In a previous paper [V. Delgado and J. G. Muga, Phys. Rev. A 56, 3425 (1997)] we introduced a self-adjoint operator $\hat {\cal T}(X)$ whose eigenstates can be used to define consistently a probability distribution of the time of arrival at a given spatial point. In the present work we show that the probability distribution previously proposed can be well understood on classical grounds in the sense that it is given by the expectation value of a certain positive definite operator $\hat J^{(+)}(X)$ which is nothing but a straightforward quantum version of the modulus of the classical current. For quantum states highly localized in momentum space about a certain momentum $p_0 \neq 0$, the expectation value of $\hat J^{(+)}(X)$ becomes indistinguishable from the quantum probability current. This fact may provide a justification for the common practice of using the latter quantity as a probability distribution of arrival times.
Relativistic free-motion time-of-arrival theory for massive spin-1/2 particles is systematically developed. Contrary to the nonrelativistic time-of-arrival operator studied thoroughly in previous literatures, the relativistic time-of-arrival operator possesses self-adjoint extensions because of the particle-antiparticle symmetry. The nonrelativistic limit of our theory is in agreement with the nonrelativistic time-of-arrival theory.
The concept and the formalization of the arrival time in quantum mechanics are discussed. Different approaches based on trajectories, quantization rules, time operators, phase space techniques, renewal equations or operational procedures are reviewed or proposed. Open questions and loose ends are pointed out.
A self-adjoint operator with dimensions of time is explicitly constructed, and it is shown that its complete and orthonormal set of eigenstates can be used to define consistently a probability distribution of the time of arrival at a spatial point.
The analysis of the model quantum clocks proposed by Aharonov et al. [Phys. Rev. A 57 (1998) 4130 - quant-ph/9709031] requires considering evanescent components, previously ignored. We also clarify the meaning of the operational time of arrival distribution which had been investigated.
The time of arrival at an arbitrary position in configuration space can be given as a function of the phase space variables for the Liouville integrable systems of classical mechanics, but only for them. We review the Jacobi-Lie transformation that explicitly implements this function of positions and momenta. We then discuss the recently developed quantum formalism for the time of arrival. We first analyze the case of free particles in one and three space dimensions. Then, we apply the quantum version of the Jacobi-Lie transformation to work out the time of arrival operator in the presence of …
We consider the arrival time distribution defined through the quantum probability current for a Gaussian wave packet representing free particles in quantum mechanics in order to explore the issue of the classical limit of arrival time. We formulate the classical analogue of the arrival time distribution for an ensemble of free particles represented by a phase space distribution function evolving under the classical Liouville's equation. The classical probability current so constructed matches with the quantum probability current in the limit of minimum uncertainty. Further, it is possible to s…
This paper compares the proposal made in previous papers for a quantum probability distribution of the time of arrival at a certain point with the corresponding proposal based on the probability current density. Quantitative differences between the two formulations are examined analytically and numerically with the aim of establishing conditions under which the proposals might be tested by experiment. It is found that quantum regime conditions produce the biggest differences between the formulations which are otherwise near indistinguishable. These results indicate that in order to discriminat…
The mean arrival time of free particles is computed using the quantum probability current. This is uniquely determined in the non-relativistic limit of Dirac equation, although the Schroedinger probability current has an inherent non-uniqueness. Since the Dirac probability current involves a spin-dependent term, an arrival time distribution based on the probability current shows an observable spin-dependent effect, even for free particles. This arises essentially from relativistic quantum dynamics, but persists even in the non-relativistic regime.
