Review Article | Open Access

V. S. Olkhovsky, "Time as a Quantum Observable, Canonically Conjugated to Energy, and Foundations of Self-Consistent Time Analysis of Quantum Processes", *Advances in Mathematical Physics*, vol. 2009, Article ID 859710, 83 pages, 2009. https://doi.org/10.1155/2009/859710

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

**Academic Editor:**Ricardo Weder

#### Abstract

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

#### 1. General Introduction

During almost ninety years (e.g., [1, 2]), it is known that time cannot be represented by a self-adjoint operator, with the possible exception of special abstract systems (such as an electrically charged particle in an infinite uniform electric field) and a system with the limited from both below and above energy spectrum (to see later)). (Namely that fact that time cannot be represented by a self-adjoint operator is known to follow from the semiboundedness of the continuous energy spectra, which are bounded from below (usually by the value zero). Only for an electrically charged particle in an infinite uniform electric field, and for other very rare special systems, the continuous energy spectrum is not bounded and extends over the whole energy axis from to .) This fact results to be in contrast with the known sircumstance that time, as well as space, in some cases plays the role just of a parameter, while in some other cases *is* a physical observable which *ought* to be represented by an operator. The list of papers devoted to the problem of time in quantum mechanics is extremely large (e.g., [3β51], 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., [12β14, 47, 50, 51]).

As to quantum mechanics, the first set of known and cited articles is [3β21]. The second set of papers on time as an observable in quantum physics [22β51] 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 [15β21]. This Naimark theorem states [52] that the nonorthogonal spectral decomposition 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 of the operator *H*.

*
Namely, by exploiting that Naimark theorem, it has been shown by Olkhovsky and Recami [15β18, 21] (more details having been added in [22β27, 32β35, 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 [23β25, 33β35, 50, 51] it was clarified that time can be introduced also for these systems as a* quantum-mechanical observable, canonically conjugate to energy, and the time operator resulted to be quasi-self-adjoint (more precisely, it can be chosen as an almost self-adjoint operator with practically almost any degree of the accuracy).

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

Then, we analyze the restricted sense of the positive operator value measure (POVM) approach, often used now (see, in particular [28β31, 36β46, 48, 49]). Finally, we do in a shorten way review our methods of time analysis and joint time-energy analysis which had already proved to be fruitful in tunnelling and nuclear processes.

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

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

For systems with continuous energy spectra, the following simple operator, canonically conjugate to energy, can be introduced for time
which is not self-adjoint, but is *Hermitian*, and acts on square-integrable space-time wave packets in representation (2.1a), and on their Fourier transforms in representation (2.1b), once the point 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 β*maximal Hermitian*β (or β*maximal symmetric*β) operator [15β18, 21] (see also [26, 27, 33β35, 52, 53]), according to Akhiezer & Glazmanβs terminology. Let us explicitly notice that, anyway, the physically reasonable boundary condition 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 [15β18, 21]. The elimination of the point is not restrictive since the βrestβ 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 of the continuous, differentiable, square-integrable functions that satisfy the conditions and the condition which is a space *dense* in the Hilbert space of functions defined (only) over the semiaxis . Obviously, the operator (2.1a) and (2.1b) is Hermitian, that is, the relation holds, only if all square-integrable functions in the space on which it is defined vanish for .

Also the operator is Hermitian, that is, the relation holds under the same conditions.

Operator has no Hermitian extension because otherwise one could find at least one function which satisfies the condition but that is inconsistent with the propriety of being Hermitian. So, according to [53], 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 bounded by a rigid wall located at . Consequently, the operator for the momentum *x*-component of the particle, which reads is defined as acting on the space of the continuous, differentiable, square-integrable functions that satisfy the conditions and the condition
which is a space *dense * in the Hilbert space of functions defined (only) over the spatial semiaxis . Therefore, operator has the same mathematical properties as operator (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 .

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 * the time operator (2.1a) and (2.1b) becomes a *really self-adjoint operator* and has a discrete time spectrum, with the βthe time quantumβ where .

