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

Authors: V. S. Olkhovsky
Journal: Advances in Mathematical Physics
 (2009)

---

## Abstract

Recent developments are reviewed and some new results are presented in the study oftime 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.


---

## Body

## 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 oftime 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(2.1a)t^=t,inthe(time)t-representation,(2.1b)t^=−iℏ∂∂E,inthe(energy)E-representation,
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 E=0 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 P of the continuous, differentiable, square-integrable functions f(E) that satisfy the conditions (2.2)∫0∞|f(E)|2dE<∞,∫0∞|∂f(E)∂E|2dE<∞,∫0∞|f(E)|2E2dE<∞, and the condition (2.3)f(0)=0,                                                                                      which is a space Pdense in the Hilbert space of L2 functions defined (only) over the semiaxis 0≤E<∞. Obviously, the operator (2.1a) and (2.1b) is Hermitian, that is, the relation   (f1,t^f2)=((t^f1),f2) holds, only if all square-integrable functions f(E) in the space on which it is defined vanish for E=0.Also the operatort^2 is Hermitian, that is, the relation   (f1,t^2f2)=((t^f1),(t^f2))=(t^2f1,f2)  holds under the same conditions.Operatort^ has no Hermitian extension because otherwise one could find at least one function f0(E) which satisfies the condition f0(0)≠0  but that is inconsistent with the propriety of being Hermitian. So, according to [53], t^ 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 (0≤x<∞)  bounded by a rigid wall located at x=0. Consequently, the operator for the momentum x-component of the particle, which reads (2.4)p^x=-iℏ∂∂x is defined as acting on the space of the continuous, differentiable, square-integrable functions f(x) that satisfy the conditions  (2.5)∫0∞|f(x)|2dx<∞,∫0∞|∂f(x)∂x|2dx<∞,∫0∞|f(x)|2x2dx<∞, and the condition
(2.6)f(0)=0, which is a space denseQ in the Hilbert space of L2 functions defined (only) over the spatial semiaxis 0≤x<∞. Therefore, operator p^x=-iℏ(∂/∂x) has the same mathematical properties as operator t^ (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      p^r=-iℏ(∂/∂r)+(1/r)(0<r<∞).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(Emin<E<Emax)  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” τ=ℏ/d where d=|Emax−Emin|.In order to considertime 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:  (2.7)W(x,t)dt=j(x,t)dt∫-∞∞j(x,t)dt, where the probabilistic interpretation of j(x,t) (namely in time) corresponds to the flux probability density of a particle passing through point x at time t  (more precisely, passing through x during a unit time interval centered at 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) and of  the continuity relation  (2.8)∂ρ(x,t)∂t+divj(x,t)=0for 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 ρ(x,t) and the flux probability density j(x,t) are  related with the wave function Ψ(x,t) by the usual definitions ρ(x,t)=|Ψ(x,t)|2 and j(x,t)=Re[Ψ*(x,t)(ℏ/iμ)∂Ψ(x,t)/∂x]. The measure (2.7) was firstly investigated in [21, 23–27, 32–35].When the flux densityj(x,t) changes its sign, the quantity W(x,t)dt  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 j(x,t)  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): (2.9)W±(x,t)dt=j±(x,t)dt∫-∞∞j±(x,t)dt,  with j±(x,t)=j(x,t)Θ(±j) where Θ(z) 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  j=j− as follows:  (2.10)∂ρ>(x,t)∂t=-∂j+(x,t)∂x,∂ρ<(x,t)∂t=-∂j-(x,t)∂x, respectively. Relations in (2.10) can be considered as formal definitions of ∂ρ>/∂t and ∂ρ</∂t. Integrating them over time t from  −∞   to  t, one obtains  (2.11)ρ>(x,t)=-∫-∞t∂j+(x,t′)∂t'dt′,ρ<(x,t)=-∫-∞t∂j-(x,t′)∂t′dt′ with the initial conditions ρ>(x,-∞)=ρ<(x,-∞)=0. Then, it is possible to introduce the quantities  (2.12)N>(x,∞;t)≡∫x∞ρ>(x′,t)dx′=∫-∞tj+(x,t′)dt′>0,N<(-∞,x;t)≡∫-∞xρ<(x′,t)dx′=-∫-∞tj-(x,t′)dt′>0,   which have the meaning of probabilities for the particle wave packet Ψ(x,t) to be located at time t on the semiaxis (x,∞)  and  (−∞,x), respectively, as functions of the flux densities  j+(−∞,t) and j−(x,t), provided that the normalization condition  ∫−∞∞ρ(x,t)dx=1 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 ρ<(x,t), and by adopting the boundary conditions  j+(−∞,t)=j−(−∞,t)=0. Then, by differentiating N>(x,∞;t) and  N<(−∞,x;t) with respect to t, one obtains  (2.13)∂N>(x,∞;t)∂t=j+(x,t)>0,∂N<(-∞,x;t)∂t=-j-(x,t)>0.      Finally, from the last four equations one can easily infer that  (2.14)W+(x,t)dt=j+(x,t)dt∫−∞∞j+(x,t)dt=∂N>(x,∞;t)/∂tN>(x,∞;∞),W-(x,t)dt=j-(x,t)dt∫−∞∞j-(x,t)dt=∂N<(-∞,x;t)/∂tN<(-∞,x;∞), which justify the abovementined probabilistic interpretation of  W±(x,t). 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(x)〉 of the time t at which a particle passes through position x (when travelling in only one positive x-direction), and  〈t±(x)〉 of the time t at which a particle passes through position x, when travelling in the positive or negative direction, respectively,   (2.15)〈t(x)〉=∫−∞∞tj(x,t)dt∫−∞∞j(x,t)dt=∫0∞dE(1/2)[G*(x,E)t^vG(x,E)+vG*(x,E)t^G(x,E)]∫0∞dEv|G(x,E)|2,                  where G(x,E) is the Fourier transform of the moving 1D wave packet  (2.16)Ψ(x,t)=∫0∞G(x,E)exp(−iEtℏ)dE=∫0∞g(E)φ(x,E)exp(−iEtℏ)dE, when going on from the time representation to the energy one,  (2.17)〈t±(x)〉=∫−∞∞tj±(x,t)dt∫−∞∞j±(x,t)dt, and also the mean durations of particle 1D transmission from xi to  xf>xi and 1D particle reflection from the region  (xi,∞) into xf≤xi:  (2.18)〈τT(xi,xf)〉=〈t+(xf)〉−〈t+(xi)〉,〈τR(xi,xf)〉=〈t−(xf)〉−〈t+(xi)〉,      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 ∫0∞dE…, similarly to  (2.15) by using the unique Fourier (Laplace) transformations and the energy expansion of    j±(x,t)=j(x,t)θ(±j), but it is evident that they result to be rather bulky.If one does now generalize the expressions (2.15) and (2.17) for 〈tn〉 with  a generic value n=2,3,…, then we will be able to write down for 〈f(t)〉 with any analytic function of time f(t),  the one-to-one relation  (2.19)〈f(t)〉=∫-∞∞j(x,t)f(t)dt∫-∞∞j(x,t)dt=∫0∞dE(1/2)[G*(x,E)f(t^)vG(x,E)+vG*(x,E)f(t^)G(x,E)]∫0∞dEv|G(x,E)|2 from the time to the energy representation. For free motion, one has G(x,E)=g(E)exp(ikx), φ(x,E)=exp(ikx),  and E=ℏ2k2/2μ=μv2/2, while the normalization condition is
(2.20)∫0∞|G(x,E)|2dE=∫0∞|g(E)|2dE=1,
and the  boundary conditions are  (2.21)[dng(E)dEn]E=0=[dng(E)dEn]E=∞=0,forn=0,1,2,….Conditions (2.21) imply a very rapid decrease till zero of the flux densities near the boundaries E=0 and E=∞:  this complies with the actual conditions of real experiments, and therefore they does not represent any restriction of generality.In (2.19), t^ 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|g(E)|2≈Kδ(E−E¯), K being a constant, quantity j(x,t) goes into ρ(x,t) and (2.19) goes into the more simple relation (2.22)〈f(t)〉≡∫-∞∞j(x,t)f(t)dt∫-∞∞j(x,t)dt≈∫-∞∞ρ(x,t)f(t)dt∫-∞∞ρ(x,t)dt≈∫0∞dEG*(x,E)f(t^)G(x,E)∫0∞dE|G(x,E)|2,  because of the relations   j(x,t)vρ(x,t)≈v¯ρ(x,t).Now, one can see that two canonically conjugate operators, the time operator (2.1a), (2.1b), and (2.26) and the energy operator  (2.23)E^={Eintheenergy(E)-representation,iℏ∂∂tinthetime(t)-representation,  satisfy the typical commutation relation  (2.24)[E^,t^]=iℏ.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 (−∞<t<∞) to the E-semiaxis (0<E<∞) 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  (2.25)ΔEΔt≥ℏ2(where the standard deviations are Δa=✓Da, quantity Da being the variance Da=〈a2〉−〈a〉2;   and a=E,t, while 〈⋯〉 denotes an average over t  by the measures W(x,t)dt or W±(x,t)dt  in the t-representation or an average over E 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,p0}, with q0=t and p0=−E, 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  t^ in (2.1a), (2.1b), and (2.26) are sufficient for considering them as quantum observables: namely, the uniqueness of the “spectral decomposition” (also called spectral function)  for operators t^, as well as for t^n(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^n(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 isno practical difference between self-adjoint and maximal Hermitian operators for systems with continuous energy spectra. Let us underline that the mathematica, properties of t^n(n>1) are quite enough for considering time asa quantum-mechanical observable (like for energy, momentum, spatial coordinates,…) without having to introduce any new physical postulates.Now let us analyse the so-calledpositive-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 (2.26)t^=(−iℏ2)∂↔∂E  (the so-called bilinear form), where the meaning of  the sign  ↔ is clear from the following definition: (f,t^g)=(f,(−iℏ/2)(∂/∂E)g)+((−iℏ/2)(∂/∂E)f,g).  For this form the direct elimination of the point E=0 is not necessary because it is eliminated automatically in (f,t^f) and in ∫-∞∞tj(x,t)dt by such bilinearity. And such an elimination of the point E=0 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 (E-) representation, with 0<E<∞,  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    −∞<k<∞:  (2.27)Ψ(x,t)=∫-∞∞dkg(k)φ(x,k)exp(−iEtℏ),  with  E=ℏ2k2/2m,k≠0. In such a case the time operator (2.1a), (2.1b), and (2.26) (acting on momentum eigenvector, defined on −∞<k<∞, is already formally self-adjoint, with the boundary conditions  (2.28)[dng(k)dkn]k=−∞=[dng(k)dkn]k=∞=0,n=0,1,2,…,    except for the fact that we have excluded point k=0; 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: (2.29)Ψ(x,t)=∫0∞dE(E)−1/2g((2mE)1/2ℏ)φ(x,(2mE)1/2ℏ)exp(−iEtℏ)+∫0∞dE(E)−1/2g(−(2mE)1/2ℏ)φ(x,−(2mE)1/2ℏ)exp(−iEtℏ). If we now introduce the weight (2.30)g˜(E)=(m2Eℏ2)1/4[g((2mE)1/2ℏ)g(−(2mE)1/2ℏ)] as a “two-dimensional” vector, then  (2.31)∫-∞∞|Ψ(x,t)|2dx=∫0∞dE|g˜(E)|2<∞, the norm being (2.32)g˜(E)=g*(E)⋅g(E)>0.If the wave packet (2.27) is one directional and g(k)≡g(k)Θ(k), then the integral ∫-∞∞dk goes on to the integral ∫0∞dk 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  E→k 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(2.33)dP(x,t)≡Z(x,t)dx=|Ψ(x,t)|2dx∫-∞∞|Ψ(x,t)|2dx 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 (x,x+dx) at the moment t, independently from the motion processes. As a consequence, the quantity  (2.34)P(xi,xf,t)=∫x1xf|Ψ(x,t)|2dx∫-∞∞|Ψ(x,t)|2dx  will have the meaning of the probability of particle dwelling in the spatial range (x1,x2) at the instant t. Taking into account the equality  (2.35)∫-∞∞j(x,t)dt=∫-∞∞|Ψ(x,t)|2dx, which evidently follows from the 1D continuity relation (2.8), the mean dwell time can be presented in the following form:
(2.36a)〈τdw(xi,xf)〉=∫-∞∞dt∫xixf|Ψx,t|2dx∫-∞∞jin(xi,t)dt with the flux density jin for the initial “dwelling” free motion through point xi. The expression (2.36a) can be rewritten in the following equivalent form
(2.36b)〈τdw(xi,xf)〉=[∫-∞∞tj(xf,t)dt−∫-∞∞tj(xi,t)dt]∫-∞∞jin(xi,t)dt,
taking into account the continuity relation (2.8) for the total flux density  j(x,t) in the interval       (xi,xf) at time  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]) (2.37)A→(r→,t)=∫k0d3kk0χ→(k→)φ(k→,r→)exp(−ik0t), where, as usual, A→(r→,t) is the electromagnetic vector potential, and r→={x,y,z}, k→={kx,ky,kz}, k0≡ω/c=ε/ℏc, k≡|k→|=k0, where the gauge condition divA→=0 is assumed.The axisx has been chosen as the propagation direction,                 χ→(k→)=∑i=yzχi(k→)e→i(k→); with e→ie→j=δij, xi,xj≡y,z, χi(k→) is the probability amplitude for the photon to have momentum k→ and polarization e→j along xj, and it is φ(k→,r→)=exp(ikxx) in the case of   plane waves, while φ(k→,r→) 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:(2.38)ρem(x,t)dx=S0dx∫S0dx,S0=∬s0dydz (s0=[E→*E→+H→*H→]/4π being the energy density, the electromagnetic field being H→=rotA→, E→=-(1/c)∂A→/∂t) of a photon to be found (localized) in the spatial intervall(x,x+dx) along axis x at the moment t, and the flux probability density(2.39)jem(x,t)dt=Sx(x,t)dt∫Sx(x,t)dt,Sx(x,t)=∬sxdydz (with sx=cRe[E→*H→]x/8π being the energy flux density, H→=rotA→)  of a photon to pass through the point (plane) x in the time interval(t,t+dt),  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:
(2.40)〈t(x)〉=∫-∞∞tJem,xdt=∫-∞∞tSx(x,t)dt∫-∞∞Sx(x,t)dt, where the form (2.1b) of time operator is valid also for photons with natural boundary conditions χi(0)=χi(∞)=0 in the energy representation  (ε=ℏck0), quite similarly to (2.1b)–(2.3) for nonrelativistic particles in the energy representation.The energy densitys0 and energy flux density sx satisfy the relevant continuity equation                                                                                                                                             (2.41)∂s0∂t+∂sx∂x=0, 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  〈t(x)〉,Dt(x) 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 quantitiesJem,x,±=Jem,xΘ(±Jem,x) 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 J by Jem).


### 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                                 (2.42)ψ(x,t)=∑n=0…gnφn(x)exp[−i(εn−ε0)tℏ], (where φn(x) are orthogonal and normalized wave functions of system bound states which satisfy  H^φn(x)=εnφn(x),H^ being the system Hamiltonian, ∑n=0…|gn|2=1,  here we factually omitted a nonsignificant phase factor  exp(−iε0t/ℏ)  as being general for all terms of the sum ∑n=0…)  for  describing the evolution of systems in the regions of the purely discrete spectrum. Without limiting the generality, we choose moment t=0 as an initial time  instant.Firstly, we will consider those systems, whose energy levels are spaced with distances for which the maximal common divisor isfactually 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)  T=2πℏ/D,D being the maximal common divisor of distances between system energy  level.In thet-representation the relevant energy operator  H^ is a self-adjoint operator acting in the space of periodical functions whereas the function   tψ(t)  does not belong to the same space. In the space of periodical functions the time operator t^, even in the eigen representation, has to be also a periodical function of time t. 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  (2.43)t^=t−T∑n=o∞Θ(t−[2n+1]T2)+T∑n=0∞Θ(−t−[2n+1]T2),  which is the so-called saw-function of  t (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 theself-adjoint energy and time operators acquires in this case (discrete energies and periodical functions) the form  (2.44)[E^,t^]=iℏ{1−T∑n=0∞δ(t−[2n+1]T)}. Let us recall (see, e.g., [70, 71]) that a generalized form of uncertainty relation holds
(2.45)(ΔA)2⋅(ΔB)2≥ℏ2[〈N〉]2
for two self-adjoint operators A^ and B^, canonically conjugate each to other by the commutator (2.46)[A^,B^]=iℏN^,N^ being a third self-adjoint operator.  One can easily obtain  (2.47)(ΔE)2⋅(Δt)2≥ℏ2[1−T|ψ(T/2+γ)|2∫−T/2+T/2|ψ(t)|2dt], where the parameter γ (with an arbitrary value between −T/2 and +T/2 ) is introduced for the univocality of calculating the integral on right part of (2.47) over dt  in the limits from −T/2 to  +T/2,  just similarly to the procedure introduced in [68] (see also [70, 71]).From (2.47) it follows that when ΔE→0 (i.e., when |gn|→δnn′) the right part of (2.47) tends to zero since ψt2 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   ΔE≫D and    ψ(T+γ)2≪T∫−T/2T/2|ψ(t)|2dt,  the periodicity condition may be inessential for  Δt≪T, that is, (2.47) passes to uncertainty relation (2.11), which is just the same one as for systems with continuous spectra.In principle, one can obtain the expression for the time operator (2.43) also in energy representation. If one will calculate the mean value 〈t(x)〉 of instants of particle passing through point x, then after a series of bulky transformations he will obtain the following expression:  (2.48)t^=iℏ2∑n;>n(-1)Nn-Nn′Δ↔n′Δn′εn in the energy representation, where  Nn=(εn−ε0)/D; the bilinear operation, denominated  by  Δ↔n, signifies (2.49)An*Δ↔n′An=An*Δn′An−AnΔn′An*,Δn′An=An′−An,〈t(x)〉=∑n=0∞gn*φn*(x)t^gnφn(x)∑n=0∞|gnφn(x)|2. (Of course, one has to average over the flux density, but for the simplicity in this case it is possible to make averaging over  Ψx,t2.)  Operator (2.43) for two levels (n=0,1) acquires the more simple form (2.50)t^=-iℏ2Δ↔Δε, and when  D=ε1−ε0→0,  the expression (2.50) passes to the differential form (2.51)t^=-iℏ2∂↔∂ε, 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, γlim) by quasicycles with quasiperiodical evolution and for sufficiently long intervals of time the motion inside such systems (however, less than ℏ/γlim) 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 ∑n=0∞ contains only one term (gn→δnn′), 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.


