Marginal picture of quantum dynamics related to intrinsic arrival times

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1 PHYSICAL REVIEW A 76, Marginal icture of uantum dynamics related to intrinsic arrival times Gabino Torres-Vega* Physics Deartment, Cinvestav, Aartado Postal 14-74, 7 México City, Distrito Federal, Mexico Received 29 November 26; ublished 5 Setember 27 We introduce a marginal icture of the evolution of uantum systems, in which the reresentation vectors are the uantities that evolve and oerators and wave ackets remain static. The reresentation vectors can be seen as robe functions that are the evolution of a function with initial suort on =X in coordinate sace. This icture of the dynamics is suited for the determination of intrinsic arrival distributions for uantum systems, roviding a clear hysical meaning to the time eigenstates used in these calculations. We also analyze Galaon et al. s confined time eigenstates Phys. Rev. Lett. 93, from this oint of view, and roose alternative robe functions for confined systems without the need of a uantized time. DOI: 1.113/PhysRevA PACS numbers: 3.65.Db I. INTRODUCTION Pauli 1 3 ointed out long ago that there is no time oerator canonical conjugate to a semibounded Hamiltonian, and Allcock 4 6 argued against the recise uantummechanical descrition of the time of arrival concet. However, there are several roosals for the calculation of intrinsic or oerational arrival-time distributions, and also there are roosals for time oerators 7 29 and conjugate airs 3, and there are even analyses and criticisms of Pauli s assertion 29,31. The existing literature on this subject is vast, and a few references are at the end of this aer, including a broad review by Muga and Leavens 32. A well-known time eigenstate for free motion is Kijowski s state, which, in momentum reresentation, is given by 24 T, = e i2 T/2m m, 1.1 where is the Heaviside ste function, and = 1 for right and left movers, resectively. The suared magnitude of this state has as classical counterart the flux that arrives at =, at time T. As was noticed by Baute et al. 12, Kijowski s time eigenstates for free motion 19 can be seen as the backward evolution of the initial state /m, which corresonds to an initial robability density /m. The arrival-time distribution is the suared modulus of the inner roduct between this state and a given initial wave acket, for which one is interested in determining the arrival time distribution, T, = T, Baute et al. have also generalized this to arbitrary otentials and ositions 12. Leavens 33 has argued that the use of the factor can lead to some aradoxical effects, but his arguments were refuted in 34. *gabino@fis.cinvestav.mx Oerator-normalized construction of a time oerator measuring the occurrence time of an effect gives similar results as using Kijowski s states or Positive Oerator Valued POV measures but with a normalization at the level of oerators 1,11. Then, we note that Kijowski s tye of arrival-time eigenstates aear, uite freuently, indeendently of the rocedure used to obtain an arrival-time distribution, intrinsic or oerational 14,2,38. Therefore, we would like to have a clear hysical meaning of these time eigenstates. In this aer, we reinterret and further develo the above ideas with the introduction of an aroach for the analysis of the dynamics that is useful, for instance, for the determination of arrival distributions for uantum nonrelativistic systems. We do not use generalizations of standard uantum mechanics, but instead we make use of robe functions that are roagated backward in time with the evolution oerator, as we do with any other state. These robe functions samle an initial robability density and then redict arrival distributions. They are well-localized functions in coordinate or momentum sace and are similar to resence and Kijowski s time eigenstates. The classical analysis, found in Ref. 39, clarifies what our uantum robe function, and Kijowski s and others, do: they ick u the art of the initial robability density that will arrive at =X at the arrival time T, and also shows that some of the difficulties found in uantum systems are also found in the classical case. The aroach used in this work does not need modification of the Hamiltonian 19,24, nor generalizations of uantum mechanics 35,36, nor uantization rules and symmetrization of multivalued nonlinear functions 4, nor the finding of conjugates to the Hamiltonian 19, nor the uantization of time 29, but, instead, it makes use only of the backward evolution of the basis vectors of the coordinate or momentum reresentations. This is a different icture of uantum mechanics in which the reresentation vectors evolve in time and robability densities and oerators do not. The robe functions are solutions of the evolution euation and then they are suited for answering time-related uestions. We will consider nonrelativistic, one-dimensional systems. We will also make use of the Husimi transform in order to make a comarison between classical and /27/763/ The American Physical Society