In order to consider *time as an observable in quantum mechanics* and to define the observable mean times and durations, one needs to introduce not only the time operator, but also, in a self-consistent way, the measure (or weight) of averaging over time. In the simple one-dimensional (1D) and one-directional motion such *measure* (weight) can be obtained by the the simple quantity: where the probabilistic interpretation of (namely in *time*) corresponds to the flux probability density of a particle passing through point at time (more precisely, passing through during a unit time interval centered at ), 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 and of the continuity relation for particle motion in the field of any hamiltonian in the desciption of the 1D Schroedinger equation. (The three-dimensional (3D) case is described in Appendix B.) Quantity is the probability of finding a moving particle inside a unit space interval, centered at point *x*, at time *t*. The probability density and the flux probability density are related with the wave function by the usual definitions and . The measure (2.7) was firstly investigated in [21, 23β27, 32β35].

When the flux density changes its sign, the quantity 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 does keep its sign. Therefore, let us introduce the two measures, by separating the positive and the negative flux-direction values (i.e., flux signs): with where is the Heaviside step function. It had been made firstly in [26, 27, 32β35]. Actually, one can rewrite the continuity relation (2.8) for those time intervals, for which or as follows: respectively. Relations in (2.10) can be considered as formal definitions of and . Integrating them over time *t* from to , one obtains with the initial conditions . Then, it is possible to introduce the quantities which have the meaning of probabilities for the particle wave packet to be located at time on the semiaxis and , respectively, as functions of the flux densities and , provided that the normalization condition 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 and , and by adopting the boundary conditions . Then, by differentiating and with respect to *t*, one obtains Finally, from the last four equations one can easily infer that which justify the abovementined probabilistic interpretation of . 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 of the time at which a particle passes through position *x* (when travelling in only one positive *x*-direction), and of the time *t* at which a particle passes through position , when travelling in the positive or negative direction, respectively, where is the Fourier transform of the moving 1D wave packet when going on from the time representation to the energy one, and also the mean durations of particle 1D transmission from to and 1D particle reflection from the region into : 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 , similarly to (2.15) by using the unique Fourier (Laplace) transformations and the energy expansion of , but it is evident that they result to be rather bulky.

If one does now generalize the expressions (2.15) and (2.17) for with a generic value then we will be able to write down for with any *analytic * function of time , the *one-to-one * relation from the time to the energy representation. For free motion, one has , , and , while the normalization condition is
and the boundary conditions are

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

In (2.19), 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 , *K* being a constant, quantity goes into and (2.19) goes into the more simple relation because of the relations .

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

Although up to now according to the Stone theorem [55] the relation (2.24) has been interpreted as holding *only* for the pair of the self-adjoint canonically conjugate operators, in both representations, and it was not directly generalized for *maximal Hermitian operators*, the difficulty of such direct generalization has in fact been by-passed by introducing with the help of the *single-valued* Fourier (Laplace) transformation from the *t*-axis to the *E*-semiaxis and by utilizing the peculiar mathematical properties of maximal symmetric operators (as in [19β21, 23β25, 33β35, 50, 51]), described in detail, for example, in [52, 53].

Actually, from (2.24) the uncertainty relation (where the standard deviations are quantity being the variance ; and , while denotes an average over by the measures or in the -representation or an average over 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 [17β21, 23β25, 33β35, 50, 51]). Moreover, relation (2.24) satisfies the Dirac βcorrespondence principle,β since the classical Poisson brackets , with and , are equal to unity [56]. In [21] (see also [23β25]) 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, 23β25] 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 in (2.1a), (2.1b), and (2.26) are *sufficient* for considering them as quantum observables: namely, the *uniqueness* of the βspectral decompositionβ (also called spectral function) for operators , as well as for 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 , corresponding with any accuracy to *real eigenvalues* of the continuous spectrum; such eigenfunctions belonging to the space of the square-integrable functions of the energy *E* with the boundary conditions (2.2)-(2.3).