## 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(2.1a)t^=t,inthe(time)t-representation,(2.1b)t^=−iℏ∂∂E,inthe(energy)E-representation,
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 E=0 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 P of the continuous, differentiable, square-integrable functions f(E) that satisfy the conditions (2.2)∫0∞|f(E)|2dE<∞,∫0∞|∂f(E)∂E|2dE<∞,∫0∞|f(E)|2E2dE<∞, and the condition (2.3)f(0)=0,                                                                                      which is a space Pdense in the Hilbert space of L2 functions defined (only) over the semiaxis 0≤E<∞. Obviously, the operator (2.1a) and (2.1b) is Hermitian, that is, the relation   (f1,t^f2)=((t^f1),f2) holds, only if all square-integrable functions f(E) in the space on which it is defined vanish for E=0.Also the operatort^2 is Hermitian, that is, the relation   (f1,t^2f2)=((t^f1),(t^f2))=(t^2f1,f2)  holds under the same conditions.Operatort^ has no Hermitian extension because otherwise one could find at least one function f0(E) which satisfies the condition f0(0)≠0  but that is inconsistent with the propriety of being Hermitian. So, according to [53], t^ 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 (0≤x<∞)  bounded by a rigid wall located at x=0. Consequently, the operator for the momentum x-component of the particle, which reads (2.4)p^x=-iℏ∂∂x is defined as acting on the space of the continuous, differentiable, square-integrable functions f(x) that satisfy the conditions  (2.5)∫0∞|f(x)|2dx<∞,∫0∞|∂f(x)∂x|2dx<∞,∫0∞|f(x)|2x2dx<∞, and the condition
(2.6)f(0)=0, which is a space denseQ in the Hilbert space of L2 functions defined (only) over the spatial semiaxis 0≤x<∞. Therefore, operator p^x=-iℏ(∂/∂x) has the same mathematical properties as operator t^ (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      p^r=-iℏ(∂/∂r)+(1/r)(0<r<∞).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(Emin<E<Emax)  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” τ=ℏ/d where d=|Emax−Emin|.In order to considertime 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:  (2.7)W(x,t)dt=j(x,t)dt∫-∞∞j(x,t)dt, where the probabilistic interpretation of j(x,t) (namely in time) corresponds to the flux probability density of a particle passing through point x at time t  (more precisely, passing through x during a unit time interval centered at 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) and of  the continuity relation  (2.8)∂ρ(x,t)∂t+divj(x,t)=0for 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 ρ(x,t) and the flux probability density j(x,t) are  related with the wave function Ψ(x,t) by the usual definitions ρ(x,t)=|Ψ(x,t)|2 and j(x,t)=Re[Ψ*(x,t)(ℏ/iμ)∂Ψ(x,t)/∂x]. The measure (2.7) was firstly investigated in [21, 23–27, 32–35].When the flux densityj(x,t) changes its sign, the quantity W(x,t)dt  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 j(x,t)  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): (2.9)W±(x,t)dt=j±(x,t)dt∫-∞∞j±(x,t)dt,  with j±(x,t)=j(x,t)Θ(±j) where Θ(z) 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  j=j− as follows:  (2.10)∂ρ>(x,t)∂t=-∂j+(x,t)∂x,∂ρ<(x,t)∂t=-∂j-(x,t)∂x, respectively. Relations in (2.10) can be considered as formal definitions of ∂ρ>/∂t and ∂ρ</∂t. Integrating them over time t from  −∞   to  t, one obtains  (2.11)ρ>(x,t)=-∫-∞t∂j+(x,t′)∂t'dt′,ρ<(x,t)=-∫-∞t∂j-(x,t′)∂t′dt′ with the initial conditions ρ>(x,-∞)=ρ<(x,-∞)=0. Then, it is possible to introduce the quantities  (2.12)N>(x,∞;t)≡∫x∞ρ>(x′,t)dx′=∫-∞tj+(x,t′)dt′>0,N<(-∞,x;t)≡∫-∞xρ<(x′,t)dx′=-∫-∞tj-(x,t′)dt′>0,   which have the meaning of probabilities for the particle wave packet Ψ(x,t) to be located at time t on the semiaxis (x,∞)  and  (−∞,x), respectively, as functions of the flux densities  j+(−∞,t) and j−(x,t), provided that the normalization condition  ∫−∞∞ρ(x,t)dx=1 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 ρ<(x,t), and by adopting the boundary conditions  j+(−∞,t)=j−(−∞,t)=0. Then, by differentiating N>(x,∞;t) and  N<(−∞,x;t) with respect to t, one obtains  (2.13)∂N>(x,∞;t)∂t=j+(x,t)>0,∂N<(-∞,x;t)∂t=-j-(x,t)>0.      Finally, from the last four equations one can easily infer that  (2.14)W+(x,t)dt=j+(x,t)dt∫−∞∞j+(x,t)dt=∂N>(x,∞;t)/∂tN>(x,∞;∞),W-(x,t)dt=j-(x,t)dt∫−∞∞j-(x,t)dt=∂N<(-∞,x;t)/∂tN<(-∞,x;∞), which justify the abovementined probabilistic interpretation of  W±(x,t). 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(x)〉 of the time t at which a particle passes through position x (when travelling in only one positive x-direction), and  〈t±(x)〉 of the time t at which a particle passes through position x, when travelling in the positive or negative direction, respectively,   (2.15)〈t(x)〉=∫−∞∞tj(x,t)dt∫−∞∞j(x,t)dt=∫0∞dE(1/2)[G*(x,E)t^vG(x,E)+vG*(x,E)t^G(x,E)]∫0∞dEv|G(x,E)|2,                  where G(x,E) is the Fourier transform of the moving 1D wave packet  (2.16)Ψ(x,t)=∫0∞G(x,E)exp(−iEtℏ)dE=∫0∞g(E)φ(x,E)exp(−iEtℏ)dE, when going on from the time representation to the energy one,  (2.17)〈t±(x)〉=∫−∞∞tj±(x,t)dt∫−∞∞j±(x,t)dt, and also the mean durations of particle 1D transmission from xi to  xf>xi and 1D particle reflection from the region  (xi,∞) into xf≤xi:  (2.18)〈τT(xi,xf)〉=〈t+(xf)〉−〈t+(xi)〉,〈τR(xi,xf)〉=〈t−(xf)〉−〈t+(xi)〉,      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 ∫0∞dE…, similarly to  (2.15) by using the unique Fourier (Laplace) transformations and the energy expansion of    j±(x,t)=j(x,t)θ(±j), but it is evident that they result to be rather bulky.If one does now generalize the expressions (2.15) and (2.17) for 〈tn〉 with  a generic value n=2,3,…, then we will be able to write down for 〈f(t)〉 with any analytic function of time f(t),  the one-to-one relation  (2.19)〈f(t)〉=∫-∞∞j(x,t)f(t)dt∫-∞∞j(x,t)dt=∫0∞dE(1/2)[G*(x,E)f(t^)vG(x,E)+vG*(x,E)f(t^)G(x,E)]∫0∞dEv|G(x,E)|2 from the time to the energy representation. For free motion, one has G(x,E)=g(E)exp(ikx), φ(x,E)=exp(ikx),  and E=ℏ2k2/2μ=μv2/2, while the normalization condition is
(2.20)∫0∞|G(x,E)|2dE=∫0∞|g(E)|2dE=1,
and the  boundary conditions are  (2.21)[dng(E)dEn]E=0=[dng(E)dEn]E=∞=0,forn=0,1,2,….Conditions (2.21) imply a very rapid decrease till zero of the flux densities near the boundaries E=0 and E=∞:  this complies with the actual conditions of real experiments, and therefore they does not represent any restriction of generality.In (2.19), t^ 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|g(E)|2≈Kδ(E−E¯), K being a constant, quantity j(x,t) goes into ρ(x,t) and (2.19) goes into the more simple relation (2.22)〈f(t)〉≡∫-∞∞j(x,t)f(t)dt∫-∞∞j(x,t)dt≈∫-∞∞ρ(x,t)f(t)dt∫-∞∞ρ(x,t)dt≈∫0∞dEG*(x,E)f(t^)G(x,E)∫0∞dE|G(x,E)|2,  because of the relations   j(x,t)vρ(x,t)≈v¯ρ(x,t).Now, one can see that two canonically conjugate operators, the time operator (2.1a), (2.1b), and (2.26) and the energy operator  (2.23)E^={Eintheenergy(E)-representation,iℏ∂∂tinthetime(t)-representation,  satisfy the typical commutation relation  (2.24)[E^,t^]=iℏ.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 (−∞<t<∞) to the E-semiaxis (0<E<∞) 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  (2.25)ΔEΔt≥ℏ2(where the standard deviations are Δa=✓Da, quantity Da being the variance Da=〈a2〉−〈a〉2;   and a=E,t, while 〈⋯〉 denotes an average over t  by the measures W(x,t)dt or W±(x,t)dt  in the t-representation or an average over E 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,p0}, with q0=t and p0=−E, 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  t^ in (2.1a), (2.1b), and (2.26) are sufficient for considering them as quantum observables: namely, the uniqueness of the “spectral decomposition” (also called spectral function)  for operators t^, as well as for t^n(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^n(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 isno practical difference between self-adjoint and maximal Hermitian operators for systems with continuous energy spectra. Let us underline that the mathematica, properties of t^n(n>1) are quite enough for considering time asa quantum-mechanical observable (like for energy, momentum, spatial coordinates,…) without having to introduce any new physical postulates.Now let us analyse the so-calledpositive-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 (2.26)t^=(−iℏ2)∂↔∂E  (the so-called bilinear form), where the meaning of  the sign  ↔ is clear from the following definition: (f,t^g)=(f,(−iℏ/2)(∂/∂E)g)+((−iℏ/2)(∂/∂E)f,g).  For this form the direct elimination of the point E=0 is not necessary because it is eliminated automatically in (f,t^f) and in ∫-∞∞tj(x,t)dt by such bilinearity. And such an elimination of the point E=0 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 (E-) representation, with 0<E<∞,  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    −∞<k<∞:  (2.27)Ψ(x,t)=∫-∞∞dkg(k)φ(x,k)exp(−iEtℏ),  with  E=ℏ2k2/2m,k≠0. In such a case the time operator (2.1a), (2.1b), and (2.26) (acting on momentum eigenvector, defined on −∞<k<∞, is already formally self-adjoint, with the boundary conditions  (2.28)[dng(k)dkn]k=−∞=[dng(k)dkn]k=∞=0,n=0,1,2,…,    except for the fact that we have excluded point k=0; 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: (2.29)Ψ(x,t)=∫0∞dE(E)−1/2g((2mE)1/2ℏ)φ(x,(2mE)1/2ℏ)exp(−iEtℏ)+∫0∞dE(E)−1/2g(−(2mE)1/2ℏ)φ(x,−(2mE)1/2ℏ)exp(−iEtℏ). If we now introduce the weight (2.30)g˜(E)=(m2Eℏ2)1/4[g((2mE)1/2ℏ)g(−(2mE)1/2ℏ)] as a “two-dimensional” vector, then  (2.31)∫-∞∞|Ψ(x,t)|2dx=∫0∞dE|g˜(E)|2<∞, the norm being (2.32)g˜(E)=g*(E)⋅g(E)>0.If the wave packet (2.27) is one directional and g(k)≡g(k)Θ(k), then the integral ∫-∞∞dk goes on to the integral ∫0∞dk 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  E→k 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(2.33)dP(x,t)≡Z(x,t)dx=|Ψ(x,t)|2dx∫-∞∞|Ψ(x,t)|2dx 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 (x,x+dx) at the moment t, independently from the motion processes. As a consequence, the quantity  (2.34)P(xi,xf,t)=∫x1xf|Ψ(x,t)|2dx∫-∞∞|Ψ(x,t)|2dx  will have the meaning of the probability of particle dwelling in the spatial range (x1,x2) at the instant t. Taking into account the equality  (2.35)∫-∞∞j(x,t)dt=∫-∞∞|Ψ(x,t)|2dx, which evidently follows from the 1D continuity relation (2.8), the mean dwell time can be presented in the following form:
(2.36a)〈τdw(xi,xf)〉=∫-∞∞dt∫xixf|Ψx,t|2dx∫-∞∞jin(xi,t)dt with the flux density jin for the initial “dwelling” free motion through point xi. The expression (2.36a) can be rewritten in the following equivalent form
(2.36b)〈τdw(xi,xf)〉=[∫-∞∞tj(xf,t)dt−∫-∞∞tj(xi,t)dt]∫-∞∞jin(xi,t)dt,
taking into account the continuity relation (2.8) for the total flux density  j(x,t) in the interval       (xi,xf) at time  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]) (2.37)A→(r→,t)=∫k0d3kk0χ→(k→)φ(k→,r→)exp(−ik0t), where, as usual, A→(r→,t) is the electromagnetic vector potential, and r→={x,y,z}, k→={kx,ky,kz}, k0≡ω/c=ε/ℏc, k≡|k→|=k0, where the gauge condition divA→=0 is assumed.The axisx has been chosen as the propagation direction,                 χ→(k→)=∑i=yzχi(k→)e→i(k→); with e→ie→j=δij, xi,xj≡y,z, χi(k→) is the probability amplitude for the photon to have momentum k→ and polarization e→j along xj, and it is φ(k→,r→)=exp(ikxx) in the case of   plane waves, while φ(k→,r→) 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:(2.38)ρem(x,t)dx=S0dx∫S0dx,S0=∬s0dydz (s0=[E→*E→+H→*H→]/4π being the energy density, the electromagnetic field being H→=rotA→, E→=-(1/c)∂A→/∂t) of a photon to be found (localized) in the spatial intervall(x,x+dx) along axis x at the moment t, and the flux probability density(2.39)jem(x,t)dt=Sx(x,t)dt∫Sx(x,t)dt,Sx(x,t)=∬sxdydz (with sx=cRe[E→*H→]x/8π being the energy flux density, H→=rotA→)  of a photon to pass through the point (plane) x in the time interval(t,t+dt),  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:
(2.40)〈t(x)〉=∫-∞∞tJem,xdt=∫-∞∞tSx(x,t)dt∫-∞∞Sx(x,t)dt, where the form (2.1b) of time operator is valid also for photons with natural boundary conditions χi(0)=χi(∞)=0 in the energy representation  (ε=ℏck0), quite similarly to (2.1b)–(2.3) for nonrelativistic particles in the energy representation.The energy densitys0 and energy flux density sx satisfy the relevant continuity equation                                                                                                                                             (2.41)∂s0∂t+∂sx∂x=0, 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  〈t(x)〉,Dt(x) 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 quantitiesJem,x,±=Jem,xΘ(±Jem,x) 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 J by Jem).


## 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                                 (2.42)ψ(x,t)=∑n=0…gnφn(x)exp[−i(εn−ε0)tℏ], (where φn(x) are orthogonal and normalized wave functions of system bound states which satisfy  H^φn(x)=εnφn(x),H^ being the system Hamiltonian, ∑n=0…|gn|2=1,  here we factually omitted a nonsignificant phase factor  exp(−iε0t/ℏ)  as being general for all terms of the sum ∑n=0…)  for  describing the evolution of systems in the regions of the purely discrete spectrum. Without limiting the generality, we choose moment t=0 as an initial time  instant.Firstly, we will consider those systems, whose energy levels are spaced with distances for which the maximal common divisor isfactually 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)  T=2πℏ/D,D being the maximal common divisor of distances between system energy  level.In thet-representation the relevant energy operator  H^ is a self-adjoint operator acting in the space of periodical functions whereas the function   tψ(t)  does not belong to the same space. In the space of periodical functions the time operator t^, even in the eigen representation, has to be also a periodical function of time t. 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  (2.43)t^=t−T∑n=o∞Θ(t−[2n+1]T2)+T∑n=0∞Θ(−t−[2n+1]T2),  which is the so-called saw-function of  t (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 theself-adjoint energy and time operators acquires in this case (discrete energies and periodical functions) the form  (2.44)[E^,t^]=iℏ{1−T∑n=0∞δ(t−[2n+1]T)}. Let us recall (see, e.g., [70, 71]) that a generalized form of uncertainty relation holds
(2.45)(ΔA)2⋅(ΔB)2≥ℏ2[〈N〉]2
for two self-adjoint operators A^ and B^, canonically conjugate each to other by the commutator (2.46)[A^,B^]=iℏN^,N^ being a third self-adjoint operator.  One can easily obtain  (2.47)(ΔE)2⋅(Δt)2≥ℏ2[1−T|ψ(T/2+γ)|2∫−T/2+T/2|ψ(t)|2dt], where the parameter γ (with an arbitrary value between −T/2 and +T/2 ) is introduced for the univocality of calculating the integral on right part of (2.47) over dt  in the limits from −T/2 to  +T/2,  just similarly to the procedure introduced in [68] (see also [70, 71]).From (2.47) it follows that when ΔE→0 (i.e., when |gn|→δnn′) the right part of (2.47) tends to zero since ψt2 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   ΔE≫D and    ψ(T+γ)2≪T∫−T/2T/2|ψ(t)|2dt,  the periodicity condition may be inessential for  Δt≪T, that is, (2.47) passes to uncertainty relation (2.11), which is just the same one as for systems with continuous spectra.In principle, one can obtain the expression for the time operator (2.43) also in energy representation. If one will calculate the mean value 〈t(x)〉 of instants of particle passing through point x, then after a series of bulky transformations he will obtain the following expression:  (2.48)t^=iℏ2∑n;>n(-1)Nn-Nn′Δ↔n′Δn′εn in the energy representation, where  Nn=(εn−ε0)/D; the bilinear operation, denominated  by  Δ↔n, signifies (2.49)An*Δ↔n′An=An*Δn′An−AnΔn′An*,Δn′An=An′−An,〈t(x)〉=∑n=0∞gn*φn*(x)t^gnφn(x)∑n=0∞|gnφn(x)|2. (Of course, one has to average over the flux density, but for the simplicity in this case it is possible to make averaging over  Ψx,t2.)  Operator (2.43) for two levels (n=0,1) acquires the more simple form (2.50)t^=-iℏ2Δ↔Δε, and when  D=ε1−ε0→0,  the expression (2.50) passes to the differential form (2.51)t^=-iℏ2∂↔∂ε, 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, γlim) by quasicycles with quasiperiodical evolution and for sufficiently long intervals of time the motion inside such systems (however, less than ℏ/γlim) 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 ∑n=0∞ contains only one term (gn→δnn′), 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 (α-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 aquantum-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 phenomenonwithout any direct classical limit (namely for particles);(iii)
there are essential physical and mathematicaldifferences 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 quitesimilarly 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 (Section3.2) 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 thex-direction, and consider a time-independent barrier in the interval (0,a);—see Figure 1, in which a larger interval (xi,xf),  containing the barrier region, is also indicated.As it is well known, in the case of a rectangular potential barrier of the heightV0, the stationary wave function for a particle with mass m and energy E<V0 has the usual form (e.g., [26, 27, 32, 47, 72–81, 90] and a lot of other papers):                  (3.1)ψ(k,x)={exp(ikx)+ARexp(-ikx),x≤0(regionI),αexp(-χx)+βexp(χx),0≤x≤a(regionII),ATexp(ikx),x≥a(regionII),                           where k=(2mE)1/2/ℏ, χ=2m(V0-E)1/2/ℏ,  AR, α, β and AT  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 numberiχ. 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 x, travelling in the positive or negative direction of the x-axis, and the variances of the distributions of these times, respectively, as       (3.2)〈t±(x)〉=∫-∞∞tj±(x,t)dt∫-∞∞j±(x,t)dt,Dt±(x)=∫-∞∞t2j±(x,t)dt,j±(x,t) being the positive or negative values, respectively, of the probability flux density j(x,t)=Re[(iℏ/m)Ψ(x,t)∂Ψ*(x,t)/∂x]  for an evolving time-dependent normalized wave packetΨ(x,t). We recall here the equivalence of canonically conjugated time and energy representations, with appropriate measures of averaging, in the following sense: 〈⋯〉t=〈⋯〉E (index t is omitted in all expressions for 〈⋯〉t 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(3.3)〈τT(xi,xf)〉=〈t+(xf)〉−〈t+(xi)〉,(3.4)DτT(xi,xf)=Dt+(xf)+Dt+(xi), with −∞<xi≤0 and a≤xf<∞.  For a pure tunnelling process one has  (3.5)〈τtun(0,a)〉=〈t+(a)〉−〈t+(0)〉,Dτtun(0,a)=Dt+(a)+Dt+(0). Similar expression we have for the penetration (into the barrier region II) temporal quantities   〈τpen(xi,xf)〉 and Dτpen(xi,xf) with 0<xf<a. For reflections in any point xf<a one has  (3.6)〈τR(xi,xf)〉=〈t−(xf)〉−〈t+(xi)〉,DτR(xi,xf)=Dt−(xf)+Dt+(xi).We stress that these definitions hold within the framework of conventional quantum mechanics,without introducing any new physical postulate.In the asymptotic cases, when|xi|≫a,  (3.7)〈τTas(xi,xf)〉=〈t(xf)〉T−〈t(xi)〉in,(3.8)〈τTas(xi,xf)〉=〈τT(xi,xf)〉+〈t+(xi)〉−〈t(xi)〉in, where 〈⋯〉T and 〈⋯〉in denote averagings over the fluxes corresponding to ψT=ATexp(ikx) and   ψin=exp(ikx), respectively.For initial wave packets(3.9)Ψin(x,t)=∫0∞G(k−k¯)exp(ikx-iEtℏ)dk,(where  E=ℏ2k2/2m,∫0∞|G(k−k¯)|2dE=1,G(0)=G(∞)=0,k>0)   with sufficiently small energy (momentum) spreads when  (3.10)∫0∞vn|GAT|2dE≅∫0∞vnG2dE,n=0,1,v=ℏkm, we get (3.11)〈τTas(xi,xf)〉≅〈τTPh(xi,xf)〉E, where (3.12)〈⋯〉E=∫0∞dEv|G(k−k¯)|2⋯∫0∞dEv|G(k−k¯)|2,τTPh(xi,xf)=(1v)(xf−xi)+ℏd(argAT)dE are the phase transmission time obtained by the stationary-phase approximation. At the same approximation and with a small contribution of Dt+(xi) into the variance  DτT(xi,xf) (that can be realized for sufficiently
large energy spreads, i.e., short wave packets) we get
(3.13)DτT(xi,xf)=ℏ2〈(d|AT|/dE)2〉E〈|AT|2〉E. 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 Dt+(xi) plus DτT(xi,xf) which is born by the barrier influence and formally is described by (3.13).At the quasimonochromatric limitG2→δ(E−E),E being ℏ2k¯2/2m, we get for  〈τTasxi,xf〉≅〈τTPhxi,xj〉E strictly the ordinary phase time, without averaging.  For a rectangular barrier with height V0 and χa≫1 (where χ=2mV0−E1/2/ℏ), the expressions (3.11) and (3.13), for xi=0,xf=a  and  am/ℏk¯≫Dt+(xi),  pass into the known expressions  (3.14)τtunPh=2vχ   (coincident with the phase time [26, 27, 47, 83]), and  (3.15)(DτtunPh)1/2=akvχ   (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.Forreal weight amplitude G(k−k¯), when 〈t0〉in=0, from (3.8) we obtain  (3.16)〈τtun(0,a)〉=〈τtunPh〉−〈t+(0)〉.By the way, if the measurement conditions are such that only the positive-momentum components of wave packets are registrated, that is,Λ+Ψxi,t=Ψin(xi,t),Λ+ being the projector onto positive-momentum states, then for anyxi  from (−∞,0) and  xf from (a,∞)(3.17)〈τT(xi,xf)〉+=〈τTPh(xi,xf)〉E,(3.18)〈τtun0,a〉+=〈τtunPh〉E,  because 〈t0〉+=〈t0〉in.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 thephase tunnelling time(3.19)τtun,emPh=2cχemforχema≫1.From (3.19), we can see that when χema>2 the effective tunnelling  velocity  (3.20)vtuneff=aτtun,emPh 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 Section2, 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  j(xi,t)=jinxi,t+jRxi,t+jintxi,t and j(xf,t)=jTxf,t with jin, jR and jT corresponding to the wave packets Ψin(xi,t),ΨR(xi,t) and ΨT(xf,t), constructed from the stationary wave functions ψin,ψR=ARexp(−ikx) and ψT, respectively, and also (3.21)jint(x,t)=Re{(iℏm)[Ψin(x,t)∂ΨR*(x,t)∂x+ΨR(x,t)∂Ψin*(x,t)∂x]},∫-∞∞jint(xi,t)dt=0.One obtains from (2.36a) and (2.36b) of
Section 2,(3.22)〈τDw(xi,xf)〉=〈T〉E〈τT(xi,xf)〉+〈R(xi)〉E〈τR(xi,xf)〉,  with 〈T〉E=〈AT2v〉E/〈v〉E,〈Rxi〉E=〈R〉E+〈rxi〉,〈R〉E=〈|AR2v〉E/〈v〉E, 〈T〉E+〈R〉E=1, and
(3.23)〈r(x)〉=∫-∞∞[J+(x,t)−Jin(x,t)]dt∫-∞∞Jin(x,t)dt. One can see that 〈r(x)〉 is negative and tends to 0 when x tends to −∞.WhenΨin(xi,t) and ΨR(xi,t) are sufficiently well separated in time, so that 〈rxi〉=0, it follows from (3.22) that the simple weighted average rule(3.24)〈τDw(xi,xf)〉=〈T〉E〈τT(xi,xf)〉+〈R〉E〈τR(xi,xi)〉                         is valid. For a rectangular barrier with χa≫1 and quasimonochromatic particles, the expressions (3.22) and (3.24) with xi=0 and xf=a pass to the known expressions (3.25)〈τDw(xi,xf)〉=〈ℏkχV0〉E,  (taking account of the interference term 〈rxi〉)   (3.26)〈τDw(xi,xf)〉=〈2χv〉E                           (when the interference term 〈rxi〉  is equal to 0).WhenAR=0, that is, a barrier is transparent, the mean dwell time (3.22) is automatically equal to  (3.27)〈τDw(xi,xf)〉=〈τT(xi,xf)〉.It is not clear how to definedirectly 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 x1 and x2.