2 GABINO TORRES-VEGA uantum uantities and then elucidate their hysical meaning. In Sec. II, we use the ideas develoed in Ref. 39 for classical densities in the uantum case and we introduce a reresentation icture of the dynamics. We study its roerties and relate it to the calculation of arrival uantities. In Sec. III, we analyze the confined arrival-time distribution of Galaon et al ,29. We find that Galaon et al. s states could be undersamling some of the uantum states. Then we roose alternative robe function for these systems. At the end there are some concluding remarks. II. QUANTUM PROBE FUNCTIONS We develo here a icture of uantum dynamics suited for arrival-time-related uestions. We recognize, analyze, and use a reresentation icture of the dynamics of uantum systems in which the reresentation vectors are the ones that evolve, and state vectors and oerators are static. Later, this icture is alied to the determination of some arrival distributions. A function that is concentrated at =X and that covers, eually weighted, the momentum values is e ix/,inthe momentum reresentation. Then a robe function for arrival at =X at time T is Q ;T;X = e itĥ,i// e ix/ = e itĥ/ X 2.1 in momentum sace. In the above definition, X is an eigenfunction of the coordinate oerator Qˆ for the articular oint =X. This function is the backward roagation of X, and is aroriate for determining resence distributions. For arrival-time distributions we need the factor /m, but this factor can be included as will be shown below. Earlier, researchers considered only free roagation or, in an indirect way, roagation in other otentials 13. Recently, Baute et al. also introduced the backward roagator e itĥ/ for arbitrary otentials 12. Actually, this tye of robe state has been in use since the early days of uantum mechanics, and constitutes another icture of uantum dynamics which has not been recognized as such before. Recall that the relationshi between coordinate and momentum reresentations of a wave acket is ;t d e = i/ ;t de = itĥ/ t = = Q t;t =. 2.2 Then, the coordinate reresentation of uantum mechanics can be seen as the samling of the initial wave function t= for arrival at at time t. Contrary to the Schrödinger or Heisenberg or interaction icture of uantum mechanics, here the basis vectors are the time-deendent uantities, and wave ackets and oerators are static. This is a comlementary icture of uantum mechanics. The only difference from the arrival-time robe states that we use is that we samle for only a secific oint =X. This allows us to assign a time value to arts of the state vector by finding the overla between the robe function and the state vector, with the origin of time assigned to the initial robe function. The same is true for the momentum reresentation ;t = P t;t =, 2.3 where P t; =e ith i/,/ e i/ is the robe function for arrival with momentum at time t. In what follows we searate into ositive and negative momentum arts right and left movers and derive some roerties of the robe functions. In uantum mechanics, the distinction between right and left movers is necessarily an aroximated concet, because the reuirement = X is not exactly comatible with the reuirement or, and we have to take some of the results with caution 44. The robe functions that we will use are O T;X = d e itĥ/ ÔPˆ,Qˆ X e itĥ/ Ô Pˆ,Qˆ X, O T;X = e itĥ/ ÔPˆ,Qˆ X = O T;X + O + T;X, where ÔPˆ,Qˆ is an observable of interest and Ô Pˆ,Qˆ d ÔPˆ,Qˆ. We will also use the notation Î d. We should kee in mind that for left movers the oint = is excluded. In what follows, we will consider only the vectors for arrival at =X, O T;X, and we can relace X by P in order to get formulas for the other robe functions, for arrival with momentum P. When ÔPˆ,Qˆ =1, we will omit the subscrit O from the robe function. Since the osition basis vectors are orthogonal, the robe states are also orthogonal with resect to X, with the same T, and PHYSICAL REVIEW A 76, X X = XX = XÎ + e itĥ/ e itĥ/ Î + X + XÎ e itĥ/ e itĥ/ Î X = T;X T;X + + T;X + T;X 2.7 =T;XT;X, T;X + T;X = These functions behave like a clock variable since if we aly the evolution oerator for a time t to the robe states, we obtain again the robe states but for a shorter arrival time, e itĥ/ O T;X = O T t;x, e itĥ/ O T;X = O T t;x