We reappraise and clarify the contradictory statements found in the literature concerning the time-of-arrival operator introduced by Aharonov and Bohm in Phys. Rev. {\bf 122}, 1649 (1961). We use Naimark's dilation theorem to reproduce the generalized decomposition of unity (or POVM) from any self-adjoint extension of the operator, emphasizing a natural one, which arises from the analogy with the momentum operator on the half-line. General time operators are set within a unifying perspective. It is shown that they are not in general related to the time of arrival, even though they may have the…
We propose a covariant algorithm for relativistic ideal measurements and for relativistic continuous measurements, its non-relativistic limit results the algorithm of the Event-Enhanced Quantum Theory. Therefore an additional intrinsic parameter, the proper time, is used. As an application we compute the time of arrival of a particle at a detector and find good agreement between the expected values of the time of arrival for weak detectors and the results of the relativistic point-mechanic over a wide range. For very high momentums there is a small probability for a negative time of arrival, s…
It is argued that the time-of-arrival cannot be precisely defined and measured in quantum mechanics. By constructing explicit toy models of a measurement, we show that for a free particle it cannot be measured more accurately then $Δt_A \sim 1/E_k$, where $E_k$ is the initial kinetic energy of the particle. With a better accuracy, particles reflect off the measuring device, and the resulting probability distribution becomes distorted. It is shown that a time-of-arrival operator cannot exist, and that approximate time-of-arrival operators do not correspond to the measurements considered here.
The arrival time problem for the free particle in one dimension may be formulated as the problem of determining a joint probability for the particle being found on opposite sides of the $x$-axis at two different times. We explore this problem using a two-time quasi-probability linear in the projection operators, a natural counterpart of the corresponding classical problem. We show that it can be measured either indirectly, by measuring its moments in different experiments, or directly, in a single experiment using a pair of sequential measurements in which the first measurement is weak (or mor…
Assorted questions: Time as a parameter in Quantum Mechanics. No-Go theorems for a time operator. Localization, time and causality. Causality violation. Localization again. Lesson 1: Evading the troubles: Im E finite. Lights and shadows of a time operator:"Table-top Spacetime" quantum mechanics. Biphotons at Berkeley. Time operator build-up. Good news - bad news. Lesson 2: We need the resonances to tell one.
We study the problem of computing the probability for the time-of-arrival of a quantum particle at a given spatial position. We consider a solution to this problem based on the spectral decomposition of the particle's (Heisenberg) state into the eigenstates of a suitable operator, which we denote as the ``time-of-arrival'' operator. We discuss the general properties of this operator. We construct the operator explicitly in the simple case of a free nonrelativistic particle, and compare the probabilities it yields with the ones estimated indirectly in terms of the flux of the Schrödinger curren…
For a quantum-mechanically spread-out particle we investigate a method for determining its arrival time at a specific location. The procedure is based on the emission of a first photon from a two-level system moving into a laser-illuminated region. The resulting temporal distribution is explicitly calculated for the one-dimensional case and compared with axiomatically proposed expressions. As a main result we show that by means of a deconvolution one obtains the well known quantum mechanical probability flux of the particle at the location as a limiting distribution.
It is well known that orthodox quantum mechanics does not make unambiguous predictions for the statistics in arrival time (or time-of-flight) experiments. Bohmian mechanics (or de Broglie-Bohm theory) offers a distinct conceptual advantage in this regard, owing to the well defined concepts of point particles and trajectories embedded in this theory. We revisit a recently proposed experiment [S. Das and D. Dürr, Sci. Rep. (2019)], the numerical analysis of which revealed a striking spin dependence in the (Bohmian) time-of-arrival distributions of a spin-1/2 particle. We present here a mathemati…
We introduced with coauthors some years ago a solution to the problem of time in quantum gravity which consists in formulating the quantum theory in terms of real clocks. It combines Page and Wootters' relational proposal with Rovelli's evolving constants of the motion. Time is associated with an operator and not a classical parameter. We show here that this construction provides a natural solution to the time of arrival problem in quantum mechanics and leads to a well defined time-energy uncertainty relation for the clocks.
The arrival time probability distribution is defined by analogy with the classical mechanics. The difficulty of requirement to have the values of non-commuting operators is circumvented using the concept of weak measurements. The proposed procedure is suitable to the free particles and to the particles subjected to an external potential, as well. It is shown that such an approach imposes an inherent limitation to the accuracy of the arrival time determination.
