall) time-dependent wave packet The sum of the wave packets (F.4) and (F.5) transforms continuously, in the passage across the initial potential wall, into the internal time-dependent wave packet inside the barrier (in region II). In the hypothesis that the tunneling packet does not feel the final potential wall, the corresponding flux is directed initially only towards the second wall, that is, the penetrated wave packet does contain only decreasing waves: by 𝛼₀=2𝑘𝑘+𝑖𝜒,𝛽^𝑛=𝛼_𝑛𝑖𝜒−𝑘𝑖𝜒+𝑘exp(−2𝜒𝑎),𝛼^𝑛+1=𝛽_𝑛𝑖𝜒−𝑘𝐴𝑖𝜒+𝑘_0𝑅,=𝑘−𝑖𝜒𝑘+𝑖𝜅𝜒,𝐴^𝑛𝑇=𝛼_𝑛2𝑖𝜒𝑖𝜒+𝑘exp(−𝜒𝑎−𝑖𝑘𝑎),𝐴^𝑅𝑛+1=𝛽_𝑛2𝑖𝜒𝑖𝜒+𝑘(F.10) we denote the coefficient of initial penetration. Then, passing from the time-dependent wave packets to the corresponding stationary wave functions, and requiring continuity at 𝛼_𝑛,𝛽_𝑛,𝐴^𝑛𝑇,, we obtain for Step 1 (entrance of the initial wave packet inside the barrier) the following two equations in the two unknowns 𝐴^𝑛𝑅(𝑛=0,1,…): where 𝑛=0 is the coefficient of the initial (external) reflection. Let us stress that the stationary flux for 𝑛≠0is equal to zero and the total flux for 𝑛=2𝜇+1integrated over time 𝛼_𝜇, is zero, too.

Step 2 (the particle crosses the second barrier wall at 𝑛=2(𝑣+1)). The wave packet, penetrated inside region II, reaches the second wall of the barrier. It therefore transforms into a wave packet, transmitted through the final wall and propagated into region III, plus a wave packet, reflected from the same wall and penetrated back into region II. Quite similarly to (F.7) we obtain for this step, by the continuity requirement at 𝛽_𝑣, the following two equations: which can be solved to yield the unknown coefficients 𝛽₀ (amplitude of the evanescent wave reflected from the second wall in region II) and 𝑛=3 (amplitude of the wave transmitted through the second wall in region III).

Step 3 (the particle, bounced back from the second wall, crosses again the first wall moving in the negative x-direction). The wave packet reflected from the second wall is incident inside the barrier upon the first wall. Then, it transforms into a wave packet transmitted through this wall (as an addition to the packet reflected back in region I), and in a wave packet reflected from the same wall forward inside region II. Again, quite similarly to (F.7) and (F.8), we get the following two equations in the unknown coefficients 𝛼₁ (amplitude of the evanescent wave reflected into region II) and 𝐴_𝑇=^∞_𝑛=00𝑥0200𝑑𝐴^𝑛𝑇=4𝑖𝑘𝜒exp(−𝜒𝑎−𝑖𝑘𝑎)𝐴_𝑅𝐹,=^∞_𝑛=00𝑥0200𝑑𝐴^𝑛𝑅=𝑘²⁰𝐷₋𝛼=_∞𝐹,_𝑛=00𝑥0200𝑑𝛼_𝑛=2𝑘(𝑘+𝑖𝜒)𝛽=_∞𝐹,_𝑛=00𝑥0200𝑑𝛽_𝑛=2𝑘(𝑖𝜒−𝑘)exp(−2𝜒𝑎)𝐹,(F.11) (amplitude of the wave transmitted through the first wall in region I):