### 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 thex direction, after tunnelling the spin develops small y and z components (see Figure 3).The Larmor timesτy,TLa  and τz,TLa are defined by the ratio of the spin-rotation angles around axes  z and y (in turn defined by the developed y- and z-spin components, resp.) to the precession (rotation) frequency.As toτz,TLa, 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 AT and coincides with the expression (3.15) (of course, at the same approximations when (3.15) is valid).For an opaque rectangular barrier withχa≫1 the expressions (3.28)〈τy,tunLa〉=〈τDw(xi,xf)〉=〈ℏkχV0〉E,(3.29)〈τz,tunLa〉=〈maℏχ〉E (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 AT for the same reasons as for 〈τz,tunLa〉.


### 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 exp[iS(x(t))/ℏ], where S is the action associated with the path x(t). 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 〈τy,tunLa〉 and 〈τz,tunLa〉, respectively.An interesting development of this approach, theinstanton 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 exp(iS/ℏ). 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 f(x,p), so that the flux at position x can be separated into positive and negative components:  (3.30)J(x)=J+(x)+J−(x),  with J+(x)=∫0∞(p/m)f(x,p)dp and J−=J−J+. 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 J± instead of our J±. The dwell time decomposition in this approach takes the form  (3.31)〈τDw(xi,xf)〉=〈T〉E〈τT(xi,xf)〉+〈RM(xi)〉E〈τR(xi,xf)〉, with RM(x)=∫0∞|J−(x,t)|dt. Asymptotically, RMx tends to 〈R〉E 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:  (3.32)τDw(xi,xf)=∫0∞dt∫x1x2|Ψ(x,t)|2dx=TτT(xi,xf)+RτR(xi,xf),with (3.33)τT(xi,xf)=∫0∞dt∫x1x2|Ψ(x,t)|2Θ(x−xc)dx/T,τR(xi,xf)=∫0∞dt∫x1x2|Ψ(x,t)|2Θ(xc−x)dx/R,where T and R are here the mean transmission and reflection probability, respectively,  the bifurcation line xc=xc(t), separating transmitted and reflected trajectories, is defined by relation  (3.34)T=∫−∞∞|Ψx,t|2Θ(x−xc)dx.                                 Factually, in addition to the difference in the temporal integration in this and our formalisms (∫0∞dt  and ∫−∞∞dt, resp.), sometimes essential, this approach gives one more alternative—in separating the flux by the line xc:  (3.35)J(x,t)=[Jx,t]T+Jx,tR,                                    with      (3.36)[Jx,t]T=J(x,t)Θ(x−xc(t)),[Jx,t]R=J(x,t)Θ(xc(t)−x).