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

Now let us analyse the so-called *positive-operator-value-measure * (POVM) approach, often used in the second set of papers on time in quantum physics (e.g., in [28β31, 36β46, 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 [28β31, 36β46, 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 Naimarkβs 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 [12β18, 21] (and independently in the papers of Holevo [19, 20], with the same principal idea) much earlier than [28β31, 36β46, 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 [12β14] one more form of the time operator (the so-called *bilinear* form), where the meaning of the sign is clear from the following definition: . For this form the *direct* elimination of the point is not necessary because it is eliminated automatically in and in by such *bilinearity*. And such an elimination of the point is not only more simple but also more physical than an elimination made in [28β31, 36β46, 48, 49], and it had been published (in [12β14]) 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 (-) representation, with , in (2.1a), (2.1b), (2.26), (2.2), (2.3), and (2.7) one can also use the momentum (-) representation, with the advantage that : with . In such a case the time operator (2.1a), (2.1b), and (2.26) (acting on momentum eigenvector, defined on , is already *formally self-adjoint*, with the boundary conditions except for the fact that we have excluded point ; an exclusion which has now only the physical meaning of nonobserving the rest (motionless) state), being inessential mathematically (this had been considered in [19, 20, 50, 51]). In fact, it is one more argument in favor of that time is an observable in the same degree as any other quantity to which a self-adjoint operator corresponds.

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

If the wave packet (2.27) is *one directional* and , then the integral goes on to the integral 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 is performed.

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

One can easily see that the weight can be considered as the meaning of the probability for a particle to be localized, or to *sojourn*, or to *dwell* in the spatial region at the moment *t *, *independently from the motion processes*. As a consequence, the quantity will have the meaning of the probability of particle dwelling in the spatial range at the instant . Taking into account the equality which evidently follows from the 1D continuity relation (2.8), the mean dwell time can be presented in the following form:
with the flux density for the initial β*dwelling*β *free* motion through point . The expression (2.36a) can be rewritten in the following equivalent form
taking into account the continuity relation (2.8) for the *total* flux density in the interval at time (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 [59β62]. 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 βwave function for a single photonβ) which is sufficient for the identical definition of mean time instants and durations (and distribution variances, etc.) of propagation, collision, tunnelling, processes for particles and photons.

In the first quantization for the 1D case, the single-photon wave function can be probabilistically described by the wave packet (see, e.g., [63, 64]) where, as usual, is the electromagnetic vector potential, and , , , , where the gauge condition is assumed.

The axis has been chosen as the propagation direction, with , , is the probability amplitude for the photon to have momentum and polarization along , and it is in the case of plane waves, while is a linear combination of evanescent (decreasing) and antievanescent (increasing) waves in the case of βphoton barriersβ (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: * ( being the energy density, the electromagnetic field being , ) *of a photon to be found (localized) in the spatial intervall along axis x at the moment t, and the flux probability density * (with being the energy flux density, ) *of a photon to pass through the point (plane) x in the time interval *, 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 [65β67]. Hence, (1) in a certain sense, for the time analysis along the motion direction, the wave packet (2.24) is quite similar to a wave packet for nonrelativistic particles and (2) similarly to the conventional nonrelativistic quantum mechanics, one can define the mean time of photon (electromagnetic wave packet) passing through point *x *:
where the form (2.1b) of time operator is valid also for photons with natural boundary conditions in the energy representation , quite similarly to (2.1b)β(2.3) for nonrelativistic particles in the energy representation.

The energy density and energy flux density satisfy the relevant continuity equation which is *Lorentz-invariant* for the spatially 1D propagation [32β35, 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 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)β(2.9), in the both time and energy representations [32β35, 47, 50, 51]. It is also possible to introduce the second measure in time averaging, quite similarly to (2.36a) and (2.36b). In other words, *in the cases of 1D photon propagations time is a quantum-physical observable also in quantum electrodynamics.*

In the case of fluxes which change their signs with time we introduce quantities 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 by ).