Step 3 corresponds, of course, to a first internal reflection. The process can be iterated by considering successive processes of internal reflections of the gradually decreasing (in consequence of preceding internal incidences with the walls) wave packet from the barrier walls (with partial transmissions through the walls outside). Such a description of the tunneling process is just the multiple internal reflection approach. It is easily seen that any of the subsequent steps can be reduced to one of the first three steps considered above. Moreover, we obtain, from the continuity requirement, the following recurrence relations: for the unknown coefficients 𝛼,𝛽,𝐴_𝑇, and 𝐴_𝑅 at the successive steps of evolution of the tunneling wave packet. The number 𝜓(𝑘,𝑥), labels the successive steps of evolution of the wave packet inside the barrier, starting from Ψ(𝑥,𝑡), (beginning of the wave-packet penetration inside the barrier). For 𝑖𝜒⟶𝑘₁,(F.12), the corresponding evolution step is the internal reflection from any barrier wall until the arriving to the other wall. Odd values 𝑘₁=[2𝑚(𝐸−𝑉₀)]^1/2/ℏ correspond to reflections from the first wall, with amplitude (𝐸>𝑉₀),, whereas even values 𝛼_𝑛,𝛽_𝑛,𝐴^𝑛𝑇,𝐴^𝑛𝑅,𝛼,𝛽,𝐴_𝑇 correspond to reflections from the second wall, with amplitude 𝐴_𝑅 (e.g., step 𝜒𝑎→∞ discussed above describes reflection back inside from the second wall with amplitude 𝐴_𝑇=0,𝐴_𝑅=𝐴^0𝑅,𝛼=𝛼₀,𝛽=0,(F.13), while step 𝑡_inc=ℏ𝜕(arg𝑔)𝜕𝐸(F.14) describes reflection forward inside from the first wall with amplitude (𝑥=0)).

The general evolution of the initial wave packet, tunneling through the barrier, is obtained by summing on all possible steps. It is easy to see that where 𝑣=ℏ𝑘/𝑚, 𝜏¹_tr=𝑡¹_tr−𝑡_inc=𝑎𝑣+ℏ𝜕arg𝐴^0𝑇𝜕𝐸=𝑣2𝜒(F.16) .

All these results for the coefficients (𝑥=𝑎). and 𝑛 coincide with those derived by a standard treatment of the tunneling process based on the general expressions for the stationary wave function 𝜏^𝑛_refl=4(𝑣+1)𝜏𝑣𝜒,𝑛=𝑣+1,𝑣=0,1,…,_𝑛tr=2(2𝜇+1)𝑣𝜒,𝑛=2𝜇+1,𝜇=0,1,….(F.17) (F.1), and for the nonstationary wave packet 𝜏_tun=𝑎𝑣+ℏ𝜕∑^∞𝑛=00𝑥0200𝑑arg𝐴^𝑛𝑇𝜕𝐸=(𝑣𝜒)⁻¹𝑘²⁰sh(2𝜒𝑎)+2𝜒𝑎𝑘²𝜒²−𝑘²𝜏4𝑘²𝜒²+𝑘²⁰sh²(𝜒𝑎)_refl,=ℏ𝜕∑^∞𝑛=00𝑥0200𝑑arg𝐴^𝑛𝑅𝜕𝐸=𝜏^tun.(F.18) (F.2)-(F.3). Moreover, by replacing where 𝜏_tun=𝜏_refl=𝜏¹_refl=𝜏¹_tr=2𝑣𝜒.(F.19) is the wave vector for the case of above-barrier energies 𝜇>0 all the expressions (F.10)-(F.11) for 𝑣≥0, and 𝛼_𝜇,𝛽_𝑣,𝐴^𝑣𝑇 transform into the corresponding ones obtained in the analysis of the ordinary particle propagation above barrier in terms of multiple internal reflections.

In the limit 𝐴^𝜇𝑅 (i.e., for infinitely wide and/or high barriers), the infinite series of multiple internal reflections reduces to only one step (the first) and we get (F.4)–( F.7) as total solution, with instead of (F.11). This case—together with the similar case of a subbarrier energy particle tunneling into a semiclosed barrier region with a dead stopper at the second wall—has been analyzed in [167].