### 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(3.37)τtunPh=ℏd(argAT+ka)dE, (which is the mean tunnelling time〈τtun〉 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], AT and E=ℏ2k2/(2m) being the transmission amplitude and the particle kinetic energy, resp.) on the barrier width a for sufficiently large a.  In particular, for a rectangular potential barrier  AT=4ikχ[(k2−χ2)D−+2ikχD+]−1exp[−(κ+iχ)a], D±=1±exp(−2χa), χ=[2m(V0−E)]1/2/ℏ,V0 being the barrier height,  and  τtunPh→2/(υχ)  when χa≫1 (υ=ℏk/m 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 themean dwell time〈τtunDw〉, the mean Larmor time〈τy,tunL〉 and the real part of the complex tunnelling time obtained by averaging over the Feynman pathsReτtunF, which are equal ℏk/(χV0) 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 themean tunnelling time within the O-R approach, is directly seen from the expression (3.38)〈τtun〉=〈t+a〉−〈t+0〉=〈τtunPh〉E−〈t+0〉,    (where 〈⋯〉E  denotes averaging over the initial wave-packet energy spread and (3.39)〈t±(x)〉=∫−∞∞tj±(x,t)dt∫−∞∞j±(x,t)dtwithj±(x,t)=Θ(±j)j(x,t),j(x,t) being the probability flux density for a wave packet moving along axis x through a barrier located in the interval (0,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 timeτz,tunL, from Sections 4.2 and 3.4 it follows that (3.40)τz,tunL=ℏ[〈(d|AT|/dE)2〉〈|AT|2〉]1/2, the Bttiker-Landauer timeτtunB−L [111–113], and the imaginary part of  the complex tunnelling timeImτtunF [92], obtained within the Feynman approach, which are equal to (3.39), they become equal to am/(ℏχ), that is, proportional to the barrier width  a, 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 [Ddynτtun]1/2 where Ddynτtun is the dynamical tunneling-time variance caused by the barrier influence only and defined by the equation   Ddynτtun=Dτtun−Dt+(0) with Dτtun=〈τtun2〉−〈τtun〉2, 〈τtun2〉=〈[t+(a)−〈t+(0)〉]2〉+Dt+(0) (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 Figure4 the dependences of the values of 〈τtun(0,a)〉 from a are presented for electronic wave packets and rectangular barriers with the same parameters as in [47] (V0=10eV; mean electron energies E=2.5,5,7.5eV with Δk=0.02A−1 (curves 1a, 2a, and 3a, resp.); energy E=5eV with k=0.04A−1 and 0.06A−1 (curves 4a and 5a, resp.)). The curves 〈τtunPh〉  corresponding to different energies and Δk merge practically into one curve 6. Since the dependence of 〈τtunPh〉 from a is very weak, the dependence of 〈τtun(0,a)〉 from a is defined  mainly by the dependence of 〈t+(0)〉 from a (curves 1b–5b, correspondent to 1a–5a,  resp.).All these calculations manifest the negative values of〈t+(0)〉. 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 a.  Since all 〈t+(0)〉 are negative, the values of 〈τtun(0,a)〉 are always positive and, moreover, larger than 〈τtunPh〉, 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  −iV1(V1>0)   to V0, then for small absorptions, when V1≪V0 and  V1m1/2vχa/2V0−E3/2≪2, 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 pointx=x0 (with |x0|≫a) at the instant t=0 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 thex axis through a rectangular potential barrier with height V0, localized in the interval [0,a]. The chosen here stationary wave function for a particle with mass m and energy E<V0 has the usual form (3.1). The time dependent wave packet ψ(x,t) is formed with wave functions (3.1):  (3.41)Ψ(x,t)=∫0∞g0(k)ψ(k,x)exp(−iEtℏ)dk, with the weight amplitude  (3.42)g(k)=g0(k)exp[−i(k−k0)x0]=Cexp{−[(k−k0)2Δk]2−i(k−k0)x0},C=[(2π)3/2Δk]−1/2 being the normalization coefficient. Here, since in calculations the only subbarrier part of the wave packet was considered, integrating over k in (3.42) had to be made at the limits from 0 to (2mV0)1/2/ℏ [117].The initial wave function turns out to be(3.43)Ψin(x,t)≅C0sin−1/2exp{−[x-(x0+νt)]22sin+ik0x−iE0tℏ},  whose center transits from point x=x0 at the instant t=0, moves along axis x from the left to the right with the velocity v and in the absence of barrier crosses the point x=0 at the instant t0=−x0/v.The flux densityj(x,t)=Re[(iℏ/2m)Ψ(x,t)∂Ψ*(x,t)/∂x], as usually in the O-R method,  has been considered as a function of the arrival  times at point x, so that the mean time instant for particles passing through point x=a (the mean time of the particle exit from the barrier), has been chosen as  (3.44)〈t(a)〉=∫0∞tj(a,t)dt∫0∞j(a,t)dt,while the mean time instant for particle passing through point x=0 (the mean time of the particle entrance into the barrier) is defined by relation  (3.45)〈t+(0)〉=∫0∞tj+(0,t)dt∫0∞j+(0,t)dt,where j+(0,t) represents the positive values of the flux density j(0,t), corresponding, therefore,  to particles moving through point x=0 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 timeτ is then defined as τ=〈t(a)〉−〈t+(0)〉.The flux densityj(a,t) contains the wave function  (3.46)Ψ(a,t)=∫0∞g(k)ψT(ka)exp(−iEtℏ)dk,where ψT(k,a) is the value of the stationary wave function (3.1) at point x=a, corresponding to the transmitted wave. Using the usual expression for the amplitude of the transmitted wave, we can ψT(k,a) represent in the form  (3.47)ψT(k,a)={2ikχD<,E<V0,2iikqD>,E>V0, where q=2m(E−V0)1/2/ℏ,  (3.48)D<=(k2−χ2)sinh(χa)+2ikχcosh(χa),D>=(k2+q2)sin(qa)+2ikqcos(qa). The normalization integral in (3.45),  (3.49)Na=∫0∞j(a,t)dt can be evaluated using  (3.42), (3.46), and (3.47):  (3.50)Na=ℏmRe{∫0∞dkg(k)ψT(k,a)∫0∞dk′k′g(k′)ψT(k′,a)∫0∞exp[-iℏ2m(k2-k′2)t]dt}. If x0 is chosen sufficiently far from the left of the barrier, the contribution of j(a,t) in the integral (3.49) from t≤0 is negligible small. In this case, the lower integration limit over t in (3.50) can be taken as −∞. Then the integration over time gives 2π(m/ℏk)δ(k−k′) and the integral (3.50) can be cast in the form  (3.51)Na=2π∫0∞g02(k)|ψT(k,a)|2dk, where  (3.52)|ψT(k,a)|2={4k2χ24k2χ2+V02sinh2(χa),E<V0,4k2q24k2q2+V02sin2(qa),E>V0.  Now we evaluate the integral  (3.53)Ia=∫0∞tj(a,t)dt.Using (3.50) and (3.51), we can write  (3.54)Ia=ℏmRe{∫0∞dkg(k)ψT(k,a)∫0∞dk′g(k′)ψT(k′,a)∫0∞k′texp[-iℏ2m(k2-k′2)t]dt}, which can be set in the form  (3.55)Ia=2πmℏ∫0∞g02(k)1k{ψ1(k,a)ψ2′(k,a)−ψ2(k,a)ψ2′(k,a)−x0|ψ(k,a)|2}dk, where ψ1(k,a)≡ReψT(k,a), ψ2(k,a)≡ImψT(k,a). After differentiating (3.47) over k, one can see the following: (3.56)ψ1ψ2′−ψ1′ψ2={−8k2χ[k2(k2−χ2)χa-V02cosh(χa)sinh(χa)][4k2χ2+V02sinh2(χa)]2,E<V0,8k2q[k2(k2+q2)qa−V02cos(qa)sin(qa)][4k2q2+V02sin2(qa)]2,E>V0. From (3.51) and (3.56) it follows that, with an appropriate choice of x0, the mean instant 〈t(a)〉 can be defined by the simple relation  (3.57)〈t(a)〉=mℏ∫0∞g021k[ψ1ψ2′−ψ2ψ1′-x0|ψT|2]dk/∫0∞g02|ψT|2dk.  As regards the calculation of τ¯0, the wave function entering the flux density j(0,t) is  (3.58)ψ(0,t)=∫0∞g(k)[1+AR(k)]exp(−iEtℏ)dk,where   (3.59)AR(k)={V0sinh(χa)D<,E<V0,V0sin(qa)D>,E>V0. Since for calculating 〈t+(0)〉 we consider only the positive values of j(0,t), the integrals I+=∫−∞∞tj+(0,t)dt and N+=∫-∞∞j+(0,t)dt in (3.45) can be obtained only numerically.Figure5 shows the results of the calculations of the tunnelling time τ as a function of the width a of the barrier, for electrons with energy E0=5eV through the rectangular barrier of potential with height 10eV and x0=−6/Δk. We can see the manifestation of the HFE for  Δk=0.01Å−1  (curve (a)), with the asymptotic behavior of the tunneling time approaching the constant value with increasing a. 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  Δk=0.05Å−1 and Δk=0.1Å−1. For  Δk=0.1Å−1 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 (3.60)Δt≅Δ0t{1+[Δv2m2ℏ]2t2}1/2, where  Δ0t~ℏ/ΔE,  Δv=ℏΔk/m. The value of Δt strongly increases in time t for large velocity spread Δv. So, the center of the initial wave packet during wave-packet approaching the barrier and increasing time t will appear even farther due to the essentially stronger increase of Δt with time t (proportional to Δk for large t) than the decrease of |x0|, proportional to 1/Δk.  For the center of the transmitted wave packet in point x=a such delay has to be considerably smaller because  the value τ is smaller than the limited value τph¯ (τph being the phase tunnelling time) due to the advance caused by the difference between j+(0,t) and jin(0,t),jin being Re[(iℏ/2m)Ψin(∂Ψin*/∂x)] (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 Δv becoming even negative.Figure2 shows qualitatively the behavior of the time shift of the outgoing wave packet with respect to the ingoing wave packet with different momentum spread Δk.t0,t0′,t0′′ represent the time instants when the wave packet peak passes through the initial point x=0 of the barrier. tf,tf′, and tf′′ represent the transit time through the final point x=a 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〈τtun(0,a)〉, 〈τtunPh〉 and 〈t+(0)〉 from a.Figure 5
The tunneling timeτ as a function of the barrier depth a, for different values of Δk: (a) Δk=0.01Å,   (b) Δk=0.05Å,  and (c) Δk=0.1Å.Figure 6
Qualitative description of the time shift between peakstf,tf′, and tf′′, and t0,t0′, and t0′′ for the probability density ρ(x=a,t) and ρ(x=0,t) 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 AppendixF (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 massm and kinetic energy E=ℏ2k2/2m=mv2/2, through two equal rectangular barriers with height V0(V0>E) and width a, the quantity l being the distance between them see Figure 7. The stationary Schroedinger equation is (3.61)−ℏ22∂2∂x2ψ(x)+V(x)ψ(x)=Eψ(x),  where V(x)=0 outside the barriers and  V(x)=V0  inside the potential barriers. In various regions I(x≤0),II(0≤x≤a),III(a≤x≤l+a),IV(l+a≤x≤l+2a), and V(x≥l+2a), the solutions of (3.61) are the following:  (3.62)ψI=eikx+A1Re−ikx,ψII=α1e−χx+β1eχx,ψIII=A1T[eikx+A2Re−ikx],ψIV=A1T[α2e−χ(x−l−a)+β2eχ(x−l−a)],ψV=ATeikx,AT≡A1TA2T, where χ=2mV0−E1/2/ℏ, and quantities A1R,A2R,A1T,A2T,α1,α2,β1, and β2 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  ψI,II,III,IV,V and their derivatives dψI,II,III,IV,V/dx at points x=0,a,l+a,l+2a. The obtained expressions for them for opaque barriers, when χa→∞, are  (3.63)A1R→ik+χik-χ,A1T→e−χae−ik(l+a)A,α1→2ikik−χ,β1→e−2χa(k−iχ)sinklχA,A2R→e2ik(l+a)ik+χik−χ,A2T→e2ik(l+a)−4ikχ(ik−χ)2,α2→eik(l+a)2ikik−χ,β2→eik(l+a)−2χa−2ik(ik+χ)(ik−χ)2, where  (3.64)A=2kχ2kχcoskl+(χ2−k2)sinkl.
From (3.63) one can derive that the phase tunneling time is  (3.65)τtunPh=ℏ∂arg[ATeik(l+2a)]∂E→2vχ, 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. Figure7), the amplitude of the transmitted wave in this case is defined by the formula [88, 129]  (3.66)AT(k)=exp(−2ika)D(k), where (3.67)D(k)=cosh2(χa)+14sinh2(χa)[σ2cos(2kl)−δ2]+isinh(χa)[δcosh(χa)+14σ2sinh(χa)sin(2kl)],δ=(χ2−k2)/kχ, and σ=(k2+χ2)/kχ. The dimensionless constants δ and σ are connected by the relation  (3.68)σ2=δ2+4.
A resonance is characterized by the condition [88, 120]  (3.69)|AT(k)|2=1, 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 k for which condition (3.69) is satisfied can be found from the equation  (3.70)cot(kl)=−12δtan(χa). We can show that from (3.70) it is possible to find out also the values of parameters a, m, and V0, at the resonance.  Indeed, from (3.67) and (3.70) it follows that at a resonance it is  (3.71)|D(k)|2=1. On introducing the functions   (3.72)u=cosh2(χa)−14δ2sinh2(χa),v=δcosh(χa)sinh(χa),w=14σ2sinh2(χa) and using (3.68), one can infer that these functions are connected by the relation  (3.73)u2+v2=(1+w)2. Then, by using functions (3.72), the denominator of (3.67) can be written in the following form:  (3.74)D=u+wcos(2kl)+i[v+wsin(2kl)]. It follows from (3.74) and (3.73) that  (3.75)D2=1+2w[1+w+ucos(2kl)+vsin(2kl)], and condition (3.71) gets transformed into (3.76)1+w+ucos(2kl)+vsin(2kl)=0. Taking (3.68) into account, we can write the term 1+w of (3.76) in the following form:  (3.77)1+w=cosh2(χa)+14δ2sinh2(χa), 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 E−Er   (quantity Er being the resonance value of energy E), we obtain  (3.78)D(k)=Dr+Cr(E-Er),where Dr=D(kr),  kr=2mEr/ℏ,  (3.79)Cr=mℏ2kDr′, and the index prime defines the derivative with respect to k. We can rewrite (3.78) in the following form:  (3.80)D(k)=Cr(E−Er+DrCr*|Cr|2). It follows from (3.70) that at the resonance it is  (3.81)cos(2krl)=−ur(1+wr),sin(2krl)=−vr(1+wr), where ur, vr and wr  are the values at E=Er of the functions u, v, and w, respectively.  On inserting (3.81) into (3.74), we obtain  (3.82)D(kr)=(ur+ivr)(1+wr). Differentiating (3.74) with respect to k, and inserting the result into (3.81), we get  (3.83)D′(kr)=[ur′−(urwr′−2lvrwr)(1+wr)]+i[vr′−(vrwr′−2lurwr)(1+wr)]. Then, on using (3.73), (3.79), (3.82), (3.83), and relation  (3.84)uu′+vv′=(1+w)w′, which follows from (3.73), one finds that at the resonance  (3.85)DrCr*=imℏ2kr[ur′vr−urvr′1+wr+2lwr], where (by having recourse to (3.73), (3.79), (3.83), and (3.84))  (3.86)|Cr|2=(mℏ2kr)2[(ur′)2+(vr′)2−(wr′)2+4lur′vr−urvr′1+wrwr+4l2wr2]. From (3.73) and (3.84)
one gets  (3.87)u′2+v′2−w′2=(u′2+v′2)2(1+w)2, while, from (3.86) and (3.87), one obtains (3.88)|Cr|2=(mℏ2kr)2(ur′vr−urvr′1+wr+2lwr). By differentiating the functions (3.72) with respect to k, one can see that  (3.89)ur′vr−urvr′=1χr(1+wr)[χrkra+2χrlwr+σr2cosh(χra)sinh(χra)], where χr,δr, and   σr are the values of χ,δ,  and σ at E=Er. On inserting (3.89) into (3.85) and (3.88), we can then rewrite (3.80) in the noteworthy form  (3.90)D(k)=Cr(E−Er+iβ), where  (3.91)β=ℏ2krχrm[χrkra+2χrlwr+σr2cosh(χra)sinh(χra)]−1, so that (3.80), (3.85), (3.88), and (3.90) yield   (3.92)|Cr|2=1β2. Finally, by inserting (3.90) into (3.66) and taking account of (3.92), we obtain that near a resonance it holds in general  (3.93)|AT(k)|2=β2(E−Er)2+β2. 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   (3.94)τtunPh=ℏdarg{ATexp[ik(2a+l)]}dE.  By using (3.66) and (3.76), we can rewrite it in the following form:
(3.95)τtunPh=mℏk[l−(D1D2′−D2D1′)|D|2], where (3.96)D1=u+wcos(2kl),D2=v+wsin(2kl) (the index prime denoting again the derivative with respect to k).  On inserting (3.96) into (3.95) and using (3.73), we get in general for the total tunneling phase time the remarkable formula:  (3.97)τtunPh=mℏkP|D|2, where  (3.98)P=(1+2w)l+u′v−uv′+(u′w−uw′)sin(2kl)+(vw′−v′w)cos(2kl).  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 τtun,rPh is given by the following expression:  (3.99)τtun,rPh=mℏkχ[σ2cosh(χa)sinh(χa)+δka+(1+2w)χl]. It follows from (3.65) and (3.66) that  (3.100)τtunPh=ℏdarg[exp(ikl)/D]/dE. Inserting (3.90) into (3.100), we get  (3.101)τtunPh=ℏddEarg[1Crexp(ikl)E−Er+iβ], so that, near a resonance, the behavior of the tunneling phase time in terms of the energy is represented by the interesting following formula:  (3.102)τtunPh≅mℏkl+ℏ=β(E−Er)2+β2 holding for any resonant tunneling through the two barriers.  The first term represents the time associated with the particle free flight over the distance l 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  (3.103)|D|2≈132σ2exp(4χa)[14σ2+(1−14δ2)cos(2kl)+δsin(2kl)]. Differentiating the functions u,v, and  w with respect to k, one can see that (3.104)u′v−uv′=1χ(1+w)[σ2cosh(χa)sinh(χa)+δka],u′w−uw′=2χ[δ(u+w)+14σ2cosh(χa)sinh(χa)],vw′−v′w=1χw[(4−δ2)cosh(χa)sinh(χa)−δka]. Therefore, from (3.72), (3.98), (3.103), and (3.104), we get that, still for χa≫1,  (3.105)P≈116σ2exp(4χa)[σ24+(1+δ24)cos(2kl)+δsin(2kl)]+18σ2lexp(2χa) and far from the resonances,  (3.106)τtunPh≈2mℏkχ+4mℏklexp(−2χa)[σ24+(1+δ24)cos(2kl)+δsin(2kl)]−1, Namely, when χa increases, the second term in (3.106) decreases as exp(−2χa); 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 τtunPh has been actually observed to increase (very slowly, almost linearly, and probably with very small oscillations) on l, 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 largeχa 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  (3.107)|AT|2≈321σ2exp(−4χa)[14σ2+(1−14δ2)cos(2kl)+δsin(2kl)]−1,  which shows that, with increasing χa, the  transmission coefficient square  |AT(k)|2 decreases as exp(−4χa), 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 ofχa which make negligible the mentioned second term in (3.106). Moreover, for the values of χa 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 χa,  results to depend very weakly on the distance l between the barriers: more precisely, it will depend almost linearly (but very slightly) on l, 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 asα-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 r>R2, represents the external region of null potential, region (II), delimited by R1 and R2, is the barrier region, and the internal region (III), with r<R1, 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 valueV1:  (3.108)rΨ0(1)(r,t)=∫0V1dEg(E)e−ikr−iEt/ℏ,(R2≤r<∞),   where k and E=ℏ2k2/2m are the wave number and the kinetic energy, respectively.
A section of the potential along ther-axis is shown in Figure 9.
When the wave encounters the barrier, it is partially reflected with wave function:(3.109)rΨR(1)(r,t)=∫0V1dEg(E)AR(ex)e−ikr−iEt/ℏ,(R2≤r<∞).                  At the same time, the wave packet tunnels through the barrier in region (II), where it is represented by the following function:  (3.110)rΨII(1)(r,t)=∫0V1dEg(E)(α1e−χr+β1eχr)e−iEt/ℏ,(R1≤r<R2),                   where χ=2m(V1−E)/ℏ2.  After the tunnelling through the barrier, the wave penetrates inside the well and in region (III) we can write  (3.111)rΨT(1)(r,t)=∫0V1dEg(E)AT(in)e−iKr−iEt/ℏ,(0<r<R1),                               where K=2m(E+V0)/ℏ2 is the wave number inside the well.
In general, we can say that the wave packet is described by the expression(3.112)rΨJ(1)(r,t)=∫0V1dEg(E)ΦJ(1)(k,r)e−iEt/ℏ,                              where  the index J is I, II, III, 0, T or R depending on the particular mode considered,  ΦJ(1) being the stationary wave functions: (3.113)ΦI(1)=e−ikr+AR(ex)eikr,R2≤r<∞,ΦII(1)=α1e−χr+β1eχr,R1≤r≤R2,ΦIII(1)=AT(in)e−iKr,0<r≤R2. Furthermore, g(E)  is a normalized amplitude weight factor such as  (3.114)∫0V1dE|g(E)|2=N,N<∞.  We normalize (3.1) by the following condition:  (3.115)4π∫R2∞r2dr|Ψ0(1)(r,t)|2=1,  which is strictly possible only in the limit V1 going to infinity.  For the sake of simplicity we neglect the contribution of the energy above the barrier, considering only cases where
E<V1.
All the previous formulas are written in the case of orbital momentuml=0, and the extension of this theory to the general case with l>0 can be obtained by replacing the terms exp(±ikr) and exp(±iKr) with the spherical Hanckel function of the first and second  kinds.
In addition, we have thatARex,α1,β1, and AT(in) 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 r=R2 and r=R1, finding for the external reflection coefficient the  expression:  (3.116)ARex=−e−2ikR2aRex,aRex=(χ+ik)(iK+χ)+e−2χ(R2−R1)(χ−ik)(iK−χ)(χ−ik)(iK−χ)+e−2χ(R2−R1)(χ+ik)(iK+χ),                   which, in the limit χ(R2−R1) tending to infinity, becomes −e−2ikR2((χ+ik)/(χ−ik)).
For the internal transmission coefficient we get(3.117)ATin=e−ikR2+iκR1aTin,aTin=4ikχe−χ(R2−R1)(χ−ik)(iK−χ)+e−2χ(R2−R1)(χ+ik)(iK+χ) 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 fluxesjI(1),jII(1), and jIII(1) in the three regions I, II, and III are in general defined as  (3.118)jn(1)=iℏ2m(rΨn(1)ddr(rΨn(1)*)−rΨn(1)*ddr(rΨn(1))),n=I,II,III.One can use the approximation  (3.119)∫0∞jn(1)(r,t)dt≈∫−∞∞jn(1)(r,t)dt.  For quasimonochromatic wave packet centred around the value E=〈E〉, that is, when the factor  |gE|2 is a delta function δ(E−〈E〉), with 〈E〉 in the open interval (0,V1), one obtains  (3.120)∫−∞∞jR(1)(r,t)dt∫−∞∞j0(1)(r,t)dt≈|AR(ex)(〈E〉)|2,∫−∞∞jII(1)(r,t)dt∫−∞∞j0(1)(r,t)dt≈|α1(〈E〉)e−χ(〈E〉)r+β1(〈E〉)eχ(〈E〉)r|2,∫−∞∞jIII(1)(r,t)dt∫−∞∞j0(1)(r,t)dt≈K(〈E〉)k(〈E〉)|AT(in)(〈E〉)|2.  Hence, from the conservation law for the probability fluxes  (3.121)∫−∞∞j0(1)(r,t)dt=∫−∞∞jR(1)(r,t)dt+∫−∞∞jT(1)(r,t)dt                                        one obtains  (3.122)|AR(ex)(〈E〉)|2+K(〈E〉)k(〈E〉)|AT(in)(〈E〉)|2=1,      that for V0=0 becomes  (3.123)|AR(ex)(〈E〉)|2+|AT(in)(〈E〉)|2=1. The transmission probability through the barrier from outwards is represented by the last of expressions (3.120).
The phase timesτTph(in) and τRph(ex) can be defined as the evident generalization of the following 1D definitions:  (3.124)τTph(in)=ℏ∂(arg(AT(in)(E)e−ikR1))∂E−−R1v=ℏ∂(arg(aT(in)(E))∂E,τRph(ex)=ℏ∂(arg(AR(ex)(E)e−iκR2))∂E−−R2v=ℏ∂(arg(aR(ex)(E))∂E,                                      where v=ℏk/m. The quantities τRph(ex) and τRph(ex), in the limit χ(R2−R1) approaching infinity, go to 2/vχ 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 barriersR1 and R2.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(3.125)rΨ(in)(r,t)=∫0V1dEG(E)eiKr−iEt/ℏ,(0≤r≤R1)             constructed from the overlapping stationary solutions propagating in the positive r-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(3.126)rΨR(2)(r,t)=∫0V1dEG(E)AR(in)e−iKr−iEt/ℏ,(0≤r≤R1).Afterwards, in the region (II) of the barrier a system of evanescent and antievanescent waves develops  (3.127)rΨII(2)(r,t)=∫0V1dEG(E)(α2e−χr+β2eχr)e−iEt/ℏ,(R1≤r≤R2),                       while in the outer region a propagating wave will exist  (3.128)rΨI(2)(r,t)=∫0V1dEG(E)AT(ex)eikr−iEt/ℏ,(R2≤r<∞).As in the previous case we have that G(E) is a normalized amplitude weight factor, and AR(in),α2,β2, and AT(ex) 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:  (3.129)ARin=−e−2ikR1aRin,aRin=(χ+iK)(ik−χ)+e2χ(R1−R2)(χ−iK)(ik+χ)(χ−iK)(ik−χ)+e2χ(R1−R2)(χ+iK)(ik+χ),(3.130)ATex=e−ikR2+iKR1aTex,aTex=4iKχe−χ(R2−R1)(χ−iK)(ik−χ)+e−2χ(R2−R1)(χ+iK)(ik+χ).Also in this case we can demonstrate the conservation of the current fluxes (3.131)|AR(in)(〈E〉)|2+k(〈E〉)K(〈E〉)|AT(ex)(〈E〉)|2=1,                                               and we can introduce the phase times: (3.132)τTph(ex)=ℏ∂(arg(AT(ex)(E)e−ikR1))∂E−−R1v=ℏ∂(arg(aT(ex)(E))∂E,τRph(in)=ℏ∂(arg(AR(in)(E)e−iκR2))∂E−−R1v=ℏ∂(arg(aR(in)(E))∂E,                    with v=ℏk/m. The quantities τTph(ex) and τRph(in), in the limit χ(R2−R1) approaching infinity, go to (1/v+1/V)/χ   (with V=ℏK/m) 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 Figure11), introducing the matrix of scattering S and considering the multiple reflections inside the potential well.
For this purpose we describe the stationary wave functions in the various regions as follows:(3.133)ΨI(k,r)=e−ikr−Seikr,ΨII(k,r)=αe−χr+βeχr,ΨIII(k,r)=A(e−iKr−eiLr).                                                        We can now find S, α, β, and A by connecting the various expression of the wave function and its derivative in r=R1 and r=R2.
Imposing the continuity conditions we have(3.134)S=e−2ikR2(KcosKR1+χsinKR1)(χ+ik)+e2χ(R1−R2)(KcosKR1−χsinKR1)(χ−ik)(KcosKR1+χsinKR1)(χ−ik)+e2χ(R1−R2)(KcosKR1−χsinKR1)(χ+ik),(3.135)A=e−2ikR22ikχe2χ(R1−R2)(KcosKR1+χsinKR1)(χ+ik)+e2χ(R1−R2)(KcosKR1−χsinKR1)(χ+ik).One can easily see that |S|=1, and that (3.136)A=AT(in)1+ARin,(3.137)S=−AR(ex)−iAAT(ex)=−AR(ex)+AT(ex)AT(in)1+AR(in).The physical meaning if the term 1/(1+ARin) is directly connected to the presence of an infinite sequence of multiple internal reflections that can be described by the stationary wave functions  (3.138)AT(in)(1−AR(in)+(AR(in))2−(AR(in))3+⋯)e−ikr=AT(in)1+AT(in)e−ikr,AT(in)(1−AR(in)+(AR(in))2−(AR(in))3+⋯)eikr=AT(in)1+AT(in)eikrfor the ingoing and the outgoing waves,  respectively.
The scattering phase timeτscph=ℏ(∂argSeikR2/∂E)−(−R2/v) that in the limit χ(R2−R1) approaching infinity goes to 2/vχ. So in this limit, the  scattering phase time coincides with τRph(ex).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 matrixS in the following form:  (3.139)S=αA1χ+ikA2A1χ−ikA2,                                                            where, from the comparison with (3.134), we set: α=e−2ikR2, A1=B1+e−2χ(R2−R1)B2, A2=B1−e−2χ(R2−R1)B2, B1=Kcos(KR1)+χsin(KR1), B2=Kcos(KR1)−χsin(KR1).
In the region of a resonance, we can developS into a series of powers of (E−Er),Er being the eigenvalue of the energy for the resonance, considered as a solution of the transcendental equation A1(Er)=0, and then we obtain the following:  (3.140)S=α(∂(A1χ)/∂E)E=Er(E−Er)+i(kA2)E=Er(∂(A1χ)/∂E)E=Er(E−Er)−i(kA2)E=Er,                                                that shows better its resonant character if written in the form  (3.141)S=αE−Er−iΓ/2E−Er+iΓ/2,                                               with (3.142)Γ2=−k(Er)A2(Er)χ(Er)(∂A1/∂E)E=Er=2k(Er)e−2χ(Er)(R2−R1)B2(Er)χ(Er)(∂A1/∂E)E=Er.If χ(R2−R1) is very large, we can neglect all the terms containing the negative exponential factor and write finally  (3.143)Γ2=2k(Er)e−2χ(Er)(R2−R1)F(Er),with  (3.144)F(Er)=ℏK(Er)χ(Er)m/2((1/V0+Er)(1+χ(Er)R1)+(1/V1−Er)(1+K(Er)R1)).The Case of A Rectangular Well with A Coulomb Barrier
In this case we can initiate from a simple Coulomb barrier(3.145)V=Z1Z2e2r,(Z1e and Z2e are the charges of the daughter nucleus and the emitted particle, resp.) instead of a rectangular potential barrier. Instead of the functions hl(1,2)(kr) we have to use the coulomb functions  Gl(k,η,r)±iFl(k,η,r) in the field of the coulomb barrier with  (3.146)F0(k,η,r)→sin(kr−ηln2kr+σ),r→∞G0(k,η,r)→cos(kr−ηln2kr+σ),r→∞ (where η=Z1Z2e2m/ℏ2k is the Sommerfeld parameter, σ=argΓ(1+iη)).
In this case, taking alsol=0, we obtain (3.147)S=[G0(k,η,R1)−iF0(k,η,R1)]KcosKR1−[G0′(k,η,R1)−iF0′(k,η,R1)]ksinKR1[G0(k,η,R1)+iF0(k,η,R1)]KcosKR1−[G0′(k,η,R1)+iF0′(k,η,R1)]ksinKR1, (one can easily see that here |S|=1 too)  (3.148)A=[2ieikR2k][G0′(k,η,R1)+iF0′(k,η,R1)]k(1−e2iKR1)+[G0(k,η,R1)+iF0(k,η,R1)]iK(1+e2iKR1)  for the scattering (see Figure 12). Here, F0′ and G0′ signify the derivatives of F0 and  G0 with respect to kR1, respectively.
Similarly, in this case we also obtain(3.149)ARex=−[G0(k,η,R1)+iF0(k,η,R1)]iK−[G0′(k,η,R1)+iF0′(k,η,R1)]k[G0(k,η,R1)+iF0(k,η,R1)]iK+[G0′(k,η,R1)+iF0′(k,η,R1)]k(3.150)ATin=2ikeiKR1[G0(k,η,R1)+iF0(k,η,R1)]iK+[G0′(k,η,R1)+iF0′(k,η,R1)]k, for the  impact from outside (see Figure 13), and (3.151)ARin=e2iKR1[G0(k,η,R1)+iF0(k,η,R1)]iK−[G0′(k,η,R1)+iF0′(k,η,R1)]k[G0(k,η,R1)+iF0(k,η,R1)]iK+[G0′(k,η,R1)+iF0′(k,η,R1)]k,(3.152)ATex=eiKR12iK[G0(k,η,R1)+iF0(k,η,R1)iK+[G0′(k,η,R1)+iF0′(k,η,R1)]k 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 F0G0′−G0F0′=1 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  1/(1+ARin) which has been made in connection with the formula (3.137).
For very smallk when k→0 (more precisely, when 2η≫kR1),  (3.153)G0→2(ρπ)1/2I0(2(2πρ)1/2)exp(πη),withI0(2(2πρ)1/2)→1,G0′→−2(2ηπ)1/2K0(2(2πρ)1/2)exp(πη),withK0(2(2πρ)1/2)→ln(1γ(2πρ)1/2),F0→(πρ)1/2I1(2(2πρ)1/2)exp(−πη),withI1(2(2πρ)1/2)→(2πρ)1/2,F0′→(2πη)1/2I0(2(2πρ)1/2)exp(−πη),γ≈1.781… being the Euler constant, and if (2k2/K2η/ρ)[lnγ−1(2πρ)−1/2]≪1,  the transmission (penetration) probability from outside through the Coulomb barrier into the internal rectangular potential well |ATin|2 becomes (3.154)|ATin|2→(πkK2R1)exp(−2πη) which contains the same exponential factor exp(−2πη) 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 R1 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.