##### 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 [23β25, 50, 51]) to introduce wave packets of the form (where are orthogonal and normalized wave functions of system bound states which satisfy being the system Hamiltonian, , here we factually omitted a nonsignificant phase factor as being general for all terms of the sum ) for describing the evolution of systems in the regions of the purely discrete spectrum. Without limiting the generality, we choose moment as an initial time instant.

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

In the *t*-representation the relevant energy operator is a self-adjoint operator acting in the space of *periodical* functions whereas the function does not belong to the same space. In the space of periodical functions the time operator , even in the eigen representation, has to be also a periodical function of time . 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 , a *periodical* function which is the so-called saw-function of (see Figure 1).

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

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

From (2.47) it follows that when (i.e., when ) the right part of (2.47) tends to zero since tends to a constant. In this case, the distribution of time instants of wave packet passing through point *x* in the limits of one PoincarΓ© cycle becomes uniform. When and , the periodicity condition may be inessential for , 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 of instants of particle passing through point , then after a series of bulky transformations he will obtain the following expression: in the energy representation, where ; the bilinear operation, denominated by , signifies (Of course, one has to average over the flux density, but for the simplicity in this case it is possible to make averaging over ) Operator (2.43) for two levels acquires the more simple form and when , the expression (2.50) passes to the differential form which coincides with (A.1) from Appendix A, that is, it is equivalemt to operator (2.1b) for the continuous energy spectra.

In general cases, for excited states of nuclei, atoms, and molecules, *level distances in discrete spectra have not strictly defined the maximal common divisor and hence, they have not the strictly defined time of the PoincarΓ© 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, ) by quasicycles with quasiperiodical evolution and for sufficiently long intervals of time the motion inside such systems (however, less than ) one can consider as *a periodical motion also with any desired accuracy*. For them one can choose (define) a time of the Poincareβ 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 contains only one term , 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 (-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., [72β80], apart from [26, 27, 32β35, 47]). And during many years, there had not been not only the consensus in the theoretical definition of the tunneling time for particles, but also there had been some declarations about the incompatibility of some approaches both quantitavely and in the physical interpretation. Among the reasons of such situation there had been the following ones:

(i)the problem of defining the tunneling time is closely connected with general fundamental problems of time as a *quantum*-*physical observable * and the general definition of quantum-collision durations, and the acquaintance with the principal solution of these problems had not got a wide prevalence yet till 2000β2004 (e.g., [47, 81]);(ii)the motion of particles inside a potential barrier is a quantum phenomenon *without* any direct classical limit (namely for particles);(iii)there are essential physical and mathematical *differences in initial, boundary, and external conditions* of various definition schemes.

Following [47, 80], we arrange the majority of approaches into several groups which are based on (1) the time-dependent wave packet description; (2) averaging over an introduced set of kinematic paths, distribution of which is supposed to describe the particle motion inside a barrier; (3) introducing a new degree of freedom, constituting a physical clock for measurements of tunnelling times. Separately, by oneβs 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 [12β14, 86, 87] and applied to tunnelling in [88, 89], and finally the Olkhovsky-Recami (O-R) method [26, 27, 32β35, 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, 23β25, 91] for atomic and nuclear collisions. The second group contains methods, utilizing the Feynman path integrals [92β98], the Wigner distribution paths [99β102], and the Bohm approach [103]. The approaches with the Larmor clock [104β107] and the oscillatory barrier [108, 109] pertain to the third group.

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

Here such a definition with the necessary formalism is presented (Section 3.2) and a brief comparison with various approaches is given (Sections 3.3β3.5), basing on the O-R formalism. In Section 3.6 the Hartman and Fletcher effect, with its generalization and its violations, is described. The tunneling through a double barrier is described in Section 3.7. The particle tunneling through three-dimensional barriers is presented in Section 3.8. The quaternion description of tunneling phenomena is mentioned in Section 3.9.

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

We confine ourselves to the simplest case of particles moving only along the -direction, and consider a time-independent barrier in the interval βsee Figure 1, in which a larger interval , containing the barrier region, is also indicated.