Intermediate Reflection and Traverse Times, Total Tunneling and Reflection Times
In order to discuss the relevant times involved in the multiple reflection description of the tunneling process, we will exploit the general definition of phase times as times of propagation of the wave packet maximum (peak) for the quasimonochromatic wave packets (F.1)–(F.3). We have, respectively:
(1)   for the incident phase time at the barrier beginning 𝜏^𝑛_tr(𝑛=1,2,…); we take this as the zero (origin) time;(2)   (with 𝑎/𝜏_tun being the mean—or group—incident velocity) for the 1-step (external) reflection phase time;(3)   for the 1-step traverse phase time at the barrier end (−∞<𝑞<∞)
Similarly, we obtain the following expressions for the Ψ¹_pen(𝑥,𝑡)=𝑑𝐸𝐺𝐸−𝐄𝛼₀exp(−𝜒𝑥)exp−𝑖𝐸𝑡=(2𝜋)₋₁ℏ^∞−∞×𝑑𝐸𝐺𝐸𝑑𝑞−𝑔_𝜂(𝑞)exp(−𝑞𝑥)𝐄𝛼₀exp−𝑖𝐸𝑡ℏ−𝜒−𝑖𝑞⁻¹exp(−𝜒𝑎−𝑖𝑞𝑎)−1,(F.20)-step reflection and traverse phase times:
Then, the total tunneling and reflection phase times are defined by:
In the limit (−𝜂,𝜂)(𝜂>0,𝜂→0⁺), (for infinitely wide and/or high barriers) one gets
We see that for 𝜀=𝐸−𝐸_𝑞 and 𝐸_𝑞=ℏ²𝑞²/2𝑚 all Ψ¹_pen(𝑥,𝑡)=(2𝜋)⁻¹^∞−∞𝑑𝑞𝑔_𝜂(𝑞)exp𝑖𝑞𝑥−𝑖𝐸^𝑞𝑡×𝑑𝐸𝐺𝐸−ℏ𝐄𝛼₀exp−𝑖𝜀𝑡ℏ−𝜒−𝑖𝑞⁻¹exp(−𝜒𝑎−𝑖𝑞𝑎)−1.(F.22), and exp(𝑖𝑞𝑥−𝑖𝐸_𝑞𝑡/ℏ) are exponentially decreasing for exp(−𝑖𝜀𝑡/ℏ) Let us stress that not only 𝜀⟶0,or𝐸_𝑞≈𝐸,(F.23), but also all 𝑡, exhibit the HFE, that is, the independence of 𝜀𝑡ℏ≤1,(F.24) from a for large a (which implies an infinite growing of the tunneling velocity 𝑞 for −𝜒≤𝑞≤𝜒.(F.25)).

Study of The Physical Meaning of The Evanescent-Wave Packets with The Help of The Virtual Momentum Fourier Expansion and The Instanton Approaches
We want now to discuss the physical meaning of the wave packet (F.6) constructed from evanescent waves by exploiting the virtual momentum Fourier expansion and the instanton approach.

The Fourier Expansion
We will expand the wave packet (F.6) constructed from evanescent waves as a virtual momentum (𝑞>0) Fourier integral: where the infinitesimally narrow-step function 𝑘 defined as within the infinitesimal interval Δ𝑘 is inserted in order to eliminate the point 𝑔(𝐸−𝐄).. By passing to the variable 𝑡→𝑖𝜏 (with 𝜒), F.20) can be rewritten as In (F.22) one can see current waves 𝜏 and the oscillating factor 𝐸→𝐸^, damped with time. The main contribution to (F.22) originates from the current waves which satisfy the condition when the dampingis practically absent even for large Ψ¹_pen(𝑥,𝑡)=𝑑𝐸𝛼₀𝐺𝐸−𝐄exp𝑖𝜅𝑥−𝐸𝜏ℏ.(F.26),—more precisely, when and also for exp−𝐸^𝜏ℏ=ℏ⁻¹(2𝜋)^−1/2^∞−∞𝐷𝐸^,𝐄́exp𝑖𝐄́𝜏ℏ𝑑𝐄́,(F.27) within the range (By the way, it is striking that generally in this approach both symmetrically forward Ψ¹_pen(𝑥,𝑡)=(2𝜋)⁻¹𝑑𝐸𝐺𝐸−𝐄𝛼₀exp𝑖𝜅−𝑖𝐸^𝜅𝜏ℏ^∞−∞𝑑𝐄́exp𝑖𝐸^𝜅+𝐄́𝜏ℏ𝐸^+𝑖𝐄́⁻¹,(F.29) and backward 𝐸_𝜅=ℏ²𝜅²/2𝑚 motions are possible.) Either condition (F.23) or (F.25) is satisfied when both 𝜏 and ±𝑑𝐸_𝜅/𝑑|𝜅|=ℏ|𝜅|/𝑚, are within the range exp[𝑖(𝐸_𝜅+𝐄́)𝜏/ℏ], covered by the weight amplitude 𝜏→∞