3 MARGINAL PICTURE OF QUANTUM DYNAMICS RELATED The inner roducts between robe states for different T and the same X, calculated in the energy reresentation, are PHYSICAL REVIEW A 76, dtt;xt;x and T;XT;X = ;XT T;X = dee SĤ it TE/ XE, 2 =Xe it TĤ/ X =2 dee,e,xxe,e,. SĤ 2.2 However, these states indeed are comlete if the summation is carried out over X, dx T;X T;X = e itĥ/ Î e itĥ/, 2.21 T;X + T;X = XÎ e it TĤ/ Î + X, 2.14 where and SĤ de indicates summation over the discrete and continuous arts of the sectrum, and takes into account the ossible degeneracy of the energy eigenvalues twofold for free motion. Then the robe states, for different T and same X, are not orthogonal, unless XE, is constant with the same value for all values of E and. In terms of energy eigenstates, we find that O T;X =SĤ dee ite/ Î E Ô Pˆ,Qˆ X, 2.15 O T;X =SĤ dee ite/ Î E ÔPˆ,Qˆ X, 2.16 where Î E E,E,. Then, the energy comonents of the time robe functions are O E;X dt e iet/ 2 O T;X = Î E Ô Pˆ,Qˆ X, 2.17 O E;X = O E;X + + O E;X = Î E ÔPˆ,Qˆ X A difference from the states of Refs. 19 and 15 is the * factor of the tye E, =X in the integrand, a factor which indicates that this is a robe state for arrival at =X. With these factors, we are considering only a narrow region in, otherwise we would be taking into account the whole real line. These exressions can be useful for aroximating the robe states in secific alications. The resence robe states do not form a comlete set when summed over T, unless XE, and E,Î X have the same constant values for all X, E, and, dt T;X T;X =2 dee,e,î SĤ XXÎ E,E,, 2.19 and dxt;xt;x = dx T;X T;X + + T;X + T;X = Î There are two ways of calculating an average with the robe functions when the system is evolving. The first otion is written in a symmetrical form and involves a single tye of robe function which deends on ÔPˆ,Qˆ, TÔPˆ,Qˆ T dx O T;X = A second otion is asymmetric in the robe functions, TÔPˆ,Qˆ T dx O T;XT;X = 2.24 = dxt;x O T;X Arrival densities for ÔPˆ,Qˆ, when the uantum system is in a state, are given by O T;X = O T;X = T;X * O T;X, O T;X = O T;X 2 =T;X * O T;X If the state vector is normalized, the above definitions are the densities of the uantity ÔPˆ,Qˆ resent at the arrival time. If ÔPˆ,Qˆ is the velocity, the integral over a time interval is the average number of arrivals in a given direction in this time interval. In some cases, as for free-article

4 GABINO TORRES-VEGA PHYSICAL REVIEW A 76, ν (; T ; X) 2 ν(; T ; X) 2 (a) (b) motion, the above densities can be normalized, and then become a robability density. Thus, it is ossible to recover a distribution like Kijowski s arrival-time distribution from 2.26 when ÔPˆ,Qˆ =Pˆ /m. The densities for arrival of ÔPˆ,Qˆ in the interval Z, when the system is in the state, are given by T,O with Z a Borel set for,. Z;X T,O T;XdT, =Z Some alications 2.3 We now will consider some examles of the use of the reresentation icture introduced above. In Fig. 1 there is a lot of the suared magnitude of the resence robe functions ;T;X and of their sum for arrival at X = 1, for free roagation and in coordinate sace. For instance, the robe function for right movers has suort mainly on X, excet for T=, when the function is like a Dirac function, with a 1/ term added to it, at =X and constant for. Let us use the Husimi transform of wave ackets in order to get a hase-sace reresentation of the free-evolution resence robe functions and thus have a better understanding of what these robe functions are. The Husimi transform is the rojection of the given uantum state onto the coherent state set Here we use the rojection in dimensionless units de, 2 /2 2 i / This function now deends on =, and can be thought of as a hase-sace function. The suared magnitudes of the Husimi transforms of the uantum resence robe functions, for free evolution, are shown in Fig. 2. They rovide a neat (c) ν + (; T ; X) 2 FIG. 1. Suared magnitude of the unnormalized resence robe functions for a free article in coordinate sace. Plots for a all movers, b left movers, and c right movers. For this calculation we have set m==1, X= 1, and T=1. Dimensionless units (a) (b) classical-like icture of the uantum functions, allowing us to make a comarison with the classical robe functions. These are densities with a small suort around the suort of the classical robe