As it is well known, in the case of a rectangular potential barrier of the height , the stationary wave function for a particle with mass and energy has the usual form (e.g., [26, 27, 32, 47, 72β81, 90] and a lot of other papers): where , , , , and are the amplitudes of the reflected, evanescent, antievanescent and transmitted waves, respectively.

Inside a barrier here we have not usual propagating waves but a superposition of an evanescent (decreasing) and antievanescent (growing) waves with an imaginary wave number . 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, 23β25, 91] and afterwards generalized in [26, 27, 32β35, 47] (see also [81]), we can eventually define the mean values of the time at which a particle passes through position , travelling in the positive or negative direction of the -axis, and the variances of the distributions of these times, respectively, as being the positive or negative values, respectively, of the probability flux density for an evolving time-dependent normalized *wave packet *. We recall here the *equivalence of canonically conjugated time and energy representations*, with appropriate measures of averaging, in the following sense: (index *t * is omitted in all expressions for for the sake of the simplicity). This equivalence is a consequence of the unique time-operator existence.

For transmissions from region I to region III we have with and . For a pure tunnelling process one has Similar expression we have for the penetration (into the barrier region II) temporal quantities and with . For reflections in any point one has

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

In the asymptotic cases, when , where and denote averagings over the fluxes corresponding to and , respectively.

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

At the quasimonochromatric limit being , we get for strictly the ordinary phase time, without averaging. For a rectangular barrier with height and (where ), the expressions (3.11) and (3.13), for and , pass into the known expressions (coincident with the phase time [26, 27, 47, 83]), and (coincident with one of the Larmor times [104β107] and the Buettiker-Landauer time [108] and also with the imaginary part of the complex time in the Feynman path-integration approach: see later Section 3.5), respectively.

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

By the way, if the measurement conditions are such that only the positive-momentum components of wave packets are registrated, that is, being the projector onto positive-momentum states, then for *any * from and from because .

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

From (3.19), we can see that when the effective tunnelling velocity is more than *c*, that is, superluminal. This result agrees with the results of the microwave-tunnelling measurements presented in [111β113] (see also [114] where moreover, the effective tunnelling velocity was identified with the group velocity of the final wave packet corresponding to a single photon).

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

In Section 2, it was analyzed the meaning of two forms of the expression for the mean dwell time (2.36a) and (2.36b) from Section 2. Taking into account that the total flux and with , and corresponding to the wave packets and constructed from the stationary wave functions and , respectively, and also One obtains from (2.36a) and (2.36b) of Section 2, with , , and One can see that is negative and tends to 0 when tends to .

When and are sufficiently well separated in time, so that , it follows from (3.22) that the simple *weighted average rule * is valid. For a rectangular barrier with and quasimonochromatic particles, the expressions (3.22) and (3.24) with and pass to the known expressions (taking account of the interference term ) (when the interference term is equal to 0).

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

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

##### 3.4. Analysis of the Larmor and Buettiker-Landauer Clocks

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

The Larmor times and are defined by the ratio of the spin-rotation angles around axes and (in turn defined by the developed - and -spin components, resp.) to the precession (rotation) frequency.

As to , in the reality it is not a precession but a jump to position βspin-upβ or βspin-downβ (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 and coincides with the expression (3.15) (of course, at the same approximations when (3.15) is valid).

For an opaque rectangular barrier with the expressions (for mean Larmor times) had been obtained in [79, 101β103].

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 for the same reasons as for .

##### 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 [92β98] 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 , where is the action associated with the path . 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 and , respectively.

An interesting development of this approach, the *instanton * version, is presented in [97, 98]. The instanton-bounce path is a stationary point of the Euclidean action. The latter is obtained by the analytic continuation to imaginary time in the Feynman-path integrals containing the factor . 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 [99β102]. 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 , so that the flux at position *x* can be separated into positive and negative components: with and . 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 instead of our . The dwell time decomposition in this approach takes the form with . Asymptotically, tends to and (3.31) takes *formally* the known form (3.24).