## 3.1. Introduction

The developments of the study of tunneling processes in nuclear physics (α-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 aquantum-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 phenomenonwithout any direct classical limit (namely for particles);(iii)
there are essential physical and mathematicaldifferences 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 quitesimilarly 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 (Section3.2) 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 thex-direction, and consider a time-independent barrier in the interval (0,a);—see Figure 1, in which a larger interval (xi,xf),  containing the barrier region, is also indicated.As it is well known, in the case of a rectangular potential barrier of the heightV0, the stationary wave function for a particle with mass m and energy E<V0 has the usual form (e.g., [26, 27, 32, 47, 72–81, 90] and a lot of other papers):                  (3.1)ψ(k,x)={exp(ikx)+ARexp(-ikx),x≤0(regionI),αexp(-χx)+βexp(χx),0≤x≤a(regionII),ATexp(ikx),x≥a(regionII),                           where k=(2mE)1/2/ℏ, χ=2m(V0-E)1/2/ℏ,  AR, α, β and AT  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 numberiχ. 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 x, travelling in the positive or negative direction of the x-axis, and the variances of the distributions of these times, respectively, as       (3.2)〈t±(x)〉=∫-∞∞tj±(x,t)dt∫-∞∞j±(x,t)dt,Dt±(x)=∫-∞∞t2j±(x,t)dt,j±(x,t) being the positive or negative values, respectively, of the probability flux density j(x,t)=Re[(iℏ/m)Ψ(x,t)∂Ψ*(x,t)/∂x]  for an evolving time-dependent normalized wave packetΨ(x,t). We recall here the equivalence of canonically conjugated time and energy representations, with appropriate measures of averaging, in the following sense: 〈⋯〉t=〈⋯〉E (index t is omitted in all expressions for 〈⋯〉t 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(3.3)〈τT(xi,xf)〉=〈t+(xf)〉−〈t+(xi)〉,(3.4)DτT(xi,xf)=Dt+(xf)+Dt+(xi), with −∞<xi≤0 and a≤xf<∞.  For a pure tunnelling process one has  (3.5)〈τtun(0,a)〉=〈t+(a)〉−〈t+(0)〉,Dτtun(0,a)=Dt+(a)+Dt+(0). Similar expression we have for the penetration (into the barrier region II) temporal quantities   〈τpen(xi,xf)〉 and Dτpen(xi,xf) with 0<xf<a. For reflections in any point xf<a one has  (3.6)〈τR(xi,xf)〉=〈t−(xf)〉−〈t+(xi)〉,DτR(xi,xf)=Dt−(xf)+Dt+(xi).We stress that these definitions hold within the framework of conventional quantum mechanics,without introducing any new physical postulate.In the asymptotic cases, when|xi|≫a,  (3.7)〈τTas(xi,xf)〉=〈t(xf)〉T−〈t(xi)〉in,(3.8)〈τTas(xi,xf)〉=〈τT(xi,xf)〉+〈t+(xi)〉−〈t(xi)〉in, where 〈⋯〉T and 〈⋯〉in denote averagings over the fluxes corresponding to ψT=ATexp(ikx) and   ψin=exp(ikx), respectively.For initial wave packets(3.9)Ψin(x,t)=∫0∞G(k−k¯)exp(ikx-iEtℏ)dk,(where  E=ℏ2k2/2m,∫0∞|G(k−k¯)|2dE=1,G(0)=G(∞)=0,k>0)   with sufficiently small energy (momentum) spreads when  (3.10)∫0∞vn|GAT|2dE≅∫0∞vnG2dE,n=0,1,v=ℏkm, we get (3.11)〈τTas(xi,xf)〉≅〈τTPh(xi,xf)〉E, where (3.12)〈⋯〉E=∫0∞dEv|G(k−k¯)|2⋯∫0∞dEv|G(k−k¯)|2,τTPh(xi,xf)=(1v)(xf−xi)+ℏd(argAT)dE are the phase transmission time obtained by the stationary-phase approximation. At the same approximation and with a small contribution of Dt+(xi) into the variance  DτT(xi,xf) (that can be realized for sufficiently
large energy spreads, i.e., short wave packets) we get
(3.13)DτT(xi,xf)=ℏ2〈(d|AT|/dE)2〉E〈|AT|2〉E. 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 Dt+(xi) plus DτT(xi,xf) which is born by the barrier influence and formally is described by (3.13).At the quasimonochromatric limitG2→δ(E−E),E being ℏ2k¯2/2m, we get for  〈τTasxi,xf〉≅〈τTPhxi,xj〉E strictly the ordinary phase time, without averaging.  For a rectangular barrier with height V0 and χa≫1 (where χ=2mV0−E1/2/ℏ), the expressions (3.11) and (3.13), for xi=0,xf=a  and  am/ℏk¯≫Dt+(xi),  pass into the known expressions  (3.14)τtunPh=2vχ   (coincident with the phase time [26, 27, 47, 83]), and  (3.15)(DτtunPh)1/2=akvχ   (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.Forreal weight amplitude G(k−k¯), when 〈t0〉in=0, from (3.8) we obtain  (3.16)〈τtun(0,a)〉=〈τtunPh〉−〈t+(0)〉.By the way, if the measurement conditions are such that only the positive-momentum components of wave packets are registrated, that is,Λ+Ψxi,t=Ψin(xi,t),Λ+ being the projector onto positive-momentum states, then for anyxi  from (−∞,0) and  xf from (a,∞)(3.17)〈τT(xi,xf)〉+=〈τTPh(xi,xf)〉E,(3.18)〈τtun0,a〉+=〈τtunPh〉E,  because 〈t0〉+=〈t0〉in.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 thephase tunnelling time(3.19)τtun,emPh=2cχemforχema≫1.From (3.19), we can see that when χema>2 the effective tunnelling  velocity  (3.20)vtuneff=aτtun,emPh 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 Section2, 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  j(xi,t)=jinxi,t+jRxi,t+jintxi,t and j(xf,t)=jTxf,t with jin, jR and jT corresponding to the wave packets Ψin(xi,t),ΨR(xi,t) and ΨT(xf,t), constructed from the stationary wave functions ψin,ψR=ARexp(−ikx) and ψT, respectively, and also (3.21)jint(x,t)=Re{(iℏm)[Ψin(x,t)∂ΨR*(x,t)∂x+ΨR(x,t)∂Ψin*(x,t)∂x]},∫-∞∞jint(xi,t)dt=0.One obtains from (2.36a) and (2.36b) of
Section 2,(3.22)〈τDw(xi,xf)〉=〈T〉E〈τT(xi,xf)〉+〈R(xi)〉E〈τR(xi,xf)〉,  with 〈T〉E=〈AT2v〉E/〈v〉E,〈Rxi〉E=〈R〉E+〈rxi〉,〈R〉E=〈|AR2v〉E/〈v〉E, 〈T〉E+〈R〉E=1, and
(3.23)〈r(x)〉=∫-∞∞[J+(x,t)−Jin(x,t)]dt∫-∞∞Jin(x,t)dt. One can see that 〈r(x)〉 is negative and tends to 0 when x tends to −∞.WhenΨin(xi,t) and ΨR(xi,t) are sufficiently well separated in time, so that 〈rxi〉=0, it follows from (3.22) that the simple weighted average rule(3.24)〈τDw(xi,xf)〉=〈T〉E〈τT(xi,xf)〉+〈R〉E〈τR(xi,xi)〉                         is valid. For a rectangular barrier with χa≫1 and quasimonochromatic particles, the expressions (3.22) and (3.24) with xi=0 and xf=a pass to the known expressions (3.25)〈τDw(xi,xf)〉=〈ℏkχV0〉E,  (taking account of the interference term 〈rxi〉)   (3.26)〈τDw(xi,xf)〉=〈2χv〉E                           (when the interference term 〈rxi〉  is equal to 0).WhenAR=0, that is, a barrier is transparent, the mean dwell time (3.22) is automatically equal to  (3.27)〈τDw(xi,xf)〉=〈τT(xi,xf)〉.It is not clear how to definedirectly 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 x1 and x2.


## 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 thex direction, after tunnelling the spin develops small y and z components (see Figure 3).The Larmor timesτy,TLa  and τz,TLa are defined by the ratio of the spin-rotation angles around axes  z and y (in turn defined by the developed y- and z-spin components, resp.) to the precession (rotation) frequency.As toτz,TLa, 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 AT and coincides with the expression (3.15) (of course, at the same approximations when (3.15) is valid).For an opaque rectangular barrier withχa≫1 the expressions (3.28)〈τy,tunLa〉=〈τDw(xi,xf)〉=〈ℏkχV0〉E,(3.29)〈τz,tunLa〉=〈maℏχ〉E (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 AT for the same reasons as for 〈τz,tunLa〉.


## 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 exp[iS(x(t))/ℏ], where S is the action associated with the path x(t). 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 〈τy,tunLa〉 and 〈τz,tunLa〉, respectively.An interesting development of this approach, theinstanton 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 exp(iS/ℏ). 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 f(x,p), so that the flux at position x can be separated into positive and negative components:  (3.30)J(x)=J+(x)+J−(x),  with J+(x)=∫0∞(p/m)f(x,p)dp and J−=J−J+. 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 J± instead of our J±. The dwell time decomposition in this approach takes the form  (3.31)〈τDw(xi,xf)〉=〈T〉E〈τT(xi,xf)〉+〈RM(xi)〉E〈τR(xi,xf)〉, with RM(x)=∫0∞|J−(x,t)|dt. Asymptotically, RMx tends to 〈R〉E 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:  (3.32)τDw(xi,xf)=∫0∞dt∫x1x2|Ψ(x,t)|2dx=TτT(xi,xf)+RτR(xi,xf),with (3.33)τT(xi,xf)=∫0∞dt∫x1x2|Ψ(x,t)|2Θ(x−xc)dx/T,τR(xi,xf)=∫0∞dt∫x1x2|Ψ(x,t)|2Θ(xc−x)dx/R,where T and R are here the mean transmission and reflection probability, respectively,  the bifurcation line xc=xc(t), separating transmitted and reflected trajectories, is defined by relation  (3.34)T=∫−∞∞|Ψx,t|2Θ(x−xc)dx.                                 Factually, in addition to the difference in the temporal integration in this and our formalisms (∫0∞dt  and ∫−∞∞dt, resp.), sometimes essential, this approach gives one more alternative—in separating the flux by the line xc:  (3.35)J(x,t)=[Jx,t]T+Jx,tR,                                    with      (3.36)[Jx,t]T=J(x,t)Θ(x−xc(t)),[Jx,t]R=J(x,t)Θ(xc(t)−x).