functions of Ref. 39, i.e., around the lines =X T/m. From the Husimi transform, we can see why these states are not orthogonal for the same X and different T. Similarly to what haens with classical densities, and contrary to what was exected earlier 4,45,46, the robe functions are not orthogonal over T because they actually overla at =, and at other oints when there is recurrence in the dynamics. For instance, a article with zero momentum and located at =X does not have a well-defined time of arrival because the article will be at =X for all time, a fact which makes this oint belong to the robe state for all time. But if we consider robe functions with different X and for the same T, they will not overla; they will scan the hase sace as we vary the value of X. In Fig. 3, there is a snashot of the robe function ν + (; T ; X) 2 FIG. 2. Density lots of the suared magnitude of the Husimi transforms of the resence robe functions of Fig. 1. Plots for a all movers, b left movers, and c right movers. Darker regions indicate that the density there is larger than in lighter regions. Dimensionless units FIG. 3. Suared magnitude of the unnormalized resence robe function + T;X for a otential barrier of height 12.5 between = 5 and = 4. In this figure, T=2 and X=, in dimensionless units. (c)

5 MARGINAL PICTURE OF QUANTUM DYNAMICS RELATED PHYSICAL REVIEW A 76, ν + (T ; X) ψ FIG. 4. Plot of the unnormalized uantum, free-evolution, resence distribution for the state E. 2.32, with X=5, =1, and K =1. Dimensionless units. T φ + 3 () 2 φ + 1 () 2 φ + 2 () 2 φ + 4 () 2 + T;X for a article exeriencing a otential barrier of height 12.5 from = 5 to = 4. For that calculation, due to the finiteness of numerical calculations, we have limited the momentum of the robe function to the range,6. This system can be used for discussing arrival-time distributions for tunneling through a barrier There is a art which is being reflected by the barrier and which comes from the negative momentum region. This function will samle the art of a wave acket that will arrive at = at time T=2. In their discussion of uantum backflow, Bracken and Melloy 51 have used the wave acket = 18 35K e /K e /2K /6,,,2.32 where = +, with a boost, and K is a ositive constant with dimensions of momentum. We have added a boost in order to not have a significant amount of robability with momentum around zero. With this state, Bracken and Melloy have shown that a wave acket which is made u of only ositive momentum comonents can have, when it roagates freely, a negative flux for some time. They even gave an uer bound of 4 for the magnitude of the backflow for any state in free motion. For comarison with the classical uantity calculated in Ref. 39, in Fig. 4 there is a lot of the resence distribution for free evolution, for the initial wave acket of E III. CONFINED ARRIVAL-TIME STATES In order to circumvent the non-self-adjointness of arrivaltime oerators, Galaon et al ,29 have introduced arrival-time eigenfunctions, ll, for confined systems which satisfy the boundary condition l=e 2i l,. For instance, for the symmetric case =, the even time eigenfunctions are given by s = B s e ir s 2 J 3/4,1/4 2 r s + 4B se ir s 4r s 1/4 J 1/4r s, 3.1 where B s is the normalization constant, J, x =x J xij x, J x is the Bessel function of the first kind, and r s are the roots of the euation J 3/4 x +2/3J 5/4 x+j 1/4 x/x=, with s=1,2,... The corresonding eigenvalues are n =l 2 /4r s, with the article s mass. The first four confined arrival-time eigenfunctions in momentum sace, for l=, can be seen in Fig. 5. They include left and right movers, they mainly samle at the edges of the system, at =l, and they also give more weight to large momentum regions. However, these functions do not samle all the range of momentum values in the same way; there are regions which are oorly samled and regions which are oversamled. Each of the time eigenstates involves many momentum eigenvalues, and the time eigenvalues do not corresond to the times associated with the momentum eigenvalues, l 2 /k. The common eigenfunctions of momentum and energy oerators, in coordinate reresentation, are k = 1 2l e i+k/l, 3.2 with eigenvalues k, =+k/l, E k = 2 k, /2, k=, 1,2,... Then, according to E. 2.15, we can get robe functions in coordinate sace, for determining the arrival-time distribution, by summing u the energy eigenfunctions 3.2 times /: where P/ ;T;X = P/ ;T;X + P/ ;T;X, 3.3 P/ FIG. 5. Suared magnitude of the unnormalized first four Galaon et al. time eigenfunctions for the symmetric case = and l=l=, in momentum sace. Dimensionless units. ;T;X = k /,k / + k 2l e it + k2 /2l 2 e i+k X/l. 3.4 For right left movers, the summation is carried out with k / k /. These functions will samle exactly the momentum values involved in the dynamics of the system