We prove an uncertainty relation for energy and arrival time, where the arrival of a particle at a detector is modeled by an absorbing term added to the Hamiltonian. In this well-known scheme the probability for the particle's arrival at the counter is identified with the loss of normalization for an initial wave packet. Under the sole assumption that the absorbing term vanishes on the initial wave function, we show that $ΔT ΔE \geq \sqrt p \hbar/2$ and $<T> ΔE\geq 1.37\sqrt p\hbar$, where $<T>e$ denotes the mean arrival time, and $p$ is the probability for the particle to be eventually absorb…
We consider the definition that might be given to the time at which a particle arrives at a given place, both in standard quantum theory and also in Bohmian mechanics. We discuss an ambiguity that arises in the standard theory in three, but not in one, spatial dimension.
We review recent progress in understanding the arrival time problem in quantum mechanics, from the point of view of the decoherent histories approach to quantum theory. We begin by discussing the arrival time problem, focussing in particular on the role of the probability current in the expected classical solution. After a brief introduction to decoherent histories we review the use of complex potentials in the construction of appropriate class operators. We then discuss the arrival time problem for a particle coupled to an environment, and review how the arrival time probability can be expres…
We extend previous work on the arrival time problem in quantum mechanics, in the framework of decoherent histories, to the case of a particle coupled to an environment. The usual arrival time probabilities are related to the probability current, so we explore the properties of the current for general open systems that can be written in terms of a master equation of Lindblad form. We specialise to the case of quantum Brownian motion, and show that after a time of order the localisation time the current becomes positive. We show that the arrival time probabilities can then be written in terms of…
We compute in a relativistic way the time-of-arrival and the traversal time through a region of a free particle with spin 1/2. We do this by applying the relativistic extension of the Event-Enhanced Quantum Theory which we have presented in a previous paper. We find a very good coincidence of the results of our formalism and the results obtained by using classical relativistic mechanics.
A number of approaches to the problem of defining arrival and dwell time probabilities in quantum theory make use of idealised models of clocks. An interesting question is the extent to which the probabilities obtained in this way are related to standard semiclassical results. In this paper we explore this question using a reasonably general clock model, solved using path integral methods. We find that in the weak coupling regime where the energy of the clock is much less than the energy of the particle it is measuring, the probability for the clock pointer can be expressed in terms of the pro…
An operational time of arrival is introduced using a realistic position and momentum measurement scheme. The phase space measurement involves the dynamics of a quantum particle probed by a measuring device. For such a measurement an operational positive operator valued measure in phase space is introduced and investigated. In such an operational formalism a quantum mechanical time operator is constructed and analyzed. A phase space time and energy uncertainty relation is derived.
Via the proper-time eigenstates (event states) instead of the proper-mass eigenstates (particle states), free-motion time-of-arrival theory for massive spin-1/2 particles is developed at the level of quantum field theory. The approach is based on a position-momentum dual formalism. Within the framework of field quantization, the total time-of-arrival is the sum of the single event-of-arrival contributions, and contains zero-point quantum fluctuations because the clocks under consideration follow the laws of quantum mechanics.
New prescription to treat position and time equally in quantum mechanics is presented. Using this prescription, we could successfully derive some interesting formulae such as time-of-arrival for a free particle and quantum tunneling formula. The physical interpretation will be discussed.
We study the construction of probability densities for time-of-arrival in quantum mechanics. Our treatment is based upon the facts that (i) time appears in quantum theory as an external parameter to the system, and (ii) propositions about the time-of-arrival appear naturally when one considers histories. The definition of time-of-arrival probabilities is straightforward in stochastic processes. The difficulties that arise in quantum theory are due to the fact that the time parameter of Schrödinger's equation does not naturally define a probability density at the continuum limit, but also becau…
Time of arrival in quantum mechanics is discussed in two versions: the classical axiomatic "time of arrival operator" introduced by J. Kijowski and the EEQT method. It is suggested that for free particles the two methods may lead to the same result. On the other hand the EEQT method can be easily geometrized within the framework of Galilei-Newton general relativistic quantum mechanics developed by M. Modugno and collaborators, and it can be applied to non-free evolutions. The way of geometrization of irreversible quantum dynamics based on dissipative Liouville equation is suggested.
