Born’s interpretation of the squared modulus of a wave function (|ψ|2) as the probability density of finding a particle within a specified region of space is a key ingredient of the standard framework of quantum mechanics, thereby implying that the standard interpretation of quantum mechanics is inherently epistemological. On the other hand, the possibility of an alternative interpretation of quantum mechanics by interpreting |ψ|2 as the probability density of a particle being actually present within a specified region was first suggested by de Broglie [1]. Later, Bohm [2-4] developed the details of such an ontological model of quantum mechanics by using the notion of an observer-independent spacetime trajectory of an individual particle T&F Cat # K11224, Chapter 8, Page 114, 15-9-2010 that is determined by its wave function through an equation of motion which is formulated in a way consistent with the Schrödinger time evolution. Bohm’s model, thus, explicitly refuted the counterarguments (such as those put forward by Pauli [5] and von Neumann [6]) that claimed to have ruled out the formulation of such a model. Subsequently, much work has been done on various aspects of the Bohmian model [7-10]. That such a model is not unique has been elaborately discussed [11,12] while different versions of the ontological model of quantum mechanics have been proposed [13-21].
In a recent paper Grot, Rovelli, and Tate (GRT) [Phys. Rev. A 54, 4676 (1996)] derived an expression for the probability distribution $\ensuremath{\pi}(T;X)$ of intrinsic arrival times
T(X) at position
x=X for a quantum particle with initial wave function $\ensuremath{\psi}(x,t=0)$ freely evolving in one dimension. This was done by quantizing the classical expression for the time of arrival of a free particle at
X, assuming a particular choice of operator ordering, and then regulating the resulting time of arrival operator. For the special case of a minimum-uncertainty-product wave packet at
t=0 with average wave number
〈k〉 and variance $\ensuremath{\Delta}k$ they showed that their analytical expression for $\ensuremath{\pi}(T;X)$ agreed with the probability current density
J(x=X,t=T) only to terms of order $\ensuremath{\Delta}k/〈k〉.$ They dismissed the probability current density as a viable candidate for the exact arrival time distribution on the grounds that it can sometimes be negative. This fact is not a problem within Bohmian mechanics where the arrival time distribution for a particle, either free or in the presence of a potential, is rigorously given by
∣J(X,T)∣ (suitably normalized) [W. R. McKinnon and C. R. Leavens, Phys. Rev. A 51, 2748 (1995); C. R. Leavens, Phys. Lett. A 178, 27 (1993); M. Daumer et al., in On Three Levels: The Mathematical Physics of Micro-, Meso-, and Macro-Approaches to Physics, edited by M. Fannes et al. (Plenum, New York, 1994); M. Daumer, in Bohmian Mechanics and Quantum Theory: An Appraisal, edited by J. T. Cushing et al. (Kluwer Academic, Dordrecht, 1996)]. The two theories are compared in this paper and a case presented for which the results could not differ more: According to GRT's theory, every particle in the ensemble reaches a point
x=X, where $\ensuremath{\psi}(x,t)$ and
J(x,t) are both zero for all
t, while no particle ever reaches
X according to the theory based on Bohmian mechanics. Some possible implications are discussed.
Abstract How to compute the probability distribution of a detection time, i.e., of the time which a detector registers as the arrival time of a quantum particle, is a long-debated problem. In this regard, Bohmian mechanics provides in a straightforward way the distribution of the time at which the particle actually does arrive at a given surface in 3-space in the absence of detectors. However, as we discuss here, since the presence of detectors can change the evolution of the wave function and thus the particle trajectories, it cannot be taken for granted that the arrival time of the Bohmian trajectories in the absence of detectors agrees with the one in the presence of detectors, and even less with the detection time. In particular, we explain why certain distributions that Das and Dürr (Sci. Rep. 9: 2242, 2019) presented as the distribution of the detection time in a case with spin, based on assuming that all three times mentioned coincide, are actually not what Bohmian mechanics predicts.
The de Broglie–Bohm interpretation of quantum mechanics with and without spin-dependence is used to determine electron trajectories, arrival-time distribution of electrons and the mean arrival time of spin-1/2 particles (including electrons) in the presence of a uniform field and a barrier separately. The difference for the mean arrival times, which are calculated with different guidance equations, is examined versus mass of the arriving particle and versus group velocity of the arriving electron wave packet and also versus the width of the barrier. Numerical calculations show that these differences are of the order of 10 −18 –10 −17 s. Another feature of the modified guidance law is that Bohmian trajectories cross each other, but this does not contradict the single-valuedness of the wavefunction.
The arrival time statistics of spin-1/2 particles governed by Pauli's equation, and defined by their Bohmian trajectories, show unexpected and very well articulated features. Comparison with other proposed statistics of arrival times that arise from either the usual (convective) quantum flux or from semiclassical considerations suggest testing the notable deviations in an arrival time experiment, thereby probing the predictive power of Bohmian trajectories. The suggested experiment, including the preparation of the wave functions, could be done with present-day experimental technology.
The computation of detection probabilities and arrival time distributions within Bohmian mechanics in general needs the explicit knowledge of a relevant sample of trajectories. Here it is shown how for one-dimensional systems and rigid inertial detectors these quantities can be computed without calculating any trajectories. An expression in terms of the wavefunction Ψ and its spatial derivative ∂ x Ψ, both restricted to the boundary of the detector's spacetime volume, is derived for the general case, where the probability current at the detector's boundary may vary its sign.