6 GABINO TORRES-VEGA ν ˆP /µ (;;) 2 ν ˆP /µ (;;) (a) (c) and do not need a discretization of time because they start a backward evolution aligned at =X. For numerical calculations, an aroximation to the above robe functions is to truncate the infinite sum. We have found that with 3 terms we obtain a function with a thin enough suort in. In Fig. 6 we show lots of the aroximated initial robe states for right and left movers. If the articular situation under study involves only the first 3 or fewer values of momentum, this aroximated function has the roerties needed for a robe function. Increasing the number of terms in the summation will increase the accuracy of the calculation. Then the robe functions introduced in this aer are also useful when the sectrum is discrete without the need of a uantization of time. Remarks It is desirable to learn how to deal with the time variable in many hysical theories. In this aer, we have introduced a marginal icture of uantum dynamics which is different from the Schrödinger, Heisenberg, or interaction ictures, (b) (d) ν + ˆP /µ (;;) 2 ν + ˆP /µ (;;) 2 FIG. 6. Aroximated unnormalized arrival robe functions E. 3.4 for Galaon s et. al. confined article, with =T=X=, in coordinate and momentum reresentations. We have considered only 3 terms of the infinite sums. a Left and b right movers in coordinate sace. Momentum reresentation for c left and d right movers. Dimensionless units. PHYSICAL REVIEW A 76, and is intended to answer uestions regarding the arrival of dynamical uantities at some oint =X. In fact, this is a icture of uantum dynamics which has not been recognized as such before; a icture in which the reresentation vectors move and the rest, namely, the state vectors and oerators, remain static. We can make contact with other works. The robe functions O T;X, with ÔPˆ,Qˆ = Pˆ /m, were mentioned by Baute et al. in Ref. 12, the crossing states, as a generalization, for arbitrary otentials, of Kijowski s time eigenstates for free motion. For X=, X=1, and for free motion E, and coincide. In this case, with ÔPˆ,Qˆ =1, the robe states become the arrival-time eigenstates t, of Delgado and Muga 19. Our arrival robe functions differ from those of Skulimowski 15, ˆ St,,X = e ixp e iĥt dee,, SĤ and Delgado and Muga 19 in that the arrival oint =X is considered inside the summation over eigenfunctions of Ĥ with the extra factor XE *. This additional factor enables a sound hysical interretation of the robe functions as backward time fronts with a classical analog 39. Probe functions reresent the motion of a given system, taking a clock as a reference for how much the system has changed. The classical icture for the robe functions is that of the roagation of the line =X in hase sace. The backward roagation of that line, labeled by T, is taken as the suort for robe functions that can samle an initial robability density and redict, then, the amount of robability that will arrive at X. With the robe functions, it is ossible to find out what art of a uantum final wave acket corresonds to a given art of the initial one. This is not like classical trajectories for single oints in hase sace, but it is close to that; we can take a robe function concentrated around the line =X and find the amount of density that is coming from another region of at time T. In this and in 39, we have introduced ictures for classical and uantum systems which allow us to treat classical and uantum arrival in a similar way, but they are marginal states since the momentum has been summed u. In 44 we use this icture of evolution, without the summation over, for the descrition of classical and uantal evolution in an energy-time sace. 1 W. Pauli, Handbüch der Physik, edited by H. Geiger and K. Scheel, Sringer-Verlag, Berlin, 1926, Vol. 23, W. Pauli, Handbüch der Physik, edited by H. Geiger and K. Scheel, Sringer-Verlag, Berlin, 1933, Vol. 24, W. Pauli, Handbüch der Physik, edited by S. Fludge Sringer-Verlag, Berlin, 1958, Vol. 5, G. R. Allcock, Ann. Phys. 53, G. R. Allcock, Ann. Phys. 53, G. R. Allcock, Ann. Phys. 53, Y. Aharonov and D. Bohm, Phys. Rev. 122, A. D. Baute, I. L. Egusuiza, J. G. Muga, and R. Sala-Mayato, Phys. Rev. A 61, Piotr Kochański and Krzysztof Wódkiewicz, Phys. Rev. A 6,

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