It is shown that a class of exponentially decaying time-of-arrival probability distributions suggested by Włodarz, Marchewka and Schuss, and Jurman and Nikolić, as well as a semiclassical distribution implicit in time-of-flight momentum measurements, do not show the expected behavior for a Gaussian wave train. This casts doubts on the physical adequacy of these arrival-time proposals. In contrast, the quantum flux distribution (a special case of the Bohmian arrival-time distribution) displays the expected behavior.
A realization of the concept of "crossing state" invoked, but not implemented, by Wigner, allows to advance in two important aspects of the time of arrival in quantum mechanics: (i) For free motion, we find that the limitations described by Aharonov et al. in Phys. Rev. A 57, 4130 (1998) for the time-of-arrival uncertainty at low energies for certain mesurement models are in fact already present in the intrinsic time-of-arrival distribution of Kijowski; (ii) We have also found a covariant generalization of this distribution for arbitrary potentials and positions.
We raise the possibility of developing a theory of constructing quantum dynamical observables independent from quantization and deriving classical dynamical observables from pure quantum mechanical consideration. We do so by giving a detailed quantum mechanical derivation of the classical time of arrival at arbitrary arrival points for a particle in one dimension.
It has always been believed that no self-adjoint and canonical time of arrival operator can be constructed within the confines of standard quantum mechanics. In this Letter we demonstrate the otherwise. We do so by pointing out that there is no a priori reason in demanding that canonical pairs form a system of imprimitivities. We then proceed to show that a class of self-adjoint and canonical time of arrival (TOA) operators can be constructed for a spatially confined free particle. And then discuss the relatiobship between the non-self-adjointess of the TOA operator for the unconfined particle…
We construct a one-particle TOA operator $\mathcal{\hat{T}}$ canonically conjugate with the Hamiltonian describing a free, charged, spin-$0$, relativistic particle in one spatial dimension and show that it is maximally symmetric. We solve for its eigenfunctions and show that they form a complete and non-orthogonal set. Plotting the time evolution of their corresponding probability densities, it implies that the eigenfunctions become more localized at the origin at the time equal to their eigenvalues. That is, a particle being described by an eigenfunction of $\mathcal{\hat{T}}$ is in a state o…
We revisit the arguments underlying two well-known arrival-time distributions in quantum mechanics, viz., the Aharonov-Bohm and Kijowski (ABK) distribution, applicable for freely moving particles, and the quantum flux (QF) distribution. An inconsistency in the original axiomatic derivation of Kijowski's result is pointed out, along with an inescapable consequence of the "negative arrival times" inherent to this proposal (and generalizations thereof). The ABK free-particle restriction is lifted in a discussion of an explicit arrival-time setup featuring a charged particle moving in a constant m…
An experimentally realizable scheme is formulated which can test any postulated quantum mechanical approach for calculating the arrival time distribution. This is specifically illustrated by using the modulus of the probability current density for calculating the arrival time distribution of spin-1/2 neutral particles at the exit point of a spin-rotator(SR) which contains a constant magnetic field. Such a calculated time distribution is then used for evaluating the distribution of spin orientations along different directions for these particles emerging from the SR. Based on this, the result o…
We investigate a fully quantum mechanical spin model for the detection of a moving particle. This model, developed in earlier work, is based on a collection of spins at fixed locations and in a metastable state, with the particle locally enhancing the coupling of the spins to an environment of bosons. Appearance of bosons from particular spins signals the presence of the particle at the spin location, and the first boson indicates its arrival. The original model used discrete boson modes. Here we treat the continuum limit, under the assumption of the Markov property, and calculate the arrival-…
We propose that measurements of time-of-arrival correlations in multi-partite systems can sharply distinguish between different approaches to the time-of-arrival problem. To show this, we construct a Positive-Operator-Valued measure for two distinct time-of-arrival measurements in a bipartite system, and we prove that the resulting probabilities differ strongly from ones defined in terms of probability currents. We also prove that time-of-arrival correlations are entanglement witnesses, a result suggesting the use of temporal observables for quantum information processing tasks. Finally, we co…
We develop a new conception for the quantum mechanical arrival time distribution from the perspective of Bohmian mechanics. A detection probability for detectors sensitive to quite arbitrary spacetime domains is formulated. Basic positivity and monotonicity properties are established. We show that our detection probability improves and generalises earlier proposals by Leavens and McKinnon. The difference between the two notions is illustrated through application to a free wave packet.