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

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

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

Firstly the Hartman and Fletcher effect (HFE) was revealed and studied in [84, 85] within the stationary-phase method for a 1D motion of quasimonochromatic nonrelativistic particles tunnelling through potential barriers. It consists in the absence of the dependence of the *phase tunnelling time * (which is the *mean tunnelling time * 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], and being the transmission amplitude and the particle kinetic energy, resp.) on the barrier width *a* for sufficiently large . In particular, for a rectangular potential barrier , , being the barrier height, and when ( is the particle velocity before entering into a barrier).

Now we will test the validity of HFE for all other known theoretical expressions for mean tunnelling times. If we firstly take from the *mean dwell time *, the *mean Larmor time * and the real part of the complex tunnelling time obtained by *averaging over the Feynman paths *, which are equal for quasimonochromatic particles and opaque rectangular barriers, we immediately easily see that also in these cases there is *no* dependence on the barrier width and consequently HFE is valid.

The validity of HFE for the *mean tunnelling time * within the O-R approach, is directly seen from the expression (where denotes averaging over the initial wave-packet energy spread and being the probability flux density for a wave packet moving along axis *x * through a barrier located in the interval and it was confirmed in [26, 27, 32β35, 47] by numerous calculations for gaussian electron wave packets with narrow momentum spreads (see also Section 3.6).

As to the other Larmor time , from Sections 4.2 and 3.4 it follows that the *Bttiker-Landauer time * [111β113], and the *imaginary part of the complex tunnelling time * [92], obtained within the Feynman approach, which are equal to (3.39), they become equal to that is, proportional to the barrier width , 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 where is the dynamical tunneling-time variance caused by the barrier influence only and defined by the equation with , (it was shown in [26, 27, 33β35]). Hence, they are not connected with the peak (or group) velocities of tunnelling particles but with the relevant tunneling *velocity distribution over the barrier region.
*

In Figure 4 the dependences of the values of from *a * are presented for electronic wave packets and rectangular barriers with the same parameters as in [47] (; mean electron energies with (curves 1a, 2a, and 3a, resp.); energy with and (curves 4a and 5a, resp.)). The curves corresponding to different energies and merge practically into one curve 6. Since the dependence of from *a * is very weak, the dependence of from is defined mainly by the dependence of from *a * (curves 1bβ5b, correspondent to 1aβ5a, resp.).

All these calculations manifest the negative values of . Such βacausalβ 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 . Since all are negative, the values of are always positive and, moreover, larger than 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 to , then for small absorptions, when and , HFE does not vanish and remains practically valid. This was confirmed experimentally for electromagnetic (microwave) tunnelling in [115].

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

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

The initial wave function turns out to be whose center transits from point at the instant , moves along axis from the left to the right with the velocity and in the absence of barrier crosses the point at the instant .

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

The tunnelling time is then defined as .

The flux density contains the wave function where is the value of the stationary wave function (3.1) at point , corresponding to the transmitted wave. Using the usual expression for the amplitude of the transmitted wave, we can represent in the form where , The normalization integral in (3.45), can be evaluated using (3.42), (3.46), and (3.47): If is chosen sufficiently far from the left of the barrier, the contribution of in the integral (3.49) from 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 and the integral (3.50) can be cast in the form where Now we evaluate the integral

Using (3.50) and (3.51), we can write which can be set in the form where , . After differentiating (3.47) over , one can see the following: From (3.51) and (3.56) it follows that, with an appropriate choice of , the mean instant can be defined by the simple relation As regards the calculation of , the wave function entering the flux density is where Since for calculating we consider only the positive values of , the integrals and in (3.45) can be obtained only numerically.

Figure 5 shows the results of the calculations of the tunnelling time as a function of the width *a* of the barrier, for electrons with energy eV through the rectangular barrier of potential with height eV and . We can see the manifestation of the HFE for (curve (a)), with the asymptotic behavior of the tunneling time approaching the constant value with increasing . 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 and . For the tunneling time is even negative. The violation of the HFE is strongly evident.