## 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(3.37)τtunPh=ℏd(argAT+ka)dE, (which is the mean tunnelling time〈τtun〉 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], AT and E=ℏ2k2/(2m) being the transmission amplitude and the particle kinetic energy, resp.) on the barrier width a for sufficiently large a.  In particular, for a rectangular potential barrier  AT=4ikχ[(k2−χ2)D−+2ikχD+]−1exp[−(κ+iχ)a], D±=1±exp(−2χa), χ=[2m(V0−E)]1/2/ℏ,V0 being the barrier height,  and  τtunPh→2/(υχ)  when χa≫1 (υ=ℏk/m 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 themean dwell time〈τtunDw〉, the mean Larmor time〈τy,tunL〉 and the real part of the complex tunnelling time obtained by averaging over the Feynman pathsReτtunF, which are equal ℏk/(χV0) 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 themean tunnelling time within the O-R approach, is directly seen from the expression (3.38)〈τtun〉=〈t+a〉−〈t+0〉=〈τtunPh〉E−〈t+0〉,    (where 〈⋯〉E  denotes averaging over the initial wave-packet energy spread and (3.39)〈t±(x)〉=∫−∞∞tj±(x,t)dt∫−∞∞j±(x,t)dtwithj±(x,t)=Θ(±j)j(x,t),j(x,t) being the probability flux density for a wave packet moving along axis x through a barrier located in the interval (0,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 timeτz,tunL, from Sections 4.2 and 3.4 it follows that (3.40)τz,tunL=ℏ[〈(d|AT|/dE)2〉〈|AT|2〉]1/2, the Bttiker-Landauer timeτtunB−L [111–113], and the imaginary part of  the complex tunnelling timeImτtunF [92], obtained within the Feynman approach, which are equal to (3.39), they become equal to am/(ℏχ), that is, proportional to the barrier width  a, 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 [Ddynτtun]1/2 where Ddynτtun is the dynamical tunneling-time variance caused by the barrier influence only and defined by the equation   Ddynτtun=Dτtun−Dt+(0) with Dτtun=〈τtun2〉−〈τtun〉2, 〈τtun2〉=〈[t+(a)−〈t+(0)〉]2〉+Dt+(0) (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 Figure4 the dependences of the values of 〈τtun(0,a)〉 from a are presented for electronic wave packets and rectangular barriers with the same parameters as in [47] (V0=10eV; mean electron energies E=2.5,5,7.5eV with Δk=0.02A−1 (curves 1a, 2a, and 3a, resp.); energy E=5eV with k=0.04A−1 and 0.06A−1 (curves 4a and 5a, resp.)). The curves 〈τtunPh〉  corresponding to different energies and Δk merge practically into one curve 6. Since the dependence of 〈τtunPh〉 from a is very weak, the dependence of 〈τtun(0,a)〉 from a is defined  mainly by the dependence of 〈t+(0)〉 from a (curves 1b–5b, correspondent to 1a–5a,  resp.).All these calculations manifest the negative values of〈t+(0)〉. 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 a.  Since all 〈t+(0)〉 are negative, the values of 〈τtun(0,a)〉 are always positive and, moreover, larger than 〈τtunPh〉, 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  −iV1(V1>0)   to V0, then for small absorptions, when V1≪V0 and  V1m1/2vχa/2V0−E3/2≪2, 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 pointx=x0 (with |x0|≫a) at the instant t=0 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 thex axis through a rectangular potential barrier with height V0, localized in the interval [0,a]. The chosen here stationary wave function for a particle with mass m and energy E<V0 has the usual form (3.1). The time dependent wave packet ψ(x,t) is formed with wave functions (3.1):  (3.41)Ψ(x,t)=∫0∞g0(k)ψ(k,x)exp(−iEtℏ)dk, with the weight amplitude  (3.42)g(k)=g0(k)exp[−i(k−k0)x0]=Cexp{−[(k−k0)2Δk]2−i(k−k0)x0},C=[(2π)3/2Δk]−1/2 being the normalization coefficient. Here, since in calculations the only subbarrier part of the wave packet was considered, integrating over k in (3.42) had to be made at the limits from 0 to (2mV0)1/2/ℏ [117].The initial wave function turns out to be(3.43)Ψin(x,t)≅C0sin−1/2exp{−[x-(x0+νt)]22sin+ik0x−iE0tℏ},  whose center transits from point x=x0 at the instant t=0, moves along axis x from the left to the right with the velocity v and in the absence of barrier crosses the point x=0 at the instant t0=−x0/v.The flux densityj(x,t)=Re[(iℏ/2m)Ψ(x,t)∂Ψ*(x,t)/∂x], as usually in the O-R method,  has been considered as a function of the arrival  times at point x, so that the mean time instant for particles passing through point x=a (the mean time of the particle exit from the barrier), has been chosen as  (3.44)〈t(a)〉=∫0∞tj(a,t)dt∫0∞j(a,t)dt,while the mean time instant for particle passing through point x=0 (the mean time of the particle entrance into the barrier) is defined by relation  (3.45)〈t+(0)〉=∫0∞tj+(0,t)dt∫0∞j+(0,t)dt,where j+(0,t) represents the positive values of the flux density j(0,t), corresponding, therefore,  to particles moving through point x=0 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 timeτ is then defined as τ=〈t(a)〉−〈t+(0)〉.The flux densityj(a,t) contains the wave function  (3.46)Ψ(a,t)=∫0∞g(k)ψT(ka)exp(−iEtℏ)dk,where ψT(k,a) is the value of the stationary wave function (3.1) at point x=a, corresponding to the transmitted wave. Using the usual expression for the amplitude of the transmitted wave, we can ψT(k,a) represent in the form  (3.47)ψT(k,a)={2ikχD<,E<V0,2iikqD>,E>V0, where q=2m(E−V0)1/2/ℏ,  (3.48)D<=(k2−χ2)sinh(χa)+2ikχcosh(χa),D>=(k2+q2)sin(qa)+2ikqcos(qa). The normalization integral in (3.45),  (3.49)Na=∫0∞j(a,t)dt can be evaluated using  (3.42), (3.46), and (3.47):  (3.50)Na=ℏmRe{∫0∞dkg(k)ψT(k,a)∫0∞dk′k′g(k′)ψT(k′,a)∫0∞exp[-iℏ2m(k2-k′2)t]dt}. If x0 is chosen sufficiently far from the left of the barrier, the contribution of j(a,t) in the integral (3.49) from t≤0 is negligible small. In this case, the lower integration limit over t in (3.50) can be taken as −∞. Then the integration over time gives 2π(m/ℏk)δ(k−k′) and the integral (3.50) can be cast in the form  (3.51)Na=2π∫0∞g02(k)|ψT(k,a)|2dk, where  (3.52)|ψT(k,a)|2={4k2χ24k2χ2+V02sinh2(χa),E<V0,4k2q24k2q2+V02sin2(qa),E>V0.  Now we evaluate the integral  (3.53)Ia=∫0∞tj(a,t)dt.Using (3.50) and (3.51), we can write  (3.54)Ia=ℏmRe{∫0∞dkg(k)ψT(k,a)∫0∞dk′g(k′)ψT(k′,a)∫0∞k′texp[-iℏ2m(k2-k′2)t]dt}, which can be set in the form  (3.55)Ia=2πmℏ∫0∞g02(k)1k{ψ1(k,a)ψ2′(k,a)−ψ2(k,a)ψ2′(k,a)−x0|ψ(k,a)|2}dk, where ψ1(k,a)≡ReψT(k,a), ψ2(k,a)≡ImψT(k,a). After differentiating (3.47) over k, one can see the following: (3.56)ψ1ψ2′−ψ1′ψ2={−8k2χ[k2(k2−χ2)χa-V02cosh(χa)sinh(χa)][4k2χ2+V02sinh2(χa)]2,E<V0,8k2q[k2(k2+q2)qa−V02cos(qa)sin(qa)][4k2q2+V02sin2(qa)]2,E>V0. From (3.51) and (3.56) it follows that, with an appropriate choice of x0, the mean instant 〈t(a)〉 can be defined by the simple relation  (3.57)〈t(a)〉=mℏ∫0∞g021k[ψ1ψ2′−ψ2ψ1′-x0|ψT|2]dk/∫0∞g02|ψT|2dk.  As regards the calculation of τ¯0, the wave function entering the flux density j(0,t) is  (3.58)ψ(0,t)=∫0∞g(k)[1+AR(k)]exp(−iEtℏ)dk,where   (3.59)AR(k)={V0sinh(χa)D<,E<V0,V0sin(qa)D>,E>V0. Since for calculating 〈t+(0)〉 we consider only the positive values of j(0,t), the integrals I+=∫−∞∞tj+(0,t)dt and N+=∫-∞∞j+(0,t)dt in (3.45) can be obtained only numerically.Figure5 shows the results of the calculations of the tunnelling time τ as a function of the width a of the barrier, for electrons with energy E0=5eV through the rectangular barrier of potential with height 10eV and x0=−6/Δk. We can see the manifestation of the HFE for  Δk=0.01Å−1  (curve (a)), with the asymptotic behavior of the tunneling time approaching the constant value with increasing a. 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  Δk=0.05Å−1 and Δk=0.1Å−1. For  Δk=0.1Å−1 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 (3.60)Δt≅Δ0t{1+[Δv2m2ℏ]2t2}1/2, where  Δ0t~ℏ/ΔE,  Δv=ℏΔk/m. The value of Δt strongly increases in time t for large velocity spread Δv. So, the center of the initial wave packet during wave-packet approaching the barrier and increasing time t will appear even farther due to the essentially stronger increase of Δt with time t (proportional to Δk for large t) than the decrease of |x0|, proportional to 1/Δk.  For the center of the transmitted wave packet in point x=a such delay has to be considerably smaller because  the value τ is smaller than the limited value τph¯ (τph being the phase tunnelling time) due to the advance caused by the difference between j+(0,t) and jin(0,t),jin being Re[(iℏ/2m)Ψin(∂Ψin*/∂x)] (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 Δv becoming even negative.Figure2 shows qualitatively the behavior of the time shift of the outgoing wave packet with respect to the ingoing wave packet with different momentum spread Δk.t0,t0′,t0′′ represent the time instants when the wave packet peak passes through the initial point x=0 of the barrier. tf,tf′, and tf′′ represent the transit time through the final point x=a 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〈τtun(0,a)〉, 〈τtunPh〉 and 〈t+(0)〉 from a.Figure 5
The tunneling timeτ as a function of the barrier depth a, for different values of Δk: (a) Δk=0.01Å,   (b) Δk=0.05Å,  and (c) Δk=0.1Å.Figure 6
Qualitative description of the time shift between peakstf,tf′, and tf′′, and t0,t0′, and t0′′ for the probability density ρ(x=a,t) and ρ(x=0,t) 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 AppendixF (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 massm and kinetic energy E=ℏ2k2/2m=mv2/2, through two equal rectangular barriers with height V0(V0>E) and width a, the quantity l being the distance between them see Figure 7. The stationary Schroedinger equation is (3.61)−ℏ22∂2∂x2ψ(x)+V(x)ψ(x)=Eψ(x),  where V(x)=0 outside the barriers and  V(x)=V0  inside the potential barriers. In various regions I(x≤0),II(0≤x≤a),III(a≤x≤l+a),IV(l+a≤x≤l+2a), and V(x≥l+2a), the solutions of (3.61) are the following:  (3.62)ψI=eikx+A1Re−ikx,ψII=α1e−χx+β1eχx,ψIII=A1T[eikx+A2Re−ikx],ψIV=A1T[α2e−χ(x−l−a)+β2eχ(x−l−a)],ψV=ATeikx,AT≡A1TA2T, where χ=2mV0−E1/2/ℏ, and quantities A1R,A2R,A1T,A2T,α1,α2,β1, and β2 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  ψI,II,III,IV,V and their derivatives dψI,II,III,IV,V/dx at points x=0,a,l+a,l+2a. The obtained expressions for them for opaque barriers, when χa→∞, are  (3.63)A1R→ik+χik-χ,A1T→e−χae−ik(l+a)A,α1→2ikik−χ,β1→e−2χa(k−iχ)sinklχA,A2R→e2ik(l+a)ik+χik−χ,A2T→e2ik(l+a)−4ikχ(ik−χ)2,α2→eik(l+a)2ikik−χ,β2→eik(l+a)−2χa−2ik(ik+χ)(ik−χ)2, where  (3.64)A=2kχ2kχcoskl+(χ2−k2)sinkl.
From (3.63) one can derive that the phase tunneling time is  (3.65)τtunPh=ℏ∂arg[ATeik(l+2a)]∂E→2vχ, 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. Figure7), the amplitude of the transmitted wave in this case is defined by the formula [88, 129]  (3.66)AT(k)=exp(−2ika)D(k), where (3.67)D(k)=cosh2(χa)+14sinh2(χa)[σ2cos(2kl)−δ2]+isinh(χa)[δcosh(χa)+14σ2sinh(χa)sin(2kl)],δ=(χ2−k2)/kχ, and σ=(k2+χ2)/kχ. The dimensionless constants δ and σ are connected by the relation  (3.68)σ2=δ2+4.
A resonance is characterized by the condition [88, 120]  (3.69)|AT(k)|2=1, 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 k for which condition (3.69) is satisfied can be found from the equation  (3.70)cot(kl)=−12δtan(χa). We can show that from (3.70) it is possible to find out also the values of parameters a, m, and V0, at the resonance.  Indeed, from (3.67) and (3.70) it follows that at a resonance it is  (3.71)|D(k)|2=1. On introducing the functions   (3.72)u=cosh2(χa)−14δ2sinh2(χa),v=δcosh(χa)sinh(χa),w=14σ2sinh2(χa) and using (3.68), one can infer that these functions are connected by the relation  (3.73)u2+v2=(1+w)2. Then, by using functions (3.72), the denominator of (3.67) can be written in the following form:  (3.74)D=u+wcos(2kl)+i[v+wsin(2kl)]. It follows from (3.74) and (3.73) that  (3.75)D2=1+2w[1+w+ucos(2kl)+vsin(2kl)], and condition (3.71) gets transformed into (3.76)1+w+ucos(2kl)+vsin(2kl)=0. Taking (3.68) into account, we can write the term 1+w of (3.76) in the following form:  (3.77)1+w=cosh2(χa)+14δ2sinh2(χa), 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 E−Er   (quantity Er being the resonance value of energy E), we obtain  (3.78)D(k)=Dr+Cr(E-Er),where Dr=D(kr),  kr=2mEr/ℏ,  (3.79)Cr=mℏ2kDr′, and the index prime defines the derivative with respect to k. We can rewrite (3.78) in the following form:  (3.80)D(k)=Cr(E−Er+DrCr*|Cr|2). It follows from (3.70) that at the resonance it is  (3.81)cos(2krl)=−ur(1+wr),sin(2krl)=−vr(1+wr), where ur, vr and wr  are the values at E=Er of the functions u, v, and w, respectively.  On inserting (3.81) into (3.74), we obtain  (3.82)D(kr)=(ur+ivr)(1+wr). Differentiating (3.74) with respect to k, and inserting the result into (3.81), we get  (3.83)D′(kr)=[ur′−(urwr′−2lvrwr)(1+wr)]+i[vr′−(vrwr′−2lurwr)(1+wr)]. Then, on using (3.73), (3.79), (3.82), (3.83), and relation  (3.84)uu′+vv′=(1+w)w′, which follows from (3.73), one finds that at the resonance  (3.85)DrCr*=imℏ2kr[ur′vr−urvr′1+wr+2lwr], where (by having recourse to (3.73), (3.79), (3.83), and (3.84))  (3.86)|Cr|2=(mℏ2kr)2[(ur′)2+(vr′)2−(wr′)2+4lur′vr−urvr′1+wrwr+4l2wr2]. From (3.73) and (3.84)
one gets  (3.87)u′2+v′2−w′2=(u′2+v′2)2(1+w)2, while, from (3.86) and (3.87), one obtains (3.88)|Cr|2=(mℏ2kr)2(ur′vr−urvr′1+wr+2lwr). By differentiating the functions (3.72) with respect to k, one can see that  (3.89)ur′vr−urvr′=1χr(1+wr)[χrkra+2χrlwr+σr2cosh(χra)sinh(χra)], where χr,δr, and   σr are the values of χ,δ,  and σ at E=Er. On inserting (3.89) into (3.85) and (3.88), we can then rewrite (3.80) in the noteworthy form  (3.90)D(k)=Cr(E−Er+iβ), where  (3.91)β=ℏ2krχrm[χrkra+2χrlwr+σr2cosh(χra)sinh(χra)]−1, so that (3.80), (3.85), (3.88), and (3.90) yield   (3.92)|Cr|2=1β2. Finally, by inserting (3.90) into (3.66) and taking account of (3.92), we obtain that near a resonance it holds in general  (3.93)|AT(k)|2=β2(E−Er)2+β2. 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   (3.94)τtunPh=ℏdarg{ATexp[ik(2a+l)]}dE.  By using (3.66) and (3.76), we can rewrite it in the following form:
(3.95)τtunPh=mℏk[l−(D1D2′−D2D1′)|D|2], where (3.96)D1=u+wcos(2kl),D2=v+wsin(2kl) (the index prime denoting again the derivative with respect to k).  On inserting (3.96) into (3.95) and using (3.73), we get in general for the total tunneling phase time the remarkable formula:  (3.97)τtunPh=mℏkP|D|2, where  (3.98)P=(1+2w)l+u′v−uv′+(u′w−uw′)sin(2kl)+(vw′−v′w)cos(2kl).  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 τtun,rPh is given by the following expression:  (3.99)τtun,rPh=mℏkχ[σ2cosh(χa)sinh(χa)+δka+(1+2w)χl]. It follows from (3.65) and (3.66) that  (3.100)τtunPh=ℏdarg[exp(ikl)/D]/dE. Inserting (3.90) into (3.100), we get  (3.101)τtunPh=ℏddEarg[1Crexp(ikl)E−Er+iβ], so that, near a resonance, the behavior of the tunneling phase time in terms of the energy is represented by the interesting following formula:  (3.102)τtunPh≅mℏkl+ℏ=β(E−Er)2+β2 holding for any resonant tunneling through the two barriers.  The first term represents the time associated with the particle free flight over the distance l 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  (3.103)|D|2≈132σ2exp(4χa)[14σ2+(1−14δ2)cos(2kl)+δsin(2kl)]. Differentiating the functions u,v, and  w with respect to k, one can see that (3.104)u′v−uv′=1χ(1+w)[σ2cosh(χa)sinh(χa)+δka],u′w−uw′=2χ[δ(u+w)+14σ2cosh(χa)sinh(χa)],vw′−v′w=1χw[(4−δ2)cosh(χa)sinh(χa)−δka]. Therefore, from (3.72), (3.98), (3.103), and (3.104), we get that, still for χa≫1,  (3.105)P≈116σ2exp(4χa)[σ24+(1+δ24)cos(2kl)+δsin(2kl)]+18σ2lexp(2χa) and far from the resonances,  (3.106)τtunPh≈2mℏkχ+4mℏklexp(−2χa)[σ24+(1+δ24)cos(2kl)+δsin(2kl)]−1, Namely, when χa increases, the second term in (3.106) decreases as exp(−2χa); 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 τtunPh has been actually observed to increase (very slowly, almost linearly, and probably with very small oscillations) on l, 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 largeχa 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  (3.107)|AT|2≈321σ2exp(−4χa)[14σ2+(1−14δ2)cos(2kl)+δsin(2kl)]−1,  which shows that, with increasing χa, the  transmission coefficient square  |AT(k)|2 decreases as exp(−4χa), 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 ofχa which make negligible the mentioned second term in (3.106). Moreover, for the values of χa 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 χa,  results to depend very weakly on the distance l between the barriers: more precisely, it will depend almost linearly (but very slightly) on l, 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 asα-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 r>R2, represents the external region of null potential, region (II), delimited by R1 and R2, is the barrier region, and the internal region (III), with r<R1, 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 valueV1:  (3.108)rΨ0(1)(r,t)=∫0V1dEg(E)e−ikr−iEt/ℏ,(R2≤r<∞),   where k and E=ℏ2k2/2m are the wave number and the kinetic energy, respectively.
A section of the potential along ther-axis is shown in Figure 9.
When the wave encounters the barrier, it is partially reflected with wave function:(3.109)rΨR(1)(r,t)=∫0V1dEg(E)AR(ex)e−ikr−iEt/ℏ,(R2≤r<∞).                  At the same time, the wave packet tunnels through the barrier in region (II), where it is represented by the following function:  (3.110)rΨII(1)(r,t)=∫0V1dEg(E)(α1e−χr+β1eχr)e−iEt/ℏ,(R1≤r<R2),                   where χ=2m(V1−E)/ℏ2.  After the tunnelling through the barrier, the wave penetrates inside the well and in region (III) we can write  (3.111)rΨT(1)(r,t)=∫0V1dEg(E)AT(in)e−iKr−iEt/ℏ,(0<r<R1),                               where K=2m(E+V0)/ℏ2 is the wave number inside the well.
In general, we can say that the wave packet is described by the expression(3.112)rΨJ(1)(r,t)=∫0V1dEg(E)ΦJ(1)(k,r)e−iEt/ℏ,                              where  the index J is I, II, III, 0, T or R depending on the particular mode considered,  ΦJ(1) being the stationary wave functions: (3.113)ΦI(1)=e−ikr+AR(ex)eikr,R2≤r<∞,ΦII(1)=α1e−χr+β1eχr,R1≤r≤R2,ΦIII(1)=AT(in)e−iKr,0<r≤R2. Furthermore, g(E)  is a normalized amplitude weight factor such as  (3.114)∫0V1dE|g(E)|2=N,N<∞.  We normalize (3.1) by the following condition:  (3.115)4π∫R2∞r2dr|Ψ0(1)(r,t)|2=1,  which is strictly possible only in the limit V1 going to infinity.  For the sake of simplicity we neglect the contribution of the energy above the barrier, considering only cases where
E<V1.
All the previous formulas are written in the case of orbital momentuml=0, and the extension of this theory to the general case with l>0 can be obtained by replacing the terms exp(±ikr) and exp(±iKr) with the spherical Hanckel function of the first and second  kinds.
In addition, we have thatARex,α1,β1, and AT(in) 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 r=R2 and r=R1, finding for the external reflection coefficient the  expression:  (3.116)ARex=−e−2ikR2aRex,aRex=(χ+ik)(iK+χ)+e−2χ(R2−R1)(χ−ik)(iK−χ)(χ−ik)(iK−χ)+e−2χ(R2−R1)(χ+ik)(iK+χ),                   which, in the limit χ(R2−R1) tending to infinity, becomes −e−2ikR2((χ+ik)/(χ−ik)).
For the internal transmission coefficient we get(3.117)ATin=e−ikR2+iκR1aTin,aTin=4ikχe−χ(R2−R1)(χ−ik)(iK−χ)+e−2χ(R2−R1)(χ+ik)(iK+χ) 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 fluxesjI(1),jII(1), and jIII(1) in the three regions I, II, and III are in general defined as  (3.118)jn(1)=iℏ2m(rΨn(1)ddr(rΨn(1)*)−rΨn(1)*ddr(rΨn(1))),n=I,II,III.One can use the approximation  (3.119)∫0∞jn(1)(r,t)dt≈∫−∞∞jn(1)(r,t)dt.  For quasimonochromatic wave packet centred around the value E=〈E〉, that is, when the factor  |gE|2 is a delta function δ(E−〈E〉), with 〈E〉 in the open interval (0,V1), one obtains  (3.120)∫−∞∞jR(1)(r,t)dt∫−∞∞j0(1)(r,t)dt≈|AR(ex)(〈E〉)|2,∫−∞∞jII(1)(r,t)dt∫−∞∞j0(1)(r,t)dt≈|α1(〈E〉)e−χ(〈E〉)r+β1(〈E〉)eχ(〈E〉)r|2,∫−∞∞jIII(1)(r,t)dt∫−∞∞j0(1)(r,t)dt≈K(〈E〉)k(〈E〉)|AT(in)(〈E〉)|2.  Hence, from the conservation law for the probability fluxes  (3.121)∫−∞∞j0(1)(r,t)dt=∫−∞∞jR(1)(r,t)dt+∫−∞∞jT(1)(r,t)dt                                        one obtains  (3.122)|AR(ex)(〈E〉)|2+K(〈E〉)k(〈E〉)|AT(in)(〈E〉)|2=1,      that for V0=0 becomes  (3.123)|AR(ex)(〈E〉)|2+|AT(in)(〈E〉)|2=1. The transmission probability through the barrier from outwards is represented by the last of expressions (3.120).
The phase timesτTph(in) and τRph(ex) can be defined as the evident generalization of the following 1D definitions:  (3.124)τTph(in)=ℏ∂(arg(AT(in)(E)e−ikR1))∂E−−R1v=ℏ∂(arg(aT(in)(E))∂E,τRph(ex)=ℏ∂(arg(AR(ex)(E)e−iκR2))∂E−−R2v=ℏ∂(arg(aR(ex)(E))∂E,                                      where v=ℏk/m. The quantities τRph(ex) and τRph(ex), in the limit χ(R2−R1) approaching infinity, go to 2/vχ 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 barriersR1 and R2.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(3.125)rΨ(in)(r,t)=∫0V1dEG(E)eiKr−iEt/ℏ,(0≤r≤R1)             constructed from the overlapping stationary solutions propagating in the positive r-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(3.126)rΨR(2)(r,t)=∫0V1dEG(E)AR(in)e−iKr−iEt/ℏ,(0≤r≤R1).Afterwards, in the region (II) of the barrier a system of evanescent and antievanescent waves develops  (3.127)rΨII(2)(r,t)=∫0V1dEG(E)(α2e−χr+β2eχr)e−iEt/ℏ,(R1≤r≤R2),                       while in the outer region a propagating wave will exist  (3.128)rΨI(2)(r,t)=∫0V1dEG(E)AT(ex)eikr−iEt/ℏ,(R2≤r<∞).As in the previous case we have that G(E) is a normalized amplitude weight factor, and AR(in),α2,β2, and AT(ex) 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:  (3.129)ARin=−e−2ikR1aRin,aRin=(χ+iK)(ik−χ)+e2χ(R1−R2)(χ−iK)(ik+χ)(χ−iK)(ik−χ)+e2χ(R1−R2)(χ+iK)(ik+χ),(3.130)ATex=e−ikR2+iKR1aTex,aTex=4iKχe−χ(R2−R1)(χ−iK)(ik−χ)+e−2χ(R2−R1)(χ+iK)(ik+χ).Also in this case we can demonstrate the conservation of the current fluxes (3.131)|AR(in)(〈E〉)|2+k(〈E〉)K(〈E〉)|AT(ex)(〈E〉)|2=1,                                               and we can introduce the phase times: (3.132)τTph(ex)=ℏ∂(arg(AT(ex)(E)e−ikR1))∂E−−R1v=ℏ∂(arg(aT(ex)(E))∂E,τRph(in)=ℏ∂(arg(AR(in)(E)e−iκR2))∂E−−R1v=ℏ∂(arg(aR(in)(E))∂E,                    with v=ℏk/m. The quantities τTph(ex) and τRph(in), in the limit χ(R2−R1) approaching infinity, go to (1/v+1/V)/χ   (with V=ℏK/m) 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 Figure11), introducing the matrix of scattering S and considering the multiple reflections inside the potential well.
For this purpose we describe the stationary wave functions in the various regions as follows:(3.133)ΨI(k,r)=e−ikr−Seikr,ΨII(k,r)=αe−χr+βeχr,ΨIII(k,r)=A(e−iKr−eiLr).                                                        We can now find S, α, β, and A by connecting the various expression of the wave function and its derivative in r=R1 and r=R2.
Imposing the continuity conditions we have(3.134)S=e−2ikR2(KcosKR1+χsinKR1)(χ+ik)+e2χ(R1−R2)(KcosKR1−χsinKR1)(χ−ik)(KcosKR1+χsinKR1)(χ−ik)+e2χ(R1−R2)(KcosKR1−χsinKR1)(χ+ik),(3.135)A=e−2ikR22ikχe2χ(R1−R2)(KcosKR1+χsinKR1)(χ+ik)+e2χ(R1−R2)(KcosKR1−χsinKR1)(χ+ik).One can easily see that |S|=1, and that (3.136)A=AT(in)1+ARin,(3.137)S=−AR(ex)−iAAT(ex)=−AR(ex)+AT(ex)AT(in)1+AR(in).The physical meaning if the term 1/(1+ARin) is directly connected to the presence of an infinite sequence of multiple internal reflections that can be described by the stationary wave functions  (3.138)AT(in)(1−AR(in)+(AR(in))2−(AR(in))3+⋯)e−ikr=AT(in)1+AT(in)e−ikr,AT(in)(1−AR(in)+(AR(in))2−(AR(in))3+⋯)eikr=AT(in)1+AT(in)eikrfor the ingoing and the outgoing waves,  respectively.
The scattering phase timeτscph=ℏ(∂argSeikR2/∂E)−(−R2/v) that in the limit χ(R2−R1) approaching infinity goes to 2/vχ. So in this limit, the  scattering phase time coincides with τRph(ex).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 matrixS in the following form:  (3.139)S=αA1χ+ikA2A1χ−ikA2,                                                            where, from the comparison with (3.134), we set: α=e−2ikR2, A1=B1+e−2χ(R2−R1)B2, A2=B1−e−2χ(R2−R1)B2, B1=Kcos(KR1)+χsin(KR1), B2=Kcos(KR1)−χsin(KR1).
In the region of a resonance, we can developS into a series of powers of (E−Er),Er being the eigenvalue of the energy for the resonance, considered as a solution of the transcendental equation A1(Er)=0, and then we obtain the following:  (3.140)S=α(∂(A1χ)/∂E)E=Er(E−Er)+i(kA2)E=Er(∂(A1χ)/∂E)E=Er(E−Er)−i(kA2)E=Er,                                                that shows better its resonant character if written in the form  (3.141)S=αE−Er−iΓ/2E−Er+iΓ/2,                                               with (3.142)Γ2=−k(Er)A2(Er)χ(Er)(∂A1/∂E)E=Er=2k(Er)e−2χ(Er)(R2−R1)B2(Er)χ(Er)(∂A1/∂E)E=Er.If χ(R2−R1) is very large, we can neglect all the terms containing the negative exponential factor and write finally  (3.143)Γ2=2k(Er)e−2χ(Er)(R2−R1)F(Er),with  (3.144)F(Er)=ℏK(Er)χ(Er)m/2((1/V0+Er)(1+χ(Er)R1)+(1/V1−Er)(1+K(Er)R1)).The Case of A Rectangular Well with A Coulomb Barrier
In this case we can initiate from a simple Coulomb barrier(3.145)V=Z1Z2e2r,(Z1e and Z2e are the charges of the daughter nucleus and the emitted particle, resp.) instead of a rectangular potential barrier. Instead of the functions hl(1,2)(kr) we have to use the coulomb functions  Gl(k,η,r)±iFl(k,η,r) in the field of the coulomb barrier with  (3.146)F0(k,η,r)→sin(kr−ηln2kr+σ),r→∞G0(k,η,r)→cos(kr−ηln2kr+σ),r→∞ (where η=Z1Z2e2m/ℏ2k is the Sommerfeld parameter, σ=argΓ(1+iη)).
In this case, taking alsol=0, we obtain (3.147)S=[G0(k,η,R1)−iF0(k,η,R1)]KcosKR1−[G0′(k,η,R1)−iF0′(k,η,R1)]ksinKR1[G0(k,η,R1)+iF0(k,η,R1)]KcosKR1−[G0′(k,η,R1)+iF0′(k,η,R1)]ksinKR1, (one can easily see that here |S|=1 too)  (3.148)A=[2ieikR2k][G0′(k,η,R1)+iF0′(k,η,R1)]k(1−e2iKR1)+[G0(k,η,R1)+iF0(k,η,R1)]iK(1+e2iKR1)  for the scattering (see Figure 12). Here, F0′ and G0′ signify the derivatives of F0 and  G0 with respect to kR1, respectively.
Similarly, in this case we also obtain(3.149)ARex=−[G0(k,η,R1)+iF0(k,η,R1)]iK−[G0′(k,η,R1)+iF0′(k,η,R1)]k[G0(k,η,R1)+iF0(k,η,R1)]iK+[G0′(k,η,R1)+iF0′(k,η,R1)]k(3.150)ATin=2ikeiKR1[G0(k,η,R1)+iF0(k,η,R1)]iK+[G0′(k,η,R1)+iF0′(k,η,R1)]k, for the  impact from outside (see Figure 13), and (3.151)ARin=e2iKR1[G0(k,η,R1)+iF0(k,η,R1)]iK−[G0′(k,η,R1)+iF0′(k,η,R1)]k[G0(k,η,R1)+iF0(k,η,R1)]iK+[G0′(k,η,R1)+iF0′(k,η,R1)]k,(3.152)ATex=eiKR12iK[G0(k,η,R1)+iF0(k,η,R1)iK+[G0′(k,η,R1)+iF0′(k,η,R1)]k 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 F0G0′−G0F0′=1 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  1/(1+ARin) which has been made in connection with the formula (3.137).
For very smallk when k→0 (more precisely, when 2η≫kR1),  (3.153)G0→2(ρπ)1/2I0(2(2πρ)1/2)exp(πη),withI0(2(2πρ)1/2)→1,G0′→−2(2ηπ)1/2K0(2(2πρ)1/2)exp(πη),withK0(2(2πρ)1/2)→ln(1γ(2πρ)1/2),F0→(πρ)1/2I1(2(2πρ)1/2)exp(−πη),withI1(2(2πρ)1/2)→(2πρ)1/2,F0′→(2πη)1/2I0(2(2πρ)1/2)exp(−πη),γ≈1.781… being the Euler constant, and if (2k2/K2η/ρ)[lnγ−1(2πρ)−1/2]≪1,  the transmission (penetration) probability from outside through the Coulomb barrier into the internal rectangular potential well |ATin|2 becomes (3.154)|ATin|2→(πkK2R1)exp(−2πη) which contains the same exponential factor exp(−2πη) 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 R1 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.