The Instanton Approach
We will now analyze the propagation of the wave packet (F.6) using the well-known method of transforming the space-time metric inside the barrier region, namely the formal inspection of the wave-packet motion along an imaginary time axis exp(𝑖𝜅𝑥−𝑖𝐸_𝜅𝜏/ℏ), which is typical of the instanton approach (see, e.g., [116, 168, 169] and references therein). In this case, exp[𝑖(𝐸_𝜅+𝐄́)𝜏/ℏ] becomes imaginary: ||𝐸_𝜅+𝐄́||𝜏ℏ≤1,(F.30) and therefore one gets nonzero fluxes along the axis 𝐄́; moreover, −𝐸≤𝐄́≤𝐸.(F.31). So, we get for 𝐸_𝜅 By introducing the virtual-energy 𝜅 Fourier transform with equation (F.26) can be rewritten as where 𝜏_tun. So, it is easily seen that along the axis 𝜏^𝑛_tr(𝑛=1,2,…) inside the barrier, the functions (F.29) correspond to wave-packet motions with group velocities 𝑥̂ accompanied by oscillations described by the factor 𝑝_𝑥̂, damped for 𝜑.
In (F.29) one can see current waves 𝐿_𝑧 and the oscillating factor ∞ damped with time. The main contribution to (F.29) originates from the current waves which satisfy the condition when the dampingis practically absent, and also for 𝑆 within the range Either condition (F.30)-(F.31) is satisfied when both E and 𝑙=0 are within the range 𝑆 (or k and 𝑒^−2𝜋𝜂 are within the range 𝑙>0, covered by the weight amplitude 𝑙, precisely as in the previous case. The only difference between these approaches is the absence of backward motion in the instanton representation.

Some Words on Multiple Internal Reflections for Particle Energies above A Single Barrier and for Two Separated Barriers with Even Particle Subbarrier Energies
For the case of the above-barrier energies, there is an evident influence of multiple internal reflections of propagating waves. In [170–174] it was presented the detailed study of such influence and it was shown the presence of a multitude of various secondary reflected and transmitted peaks in addition to two main treflected and transmitted wave packet peaks. In [175], the influence of multiple internal reflections was extended for transmissions of relativistic Dirac wave packets with energies above a single barrier.
In [176], the influence of multiple internal reflections was further extended, taking into account multiple internal reflections in the motion of free (nonrelativisctic) propagating wave packets between two separated barriers. It was shown the existence of multiple peaks, due to multiple reflections in the free region between two barriers, in addition to the assumption of a single outgoing peak in the case of neglecting such multiple reflections as it was made in Section 3.7 on the base of [127–129].

The Case of Photon Tunneling
It follows from [165, 166] (see also [47] and Section 2.4) that all the results and conclusions of Appendix F for subbarier energies, concerning both multiple internal reflections and total phase tunneling and reflection times, are also valid for the tunneling of photons in a 1D rectangular wave guide (with the only substitution 𝑘 in (2.27)–(2.31)).
In the particular case of quasimonochromatic wave packets, using the stationary-phase method under the same boundary and measurement conditions as considered for particles in Section 3.2, we obtained there the identical expression (3.19) which manifests the superluminal photon effective tunneling velocity. This result agrees with the experimental findings of the microwave-tunneling measurements presented in [170–174]. Analyzing now the multiple internal reflections for photons, one can also see that (as in the particle case) not only 𝑒^−2𝜋𝜂 but also all 𝐴ⁱⁿ_𝑅⟶𝑒^{2𝑖𝐾𝑅1},11+𝐴ⁱⁿ_𝑅⟶1+exp2𝑖𝑘𝑅₁⁻¹(F.32) manifest the HFE.

Summary: Conclusions and Perspectives

(1) The time operator (2.1a), (2.1b), and (2.26) is general for any quantum collision and process in the continuum energy spectrum within nonrelativistic quantum mechanics and quantum electrodynamics (with self-consistent definition of averaging measures over time for the 1D particle and photon motion). Of course, it cannot be defined in the cases with zero fluxes and unmoving particles, but for the same cases apriori there is no evolution processes at all. The uniqueness of the maximal symmetric time operator (2.1a), (2.1b), and (2.26) does directly follow from the uniqueness of the Fourier transformations from time representation to energy one.