This effect can be explained with the following reasons [117]: the time spread of the Gaussian wave packet (3.43) is described by the relation where , . The value of strongly increases in time for large velocity spread . So, the center of the initial wave packet during wave-packet approaching the barrier and increasing time will appear even farther due to the essentially stronger increase of with time (proportional to for large ) than the decrease of proportional to . For the center of the transmitted wave packet in point such delay has to be considerably smaller because the value is smaller than the limited value ( being the phase tunnelling time) due to the advance caused by the difference between and being (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 becoming even negative.

Figure 2 shows qualitatively the behavior of the time shift of the outgoing wave packet with respect to the ingoing wave packet with different momentum spread represent the time instants when the wave packet peak passes through the initial point of the barrier. , and represent the transit time through the final point of the barrier.

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

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

Some authors (see, e.g., [118β121]) 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 [122β126]). Its origin is now usually described as the electropositron pair production for large potential step but it is not possible to develop a simple relationship between the *time-dependent* pair production process with a finite lifetime and the *time-independent* transmission coefficient in general. The problem of the Klein paradox and the Klein tunneling has to be studied in a self-consistent way and reviewed separately in another paper.

##### 3.7. Tunneling through a Double Barrier

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

*Phase Time of Nonresonant Tunneling through Two Opaque Barriers*

Now let us consider the stationary solution for 1D tunneling of a particle with mass and kinetic energy , through two equal rectangular barriers with height and width , the quantity being the distance between them see Figure 7. The stationary Schroedinger equation is where outside the barriers and inside the potential barriers. In various regions and , the solutions of (3.61) are the following: where , and quantities , and are the reflection amplitudes, the transmission amplitudes, and the coefficients of the βevanescentβ (decreasing) and βantievanescentβ (increasing) waves for barriers 1 and 2, respectively. These 8 quantities can be easily obtained from 8 matching (continuity) conditions for the functions and their derivatives at points . The obtained expressions for them for opaque barriers, when are where

From (3.63) one can derive that the phase tunneling time is which is precisely the same as for one barrier and *does not depend * not only on the width *a* of the opaque barrier, but also on the distance *l * between two opaque barriers. This result is a striking generalization of the HFE, firstly obtained in [127]. It is important to stress that this result holds, however, for nonresonant tunneling, that is, for energies far from the resonances.

Esposito in [128] has generalized this result for the tunneling through an arbitrary number of finite rectangular opaque barriers and it has been shown that the total tunneling phase time depends neither on the barrier thickness nor on the interbarrier separation. It has been also shown the independence of the phase transit time (for nonresonant tunneling).

Now, we will consider the cases of the resonances (between two barriers) and the influence of not very far resonances.

*The Resonant Tunneling*

If one takes two arbitrary (not necessarily opaque) barriers (cf. Figure 7), the amplitude of the transmitted wave in this case is defined by the formula [88, 129] where , and . The dimensionless constants and are connected by the relation

A resonance is characterized by the condition [88, 120] that is, the double barrier becomes totally transparent (without any reflections).

It is easy to see (cf. (3.67), and also [88, 129]) that the values of the wave number for which condition (3.69) is satisfied can be found from the equation We can show that from (3.70) it is possible to find out also the values of parameters , , and , at the resonance. Indeed, from (3.67) and (3.70) it follows that at a resonance it is On introducing the functions and using (3.68), one can infer that these functions are connected by the relation Then, by using functions (3.72), the denominator of (3.67) can be written in the following form: It follows from (3.74) and (3.73) that and condition (3.71) gets transformed into Taking (3.68) into account, we can write the term of (3.76) in the following form: and this (last) equation can be easily transformed, afterwards, into (3.70). There are no other solutions to (3.76). Hence, (3.70), as well as (3.69) or (3.71) is a general resonance condition. In the region of a resonance, if we limit ourselves to the first two terms of the expansion of (3.67) into a series of powers of (quantity being the resonance value of energy ), we obtain where , , and the index *prime* defines the derivative with respect to . We can rewrite (3.78) in the following form: It follows from (3.70) that at the resonance it is where , and are the values at of the functions