## 3.8.1. 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 valueV1:  (3.108)rΨ0(1)(r,t)=∫0V1dEg(E)e−ikr−iEt/ℏ,(R2≤r<∞),   where k and E=ℏ2k2/2m are the wave number and the kinetic energy, respectively.
A section of the potential along ther-axis is shown in Figure 9.
When the wave encounters the barrier, it is partially reflected with wave function:(3.109)rΨR(1)(r,t)=∫0V1dEg(E)AR(ex)e−ikr−iEt/ℏ,(R2≤r<∞).                  At the same time, the wave packet tunnels through the barrier in region (II), where it is represented by the following function:  (3.110)rΨII(1)(r,t)=∫0V1dEg(E)(α1e−χr+β1eχr)e−iEt/ℏ,(R1≤r<R2),                   where χ=2m(V1−E)/ℏ2.  After the tunnelling through the barrier, the wave penetrates inside the well and in region (III) we can write  (3.111)rΨT(1)(r,t)=∫0V1dEg(E)AT(in)e−iKr−iEt/ℏ,(0<r<R1),                               where K=2m(E+V0)/ℏ2 is the wave number inside the well.
In general, we can say that the wave packet is described by the expression(3.112)rΨJ(1)(r,t)=∫0V1dEg(E)ΦJ(1)(k,r)e−iEt/ℏ,                              where  the index J is I, II, III, 0, T or R depending on the particular mode considered,  ΦJ(1) being the stationary wave functions: (3.113)ΦI(1)=e−ikr+AR(ex)eikr,R2≤r<∞,ΦII(1)=α1e−χr+β1eχr,R1≤r≤R2,ΦIII(1)=AT(in)e−iKr,0<r≤R2. Furthermore, g(E)  is a normalized amplitude weight factor such as  (3.114)∫0V1dE|g(E)|2=N,N<∞.  We normalize (3.1) by the following condition:  (3.115)4π∫R2∞r2dr|Ψ0(1)(r,t)|2=1,  which is strictly possible only in the limit V1 going to infinity.  For the sake of simplicity we neglect the contribution of the energy above the barrier, considering only cases where
E<V1.
All the previous formulas are written in the case of orbital momentuml=0, and the extension of this theory to the general case with l>0 can be obtained by replacing the terms exp(±ikr) and exp(±iKr) with the spherical Hanckel function of the first and second  kinds.
In addition, we have thatARex,α1,β1, and AT(in) 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 r=R2 and r=R1, finding for the external reflection coefficient the  expression:  (3.116)ARex=−e−2ikR2aRex,aRex=(χ+ik)(iK+χ)+e−2χ(R2−R1)(χ−ik)(iK−χ)(χ−ik)(iK−χ)+e−2χ(R2−R1)(χ+ik)(iK+χ),                   which, in the limit χ(R2−R1) tending to infinity, becomes −e−2ikR2((χ+ik)/(χ−ik)).
For the internal transmission coefficient we get(3.117)ATin=e−ikR2+iκR1aTin,aTin=4ikχe−χ(R2−R1)(χ−ik)(iK−χ)+e−2χ(R2−R1)(χ+ik)(iK+χ) 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 fluxesjI(1),jII(1), and jIII(1) in the three regions I, II, and III are in general defined as  (3.118)jn(1)=iℏ2m(rΨn(1)ddr(rΨn(1)*)−rΨn(1)*ddr(rΨn(1))),n=I,II,III.One can use the approximation  (3.119)∫0∞jn(1)(r,t)dt≈∫−∞∞jn(1)(r,t)dt.  For quasimonochromatic wave packet centred around the value E=〈E〉, that is, when the factor  |gE|2 is a delta function δ(E−〈E〉), with 〈E〉 in the open interval (0,V1), one obtains  (3.120)∫−∞∞jR(1)(r,t)dt∫−∞∞j0(1)(r,t)dt≈|AR(ex)(〈E〉)|2,∫−∞∞jII(1)(r,t)dt∫−∞∞j0(1)(r,t)dt≈|α1(〈E〉)e−χ(〈E〉)r+β1(〈E〉)eχ(〈E〉)r|2,∫−∞∞jIII(1)(r,t)dt∫−∞∞j0(1)(r,t)dt≈K(〈E〉)k(〈E〉)|AT(in)(〈E〉)|2.  Hence, from the conservation law for the probability fluxes  (3.121)∫−∞∞j0(1)(r,t)dt=∫−∞∞jR(1)(r,t)dt+∫−∞∞jT(1)(r,t)dt                                        one obtains  (3.122)|AR(ex)(〈E〉)|2+K(〈E〉)k(〈E〉)|AT(in)(〈E〉)|2=1,      that for V0=0 becomes  (3.123)|AR(ex)(〈E〉)|2+|AT(in)(〈E〉)|2=1. The transmission probability through the barrier from outwards is represented by the last of expressions (3.120).
The phase timesτTph(in) and τRph(ex) can be defined as the evident generalization of the following 1D definitions:  (3.124)τTph(in)=ℏ∂(arg(AT(in)(E)e−ikR1))∂E−−R1v=ℏ∂(arg(aT(in)(E))∂E,τRph(ex)=ℏ∂(arg(AR(ex)(E)e−iκR2))∂E−−R2v=ℏ∂(arg(aR(ex)(E))∂E,                                      where v=ℏk/m. The quantities τRph(ex) and τRph(ex), in the limit χ(R2−R1) approaching infinity, go to 2/vχ 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 barriersR1 and R2.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(3.125)rΨ(in)(r,t)=∫0V1dEG(E)eiKr−iEt/ℏ,(0≤r≤R1)             constructed from the overlapping stationary solutions propagating in the positive r-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(3.126)rΨR(2)(r,t)=∫0V1dEG(E)AR(in)e−iKr−iEt/ℏ,(0≤r≤R1).Afterwards, in the region (II) of the barrier a system of evanescent and antievanescent waves develops  (3.127)rΨII(2)(r,t)=∫0V1dEG(E)(α2e−χr+β2eχr)e−iEt/ℏ,(R1≤r≤R2),                       while in the outer region a propagating wave will exist  (3.128)rΨI(2)(r,t)=∫0V1dEG(E)AT(ex)eikr−iEt/ℏ,(R2≤r<∞).As in the previous case we have that G(E) is a normalized amplitude weight factor, and AR(in),α2,β2, and AT(ex) 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:  (3.129)ARin=−e−2ikR1aRin,aRin=(χ+iK)(ik−χ)+e2χ(R1−R2)(χ−iK)(ik+χ)(χ−iK)(ik−χ)+e2χ(R1−R2)(χ+iK)(ik+χ),(3.130)ATex=e−ikR2+iKR1aTex,aTex=4iKχe−χ(R2−R1)(χ−iK)(ik−χ)+e−2χ(R2−R1)(χ+iK)(ik+χ).Also in this case we can demonstrate the conservation of the current fluxes (3.131)|AR(in)(〈E〉)|2+k(〈E〉)K(〈E〉)|AT(ex)(〈E〉)|2=1,                                               and we can introduce the phase times: (3.132)τTph(ex)=ℏ∂(arg(AT(ex)(E)e−ikR1))∂E−−R1v=ℏ∂(arg(aT(ex)(E))∂E,τRph(in)=ℏ∂(arg(AR(in)(E)e−iκR2))∂E−−R1v=ℏ∂(arg(aR(in)(E))∂E,                    with v=ℏk/m. The quantities τTph(ex) and τRph(in), in the limit χ(R2−R1) approaching infinity, go to (1/v+1/V)/χ   (with V=ℏK/m) 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 Figure11), introducing the matrix of scattering S and considering the multiple reflections inside the potential well.
For this purpose we describe the stationary wave functions in the various regions as follows:(3.133)ΨI(k,r)=e−ikr−Seikr,ΨII(k,r)=αe−χr+βeχr,ΨIII(k,r)=A(e−iKr−eiLr).                                                        We can now find S, α, β, and A by connecting the various expression of the wave function and its derivative in r=R1 and r=R2.
Imposing the continuity conditions we have(3.134)S=e−2ikR2(KcosKR1+χsinKR1)(χ+ik)+e2χ(R1−R2)(KcosKR1−χsinKR1)(χ−ik)(KcosKR1+χsinKR1)(χ−ik)+e2χ(R1−R2)(KcosKR1−χsinKR1)(χ+ik),(3.135)A=e−2ikR22ikχe2χ(R1−R2)(KcosKR1+χsinKR1)(χ+ik)+e2χ(R1−R2)(KcosKR1−χsinKR1)(χ+ik).One can easily see that |S|=1, and that (3.136)A=AT(in)1+ARin,(3.137)S=−AR(ex)−iAAT(ex)=−AR(ex)+AT(ex)AT(in)1+AR(in).The physical meaning if the term 1/(1+ARin) is directly connected to the presence of an infinite sequence of multiple internal reflections that can be described by the stationary wave functions  (3.138)AT(in)(1−AR(in)+(AR(in))2−(AR(in))3+⋯)e−ikr=AT(in)1+AT(in)e−ikr,AT(in)(1−AR(in)+(AR(in))2−(AR(in))3+⋯)eikr=AT(in)1+AT(in)eikrfor the ingoing and the outgoing waves,  respectively.
The scattering phase timeτscph=ℏ(∂argSeikR2/∂E)−(−R2/v) that in the limit χ(R2−R1) approaching infinity goes to 2/vχ. So in this limit, the  scattering phase time coincides with τRph(ex).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 matrixS in the following form:  (3.139)S=αA1χ+ikA2A1χ−ikA2,                                                            where, from the comparison with (3.134), we set: α=e−2ikR2, A1=B1+e−2χ(R2−R1)B2, A2=B1−e−2χ(R2−R1)B2, B1=Kcos(KR1)+χsin(KR1), B2=Kcos(KR1)−χsin(KR1).
In the region of a resonance, we can developS into a series of powers of (E−Er),Er being the eigenvalue of the energy for the resonance, considered as a solution of the transcendental equation A1(Er)=0, and then we obtain the following:  (3.140)S=α(∂(A1χ)/∂E)E=Er(E−Er)+i(kA2)E=Er(∂(A1χ)/∂E)E=Er(E−Er)−i(kA2)E=Er,                                                that shows better its resonant character if written in the form  (3.141)S=αE−Er−iΓ/2E−Er+iΓ/2,                                               with (3.142)Γ2=−k(Er)A2(Er)χ(Er)(∂A1/∂E)E=Er=2k(Er)e−2χ(Er)(R2−R1)B2(Er)χ(Er)(∂A1/∂E)E=Er.If χ(R2−R1) is very large, we can neglect all the terms containing the negative exponential factor and write finally  (3.143)Γ2=2k(Er)e−2χ(Er)(R2−R1)F(Er),with  (3.144)F(Er)=ℏK(Er)χ(Er)m/2((1/V0+Er)(1+χ(Er)R1)+(1/V1−Er)(1+K(Er)R1)).The Case of A Rectangular Well with A Coulomb Barrier
In this case we can initiate from a simple Coulomb barrier(3.145)V=Z1Z2e2r,(Z1e and Z2e are the charges of the daughter nucleus and the emitted particle, resp.) instead of a rectangular potential barrier. Instead of the functions hl(1,2)(kr) we have to use the coulomb functions  Gl(k,η,r)±iFl(k,η,r) in the field of the coulomb barrier with  (3.146)F0(k,η,r)→sin(kr−ηln2kr+σ),r→∞G0(k,η,r)→cos(kr−ηln2kr+σ),r→∞ (where η=Z1Z2e2m/ℏ2k is the Sommerfeld parameter, σ=argΓ(1+iη)).
In this case, taking alsol=0, we obtain (3.147)S=[G0(k,η,R1)−iF0(k,η,R1)]KcosKR1−[G0′(k,η,R1)−iF0′(k,η,R1)]ksinKR1[G0(k,η,R1)+iF0(k,η,R1)]KcosKR1−[G0′(k,η,R1)+iF0′(k,η,R1)]ksinKR1, (one can easily see that here |S|=1 too)  (3.148)A=[2ieikR2k][G0′(k,η,R1)+iF0′(k,η,R1)]k(1−e2iKR1)+[G0(k,η,R1)+iF0(k,η,R1)]iK(1+e2iKR1)  for the scattering (see Figure 12). Here, F0′ and G0′ signify the derivatives of F0 and  G0 with respect to kR1, respectively.
Similarly, in this case we also obtain(3.149)ARex=−[G0(k,η,R1)+iF0(k,η,R1)]iK−[G0′(k,η,R1)+iF0′(k,η,R1)]k[G0(k,η,R1)+iF0(k,η,R1)]iK+[G0′(k,η,R1)+iF0′(k,η,R1)]k(3.150)ATin=2ikeiKR1[G0(k,η,R1)+iF0(k,η,R1)]iK+[G0′(k,η,R1)+iF0′(k,η,R1)]k, for the  impact from outside (see Figure 13), and (3.151)ARin=e2iKR1[G0(k,η,R1)+iF0(k,η,R1)]iK−[G0′(k,η,R1)+iF0′(k,η,R1)]k[G0(k,η,R1)+iF0(k,η,R1)]iK+[G0′(k,η,R1)+iF0′(k,η,R1)]k,(3.152)ATex=eiKR12iK[G0(k,η,R1)+iF0(k,η,R1)iK+[G0′(k,η,R1)+iF0′(k,η,R1)]k 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 F0G0′−G0F0′=1 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  1/(1+ARin) which has been made in connection with the formula (3.137).
For very smallk when k→0 (more precisely, when 2η≫kR1),  (3.153)G0→2(ρπ)1/2I0(2(2πρ)1/2)exp(πη),withI0(2(2πρ)1/2)→1,G0′→−2(2ηπ)1/2K0(2(2πρ)1/2)exp(πη),withK0(2(2πρ)1/2)→ln(1γ(2πρ)1/2),F0→(πρ)1/2I1(2(2πρ)1/2)exp(−πη),withI1(2(2πρ)1/2)→(2πρ)1/2,F0′→(2πη)1/2I0(2(2πρ)1/2)exp(−πη),γ≈1.781… being the Euler constant, and if (2k2/K2η/ρ)[lnγ−1(2πρ)−1/2]≪1,  the transmission (penetration) probability from outside through the Coulomb barrier into the internal rectangular potential well |ATin|2 becomes (3.154)|ATin|2→(πkK2R1)exp(−2πη) which contains the same exponential factor exp(−2πη) 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 R1 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→ Exponential Decay