Two measures of averaging over time and connection between them are analyzed. The foundations of the self-consistent time analysis for quantum (in particular, tunneling and nuclear) processes are in fact developed on the base of the time operator with the proper measure of averaging over time (2.7) and/or (2.9). They include the mathematically rigorous and one-to-one time and energy representations of the Olkhovsky-Recami definition of mean durations and variances in distributions of durations for all really known Hamiltonians in various quantum processes and collisions (including all kinds of multiple internal reflections between barriers and inside barriers among them).

An actual perspective for the nearest future is opened for generalizing the time analysis of quantum processes for more complicated particle and photon motions (e.g., such as along helixes and motions through 2D and 3D (nonspherical) potentials and barriers).

As to systems with the discrete energy spectra, the form (2.43) for the time operator corresponds to the class of bound-state wave functions just similarly to the situation with the azimuth-angle operator. It is general for processes in the discrete energy spectrum (or a system has a purely discrete energy spectrum either it has a discrete spectrum as a part of its total energy spectrum). The time operator cannot be defined in the case of one bound state with zero flux and unmoving particle, and there is no evolution here too.

Once more we underline that at the limit of infinitesimally close levels formula (2.50) passes to formula (2.1b) for systems with continuous spectra.

All commutation relations, analyzed here, (2.24) and (2.44) and also uncertainty relations (2.25) and (2.47) are set side by side with the similar relations for other pairs of canonically conjugate observables (such as, for coordinate 𝑆 and momentum ¹²C in the case of (2.25), and for azimuthal angle ¹⁴N and angular momentum [𝐿_𝑍,𝜑] in the case of (2.47)). Our relations do not replace, but essentially extend the meaning of the time and energy uncertainties given in [177, 178]. Also they are consistent with the conclusions of [179, 180] where, in particular, it was directly said about absence of objections against those limitations on the time and energy measurements which can be derived from the mathematic formalism with introducing the adequate energy and time operators and the corresponding statistical fluctuations. We hope that these relations which are deduced with applying the properties of the time operator can help to attenuate endless debates about the status of the time-energy uncertainty relation.

(2) Finally, we state that not only time but any other quantities, to which maximal symmetrical (Hermitian) operators correspond (e.g., a momentum in a semispace with a rigid wall and a radial momentum, both are defined over the semibounded axis from 0 to 𝛼) can be considered as quantum-physical observables in the same degree as the quantities to which self-adjoint operators correspond, without introducing any new physical postulates. The same conclusion is valid for quasi-self-adjoint operators like (2.37) and (2.43).

(3) Similar derivations and conclusions with quite evident generalization can be carried out for time operator in relativistic quantum mechanics (the Klein-Gordon case and the Dirac case). It is rather perspective (but, of course, not always simple) to develop the analysis of three- and four-position operators for other relativistic cases, especially to analyze the localization problems (the Dirac particles, 2D and 3D particles and photon motions, etc.)

(4) Actually, the time operator (2.1a), (2.1b), and (2.26) has been rather fruitfully used in the case of the tunneling times (see [26, 27, 32–35, 47]). We have established that practically all earlier known particular tunneling times appear to be the special cases of the mean tunneling time or of the square root of the variance in the tunneling-time distribution (or pass into them under some boundary conditions), defined within the general O-R approach. It had been carried out in some reviews (in particular, in [26, 27, 32, 47, 81]) also the connection of other earlier known approaches or simultaneously elaborated approaches with the O-R approach which had been recognized as the most self-consistent definition of the tunnelling time within the conventional quantum mechanics (see, e.g., [81]).

It is meaningful to stress also that, although any direct classical limit for particle tunnelling through potential barrier with subbarrier energies is really absent, there is the direct classical limit for wave packet tunneling. Let us recall real evanescent and antievanescent waves, well known in classical optics and in classical acoustics (see, e.g., [176, 181–185]).

(5) The HFE is now extended for all expressions of mean tunneling times, however, with sufficiently narrow momentum spreads of initial particle wave packets (and, of course, for quasimonochromatical particles). The violations of the HFE are revealed and explained for the presence of the absorption and also for the cases of the rather large momentum spreads of initial particle wave packets.

(6) It is elaborated the rigorous combined resonant and nonresonant description of the 1D particle tunneling through two potential rectangular barriers.