We examine the measurability of the temporal ordering of two events, as well as event coincidences. In classical mechanics, a measurement of the order-of-arrival of two particles is shown to be equivalent to a measurement involving only one particle (in higher dimensions). In quantum mechanics, we find that diffraction effects introduce a minimum inaccuracy to which the temporal order-of-arrival can be determined unambiguously. The minimum inaccuracy of the measurement is given by dt=1/E where E is the total kinetic energy of the two particles. Similar restrictions apply to the case of coincid…
In the arrival time problem in quantum mechanics, a standard formula that frequently emerges as the probability for crossing the origin during a given time interval is the current integrated over that time interval. This is semiclassically correct but can be negative due to backflow. Here, we show that this formula naturally arises in a decoherent histories analysis of the arrival time problem. For a variety of initial states, we show that histories crossing during different time intervals are approximately decoherent. Probabilities may therefore be assigned and coincide with the standard form…
The classical limit problem of quantum mechanics is revisited on the basis of a scheme that enables a quantitative study of the way the quantum-classical agreement emerges while going through the intermediate mass range between the microscopic and the macroscopic domains. As a specific application of such a scheme, we investigate the classical limit of a quantum time distribution - an area of study that has remained largely unexplored. For this purpose, we focus on the arrival time distribution in order to examine the way the observable results pertaining to the quantum arrival time distributi…
The model of weak measurements is applied to various problems, related to the time problem in quantum mechanics. The review and generalization of the theoretical analysis of the time problem in quantum mechanics based on the concept of weak measurements are presented. A question of the time interval the system spends in the specified state, when the final state of the system is given, is raised. Using the concept of weak measurements the expression for such time is obtained. The results are applied to the tunneling problem. A procedure for the calculation of the asymptotic tunneling and reflec…
Arrival Time\textemdash Classical Parameter or Quantum Operator?
University of Qom（科姆大学）；Mana Research Institute for Social and Cultural Studies（玛娜社会文化研究研究所）；Farhangian University（法尔干甘大学）；University of Cape Town（开普敦大学）；The National Institute for Theoretical and Computational Sciences（国家理论与计算科学研究所）；University of Lethbridge（莱思布里奇大学）
We present a decomposition of the general quantum mechanical evolution operator, that corresponds to the path decomposition expansion, and interpret its constituents in terms of the quantum Zeno effect (QZE). This decomposition is applied to a finite dimensional example and to the case of a free particle in the real line, where the possibility of boundary conditions more general than those hitherto considered in the literature is shown. We reinterpret the assignment of consistent probabilities to different regions of spacetime in terms of the QZE. The comparison of the approach of consistent h…
Possible theoretical frameworks for measurement of (arrival) time in the nonrelativistic quantum mechanics are reviewed. It is argued that the ambiguity between indirect measurements by a suitably introduced time operator and direct measurements by a physical clock particle has a counterpart in the corresponding classical framework of measurement of the Newtonian time based on the Hamiltonian mechanics.
We introduce a formalism for the calculation of the time of arrival t at a space point for particles traveling through interacting media. We develop a general formulation that employs quantum canonical transformations from the free to the interacting cases to construct t in the context of the Positive Operator Valued Measures. We then compute the probability distribution in the times of arrival at a point for particles that have undergone reflection, transmission or tunneling off finite potential barriers. For narrow Gaussian initial wave packets we obtain multimodal time distributions of the …
We introduce a formalism for the calculation of the time of arrival t at a detector of particles traveling through interacting environments. We develop a general formulation that employs quantum canonical transformations from the free to the interacting cases to compute t. We interpret our results in terms of a Positive Operator Valued Measure. We then compute the probability distribution in times of arrival at a detector of those particles that, after their initial preparation, have undergone quantum tunneling or reflection due to the presence of potential barriers. We obtain the expected ret…