Let us firstly explain how a typical isolated Lorentzian (Breit-Wigner) resonance, in the cross-section of a quantum collision or nuclear reactionα→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  fαβ(E) as   (4.1)fαβ(E)=CαβE−Er+iΓ/2, where Cαβ is a constant or a smooth function of the final-particle kinetic energy E in the region (Er−Γ/2,Er+Γ/2),Er and Γ being the resonance energy and width, respectively.The final-particle wave packet in the 1D radial asymptotic limit is described by(4.2)Ψβ(rβ,t)=rβ−1∫0∞dEg(E)fαβ(E)exp[ikrβ−iEtℏ], where g(E) is a smooth weight amplitude with an energy spread ΔE (usually ΔE≪Er),mβ and rβ are the final-particle mass and radial coordinate, respectively, and k=2mβE/ℏ. For short-ranged interactions (including also screened Coulomb potentials), it can be rewritten as  (4.3)Ψβ(zβ,t)=∫0∞dEg(E)Tαβ(E)exp[ikzβ−iEtℏ], with Tαβ(E)=Nβ(E)fαβ(E),Tαβ(E) is the T-matrix elements connected with the S-matrix elements by known relation Tαβ=δαβ−Sαβ,Nαβ(E) is an unessential smooth function of E,zβ is the axis along the direction of the final-particle emission imposed by the registration geometry, zβ≥Rβ,Rβ is the interaction radius in the final channel. In the simplest case one can fix zβ=Rβ and  (4.4)Ψβ(zβ,t)=∫0∞dEg(E)T˜αβ(E)exp[−iEtℏ], where T˜αβ(E)=Tαβexp(ikRβ) is a smooth function of E: in accordance with the analytical S-matrix theory, Tαβ(E) contains the factor exp(−ikRβ) and consequently this factor is being cancelled by exp(ikRβ) in T˜αβ(E). For condition     (4.5)Γ≪ΔE≪Er,one can rewrite (4.4) in the following simplified form:     (4.6)Ψβ(Rβ,t)=A∫0∞dEexp[−iEt/ℏ]E−Er+iΓ/2, where A is a constant. In the approximation (4.5) and Γ=  constant one obtain  (4.7)Ψβ(Rβ,t)={Bexp[−iErtℏ-(Γ/2ℏ)t],fort>0,0,fort<0,                (moving the lower integration limit in (4.15) from 0 to −∞ and utilizing the residue theorem). Here B is a constant and more precisely there must be t−tin  (with tin=ℏ(∂argg/∂E)) instead of t.  The form (4.7) is valid also with slight modifications in the cases when  Γ~E1/2 or  Γ is a linear function of E (see, e.g., [149]).The evolution of the particleβ passing through position zβ during the unitary time interval, centered at t, is described by the probability flux density jβ(zβ,t)=Re[Ψβ(zβ,t)(iℏ/2m)∂Ψβ*(zβ,t)/∂zβ] with the adequate normalization ∫−∞∞jβ(zβ,t)dt=1. The emission probability (per a time unit) in the vicinity of the compound nucleus (near zβ=Rβ)  (4.8)I(t)=jβ(Rβ,t)∫−∞∞dtjβ(Rβ,t)  is equal to  (4.9)I(t)=(Γℏ)exp(−Γtℏ). (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          (4.10)limzβ→Rβ(−iℏmβ)Tαβ∂[exp(ikzβ)]∂zβ=vT˜αβ(v=ℏk/mβ).    If Ψβ(Rβ,t) has a form (4.7), the Fourier transform of Ψβ is equal to  (4.11)∫0∞dtΨβ(Rβ,t)exp(−iEtℏ)=B∫0∞dtexp[−i(E−Er)t/ℏ−Γt/2ℏ]=iBE−Er+iΓ/2,  which is proportional to the amplitude
from (4.1).Forzβ>Rβ one can rewrite (4.3) in a following way:          (4.12)Ψβ(zβ,t)=∫0∞dkG(k)D(k)(k-k0)(k+k0)exp(ikzβ-iEtℏ),with                       (4.13)G(k)=g(ℏ2k22mβ)(dEdk),D(k)=(2mβℏ2)Nβ(E)Cαβ,k0=(12)(kr4+γ2+kr2−ikr4+γ2−kr2),kr=2mβErℏ,γ=Γmβℏ2.  Since G(k) and D(k) are smooth functions of k(4.14)Ψβ(zβ,t)≅∫−∞∞dkG(k)D(k)(k−k0)(k+k0)exp(ikzβ−iEtℏ), under the condition (4.5), then, introducing in (4.14) a new variable  (4.15)y=iℏt2mβ(k−mβzβℏt),                                                       and performing the transformations quite similar to those which were used in [30–32], it is possible to obtain  (4.16)Ψβ(Rβ,t)={0,forzβ>vrt,const⋅exp[ikrzβ−iErtℏ−(Γ2ℏ)(t−zβvr)],forzβ≤vrt,             with vr=ℏk/mβ. The wave function (4.16) can be applied for macroscopic distances zβ, near a detector which registers particles 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 Er are completely populated in the large region with the width ΔE≫Γ. If, on the contrary,  (4.17)ΔE≪Γ.The emission probability is nonexponential and does essentially depend on ΔE and the form of g(E). If one will take the Lorentzian form also for g(E), that is,  (4.18)g(E′)=g0E′−E+iΓ/2,with g0= const or smooth inside ΔE under conditions (4.5) and (4.17), instead of (4.7) the expression  (4.19)Ψβ(Rβ,t)={0,forzβ>vr(t-tin0);constE−Er+iΓ/2⋅exp[ikrzβ-iErtℏ-(ΔE2ℏ)(t−zβvr−tin0)],forzβ≤vr(t-tin0)   (with v=ℏk/mβ and tin0=ℏ(∂argg0/∂E)) will be obtained. The cross-section σαβ, which is proportional to  (4.20)σαβ~∫tmin∞dtjβ(zβ,t),  where tmin=zβ/v+tin0,(tmin,∞) being the operative registration time interval of detector, after integrating in (4.20) acquires the Breit-Wigner form  (4.21)σαβ=|fαβ|2=const(E−Er)2+Γ2/4.


### 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 amplitudeF(E,θ) is  (4.22)FE,θ=fE,θ+fl,res(E,θ), with  (4.23)f(E,θ)=fcoul(E,θ)+2ik-1∑λ≠l(2λ+1)Pλ(cosθ)exp(2iηλ)[exp(2iδλb)−1],fl,res(E,θ)=2ik-1(2λ+1)Pλ(cosθ)exp(2iηλ)[exp(2iδlb)E−Eres−iΓ/2E−Eres+iΓ/2−1] is the Coulomb scattering amplitude, δlb and  ηl  being the background nuclear l-scattering phase-shift and the Coulomb l-scattering phase shift, respectively, k is the wave number, θ is the scattering angle in the c.m.s. Rewriting  (4.22) in the form  (4.24)F(E,θ)=[A(E−Eres)+iBΓ2](E−Eres+iΓ2)−1 where (4.25)A=f(E,θ)+k−12l+1Pl(cosθ)exp(2iηl+iδlb)sinδlb,B=f(E,θ)+ik−12l+1Pl(cosθ)exp(2iηl+iδlb)cosδlb,we obtain (as it was firstly made in [150]) the following expression for the quasimonochromatic total scattering durationτ(E,θ):  (4.26)τ(E,θ)=2Rv+ℏ∂argF∂E≡2Rv+Δτ(E,θ), where  v=ℏk/μ   is the projectile velocity and  R is the interaction radius,  (4.27)Δτ(E,θ)=(ℏΓ2)[(E−Eres)2+Γ24]−1−(ℏReα2)[(E−Eres−Imα2)2+(Reα)24]−1      is the time delay, α=ΓB/A.  When 0<Re(B/A)<1 and |Imα/2|≪Eres, near E=Eres  the time delay Δτ 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 radiusR; it lies between the minimal value, determined by the equality of the energy E 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≥0) in the case of negative values of Δτ.The time analysis of proton scattering by nucleiC12 and N14 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, whenΔE≫Γ¯,D (ΔE,Γ¯, and D 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 fαβ(E) by the form  (4.28)fαβ(E)=Cαβnexp(−Eτn2ℏ+iEtnℏ),where τn and tn are constants (with the time dimension), τn and tn determine the exponential dependence on energy for the correspondent cross-section and the  linear dependence on energy for the amplitude phase, respectively, Cαβn is a constant or a very smooth function (inside ΔE) of the final-particle energy E, then the correspondent cross-section and the emission probability under the condition  (4.29)ΔE≪2ℏτnwill be (4.30)σαβ=|fαβ|2=const⋅exp(−Eτnℏ),(4.31)I(t)=(τn2π)1(t−tn)2+τn4/4, 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:  (4.32)Lc(t)=1−∫(0)tdtI(t). From (4.31)-(4.32), one can deduce the strongly nonexponential form of Lc(t) and I(t), like depicted in Figure 15.Figure 15
Lc(t) (curve 1) and I(t) (curve 2).Whenfαβ has a more general form like  (4.33)fαβ(E)=∑n=1vCαβnexp(−Eτn2ℏ+iEtnℏ),with several terms (v=2,3,…), the cross-section σαβ=|fαβ|2 contains not only exponentially decreasing terms but also oscillating terms with  factors  cos[E(tn−tn′)/ℏ]  and   sin[E(tn−tn′)/ℏ]. In the case of two terms (v=2) in (4.33) formula (4.30) can be rewritten as (4.34)σαβ=|fαβ1|2exp(−Eτ1ℏ)+|fαβ2|2exp(−Eτ2ℏ)+2Re{fαβ1fαβ2*exp[iE(t1−t2)ℏ−E(τ1+τ2)2ℏ]}, where the terms with ΔE are neglected if the conditions ΔEtn≪EτnandΔEτn≪Etn are supposed.In particular, for inclusive energy spectra of thekth final fragment we will use the following expression: (4.35)σinc,k(Ek)=∑n=12|Cnexp[(itn−τn2)Ekℏ]|2=∑n=12|Cn|2exp(−Ekτnℏ)+2ReC1*C2exp{[i(t2−t1)−(τ1+τ2)2]Ekℏ}.  In Figure 16, an example of the calculated inclusive energy spectra σinc,k(Ek), in arbitrary units and in semilogarithmic scale, is presented in comparison with the experimental data (see from [148]).Inclusive processp+C12→Be7 (2.1 GeV protons).
(a)
C1=0.04,C2=0.36(θ=90°)(b)
C1=0.35,C2=0.05(θ=160°)


## 4.1. Narrow Resonance→ Exponential Decay

Let us firstly explain how a typical isolated Lorentzian (Breit-Wigner) resonance, in the cross-section of a quantum collision or nuclear reactionα→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  fαβ(E) as   (4.1)fαβ(E)=CαβE−Er+iΓ/2, where Cαβ is a constant or a smooth function of the final-particle kinetic energy E in the region (Er−Γ/2,Er+Γ/2),Er and Γ being the resonance energy and width, respectively.The final-particle wave packet in the 1D radial asymptotic limit is described by(4.2)Ψβ(rβ,t)=rβ−1∫0∞dEg(E)fαβ(E)exp[ikrβ−iEtℏ], where g(E) is a smooth weight amplitude with an energy spread ΔE (usually ΔE≪Er),mβ and rβ are the final-particle mass and radial coordinate, respectively, and k=2mβE/ℏ. For short-ranged interactions (including also screened Coulomb potentials), it can be rewritten as  (4.3)Ψβ(zβ,t)=∫0∞dEg(E)Tαβ(E)exp[ikzβ−iEtℏ], with Tαβ(E)=Nβ(E)fαβ(E),Tαβ(E) is the T-matrix elements connected with the S-matrix elements by known relation Tαβ=δαβ−Sαβ,Nαβ(E) is an unessential smooth function of E,zβ is the axis along the direction of the final-particle emission imposed by the registration geometry, zβ≥Rβ,Rβ is the interaction radius in the final channel. In the simplest case one can fix zβ=Rβ and  (4.4)Ψβ(zβ,t)=∫0∞dEg(E)T˜αβ(E)exp[−iEtℏ], where T˜αβ(E)=Tαβexp(ikRβ) is a smooth function of E: in accordance with the analytical S-matrix theory, Tαβ(E) contains the factor exp(−ikRβ) and consequently this factor is being cancelled by exp(ikRβ) in T˜αβ(E). For condition     (4.5)Γ≪ΔE≪Er,one can rewrite (4.4) in the following simplified form:     (4.6)Ψβ(Rβ,t)=A∫0∞dEexp[−iEt/ℏ]E−Er+iΓ/2, where A is a constant. In the approximation (4.5) and Γ=  constant one obtain  (4.7)Ψβ(Rβ,t)={Bexp[−iErtℏ-(Γ/2ℏ)t],fort>0,0,fort<0,                (moving the lower integration limit in (4.15) from 0 to −∞ and utilizing the residue theorem). Here B is a constant and more precisely there must be t−tin  (with tin=ℏ(∂argg/∂E)) instead of t.  The form (4.7) is valid also with slight modifications in the cases when  Γ~E1/2 or  Γ is a linear function of E (see, e.g., [149]).The evolution of the particleβ passing through position zβ during the unitary time interval, centered at t, is described by the probability flux density jβ(zβ,t)=Re[Ψβ(zβ,t)(iℏ/2m)∂Ψβ*(zβ,t)/∂zβ] with the adequate normalization ∫−∞∞jβ(zβ,t)dt=1. The emission probability (per a time unit) in the vicinity of the compound nucleus (near zβ=Rβ)  (4.8)I(t)=jβ(Rβ,t)∫−∞∞dtjβ(Rβ,t)  is equal to  (4.9)I(t)=(Γℏ)exp(−Γtℏ). (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          (4.10)limzβ→Rβ(−iℏmβ)Tαβ∂[exp(ikzβ)]∂zβ=vT˜αβ(v=ℏk/mβ).    If Ψβ(Rβ,t) has a form (4.7), the Fourier transform of Ψβ is equal to  (4.11)∫0∞dtΨβ(Rβ,t)exp(−iEtℏ)=B∫0∞dtexp[−i(E−Er)t/ℏ−Γt/2ℏ]=iBE−Er+iΓ/2,  which is proportional to the amplitude
from (4.1).Forzβ>Rβ one can rewrite (4.3) in a following way:          (4.12)Ψβ(zβ,t)=∫0∞dkG(k)D(k)(k-k0)(k+k0)exp(ikzβ-iEtℏ),with                       (4.13)G(k)=g(ℏ2k22mβ)(dEdk),D(k)=(2mβℏ2)Nβ(E)Cαβ,k0=(12)(kr4+γ2+kr2−ikr4+γ2−kr2),kr=2mβErℏ,γ=Γmβℏ2.  Since G(k) and D(k) are smooth functions of k(4.14)Ψβ(zβ,t)≅∫−∞∞dkG(k)D(k)(k−k0)(k+k0)exp(ikzβ−iEtℏ), under the condition (4.5), then, introducing in (4.14) a new variable  (4.15)y=iℏt2mβ(k−mβzβℏt),                                                       and performing the transformations quite similar to those which were used in [30–32], it is possible to obtain  (4.16)Ψβ(Rβ,t)={0,forzβ>vrt,const⋅exp[ikrzβ−iErtℏ−(Γ2ℏ)(t−zβvr)],forzβ≤vrt,             with vr=ℏk/mβ. The wave function (4.16) can be applied for macroscopic distances zβ, near a detector which registers particles 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 Er are completely populated in the large region with the width ΔE≫Γ. If, on the contrary,  (4.17)ΔE≪Γ.The emission probability is nonexponential and does essentially depend on ΔE and the form of g(E). If one will take the Lorentzian form also for g(E), that is,  (4.18)g(E′)=g0E′−E+iΓ/2,with g0= const or smooth inside ΔE under conditions (4.5) and (4.17), instead of (4.7) the expression  (4.19)Ψβ(Rβ,t)={0,forzβ>vr(t-tin0);constE−Er+iΓ/2⋅exp[ikrzβ-iErtℏ-(ΔE2ℏ)(t−zβvr−tin0)],forzβ≤vr(t-tin0)   (with v=ℏk/mβ and tin0=ℏ(∂argg0/∂E)) will be obtained. The cross-section σαβ, which is proportional to  (4.20)σαβ~∫tmin∞dtjβ(zβ,t),  where tmin=zβ/v+tin0,(tmin,∞) being the operative registration time interval of detector, after integrating in (4.20) acquires the Breit-Wigner form  (4.21)σαβ=|fαβ|2=const(E−Er)2+Γ2/4.


## 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 amplitudeF(E,θ) is  (4.22)FE,θ=fE,θ+fl,res(E,θ), with  (4.23)f(E,θ)=fcoul(E,θ)+2ik-1∑λ≠l(2λ+1)Pλ(cosθ)exp(2iηλ)[exp(2iδλb)−1],fl,res(E,θ)=2ik-1(2λ+1)Pλ(cosθ)exp(2iηλ)[exp(2iδlb)E−Eres−iΓ/2E−Eres+iΓ/2−1] is the Coulomb scattering amplitude, δlb and  ηl  being the background nuclear l-scattering phase-shift and the Coulomb l-scattering phase shift, respectively, k is the wave number, θ is the scattering angle in the c.m.s. Rewriting  (4.22) in the form  (4.24)F(E,θ)=[A(E−Eres)+iBΓ2](E−Eres+iΓ2)−1 where (4.25)A=f(E,θ)+k−12l+1Pl(cosθ)exp(2iηl+iδlb)sinδlb,B=f(E,θ)+ik−12l+1Pl(cosθ)exp(2iηl+iδlb)cosδlb,we obtain (as it was firstly made in [150]) the following expression for the quasimonochromatic total scattering durationτ(E,θ):  (4.26)τ(E,θ)=2Rv+ℏ∂argF∂E≡2Rv+Δτ(E,θ), where  v=ℏk/μ   is the projectile velocity and  R is the interaction radius,  (4.27)Δτ(E,θ)=(ℏΓ2)[(E−Eres)2+Γ24]−1−(ℏReα2)[(E−Eres−Imα2)2+(Reα)24]−1      is the time delay, α=ΓB/A.  When 0<Re(B/A)<1 and |Imα/2|≪Eres, near E=Eres  the time delay Δτ 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 radiusR; it lies between the minimal value, determined by the equality of the energy E 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≥0) in the case of negative values of Δτ.The time analysis of proton scattering by nucleiC12 and N14 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, whenΔE≫Γ¯,D (ΔE,Γ¯, and D 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 fαβ(E) by the form  (4.28)fαβ(E)=Cαβnexp(−Eτn2ℏ+iEtnℏ),where τn and tn are constants (with the time dimension), τn and tn determine the exponential dependence on energy for the correspondent cross-section and the  linear dependence on energy for the amplitude phase, respectively, Cαβn is a constant or a very smooth function (inside ΔE) of the final-particle energy E, then the correspondent cross-section and the emission probability under the condition  (4.29)ΔE≪2ℏτnwill be (4.30)σαβ=|fαβ|2=const⋅exp(−Eτnℏ),(4.31)I(t)=(τn2π)1(t−tn)2+τn4/4, 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:  (4.32)Lc(t)=1−∫(0)tdtI(t). From (4.31)-(4.32), one can deduce the strongly nonexponential form of Lc(t) and I(t), like depicted in Figure 15.Figure 15
Lc(t) (curve 1) and I(t) (curve 2).Whenfαβ has a more general form like  (4.33)fαβ(E)=∑n=1vCαβnexp(−Eτn2ℏ+iEtnℏ),with several terms (v=2,3,…), the cross-section σαβ=|fαβ|2 contains not only exponentially decreasing terms but also oscillating terms with  factors  cos[E(tn−tn′)/ℏ]  and   sin[E(tn−tn′)/ℏ]. In the case of two terms (v=2) in (4.33) formula (4.30) can be rewritten as (4.34)σαβ=|fαβ1|2exp(−Eτ1ℏ)+|fαβ2|2exp(−Eτ2ℏ)+2Re{fαβ1fαβ2*exp[iE(t1−t2)ℏ−E(τ1+τ2)2ℏ]}, where the terms with ΔE are neglected if the conditions ΔEtn≪EτnandΔEτn≪Etn are supposed.In particular, for inclusive energy spectra of thekth final fragment we will use the following expression: (4.35)σinc,k(Ek)=∑n=12|Cnexp[(itn−τn2)Ekℏ]|2=∑n=12|Cn|2exp(−Ekτnℏ)+2ReC1*C2exp{[i(t2−t1)−(τ1+τ2)2]Ekℏ}.  In Figure 16, an example of the calculated inclusive energy spectra σinc,k(Ek), in arbitrary units and in semilogarithmic scale, is presented in comparison with the experimental data (see from [148]).Inclusive processp+C12→Be7 (2.1 GeV protons).
(a)
C1=0.04,C2=0.36(θ=90°)(b)
C1=0.35,C2=0.05(θ=160°)



---

*Source: 859710-2009-04-27.xml*