(7) There are derived and analyzed new general expressions for the elastic scattering 16_O+p-matrix, for internal and external transmissions and reflection probability amplitudes and also the connection between them in a 3D tunnelling process through spherically symmetric potential (rectangular and Coulomb) barriers, taking into account the multiple internal reflections from internal barrier wall into the potential well. In the case of a rectangular barrier, there are also derived the expressions for the tunnelling and reflection phase time, and it is shown the occurrence of the HFE. Of course, in the realistic 3D situations in nuclear physics one has to use typical phenomenological potentials (like the Saxon-Woods well) and consider also the charge distribution inside and at the surface of the daughter nucleus.

We have also taken into account the resonances and written in this case the explicit analytical expression for the resonant 12_C+p-matrix, with the resonance width proportional to the very small exponential factor (in the case of a rectangular barrier with ). Similarly one can write the resonant -matrix for the Coulomb barrier, with the resonance width proportional to the very small exponential factor and for cases with .

The presented method can be used as the initial phase for the joint time-dependent study of nuclear reactions (beginning from the scattering) and decays for any value of , and not only for spherically symmetric interactions but also for nonspherical cases (beginning from the coupled-channel method). Also the results for the case of the Coulomb barrier can be used as an initial phase for analysis of the subbarrier low-energy astrophysical nuclear-fusion reactions (near the Gamow energy). In distinction from the 1D WKB approximation which is used till now for such analysis, in the 3D case it is important even for very small to take into account not only the exponential factor but also the multiple internal reflections with for both resonance and nonresonant collisions.

Then, for this same system, we have derived the elastic scattering -matrix (and its connection with the transmission and reflection probability amplitudes) and all quantities related taking into account the multiple internal reflection.

The results we have obtained can be used for the study of the alpha and proton radioactivity decay, if one will use the Coulomb barrier instead of the rectangular one and may be introduce the hard core inside the internal well and then modify all the potentials into more realistic (“smoothed” like Woods-Saxon well, etc.) potentials. Also they can be used in another field, with the appropriate modifications, for the study of the photon emission from a glass sphere surrounded by a spherical air layer and externally by another glass sphere.

(8) The analysis of the possibility and the study of the multiple internal reflections, of wave packets with subbarrier energies, not only between barriers but also inside barriers (and not only for particles but also for photons) by several approaches is carried out in Appendix F. Inside barriers we have used (i) the analytic continuation from axis of real momentum values to axis of imaginary momentum values, (ii) the Fourier expansion, and (iii) the instanton approach.

One can see an interesting perspective to research the multiple internal reflections also experimentally (in particular, by frustrated total internal reflections by 2D and 3D barriers in photon (see [181–183] where namely such optical experiments are firstly described) and acoustical tunneling) with presented here theoretic foundations of time analysis of quantum processes.

Another interesting perspective, firstly in theoretical reseach, is revealed in [186, 187] by establishing of not only incoherent multiple internal reflections for particles and also photons but also coherent multiple internal reflections from barriers inside wells between barriers––in the cases of the tunneling of small bound systems through certain Coulomb barriers during long decays of more complex systems (e.g., long-living alpha-radioactive nuclei and heavy nuclei, undergoing the spontaneous fission, etc.)

(9) Some results of the time analysis of nuclear collisions and decays are briefly reviewed. In the regions of isolated resonances, distorted by the nonresonant background, the principal possibility of the negative values of time delays (i.e., advances) is shown and concretely the time analysis of proton scattering by nuclei and at the range of isolated resonances distorted by the nonresonant background, accompanied by the bremsstrahlung, resulted in the discovery of the real possibility of the delay-advance phenomenon in the proton emission during scattering [151, 152].

The time analysis of high-energy nuclear reactions, with highly excited (at the range of very dense overlapping energy resonances) final compound-fragment formations, resulted in the revealing of the new phenomenon of time resonances (or explosions) of such formations [148].

Results of the time analysis for other nuclear processes are presented in [21, 23–25, 188–191].

Acknowledgments

The author is deeply indebted to L. S. Emelyanova for the encouraging stimulation to write this review. He acknowledges the strong cooperation, collaboration, and useful discussions received along years by E. Recami and also V. Petrillo and R. Mignani. The author thanks N. V. Eremin, A. S. Holevo, G. Giardina, G. Fazio, V. L. Lyuboshitz, and B. N. Zakharyev for the useful discussions and friendly support. He is also grateful to his former postgraduate students A. K. Zaichenko, S. P. Maydanyuk, M. E. Dolinska, and V. V. Davydovsky for the help.