Observability of the arrival time distribution using spin-rotator as a quantum clock

Abstract

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 of spin measurement along any arbitrary direction for such an ensemble is predicted. Introduction.-Of late, the question of calculating the arrival or transit time distribution in quantum mechanics has been a topic of much interest. For comprehensive reviews see, for example, Muga and Leavens [1], and Muga et al. [2]. A number of schemes [3,4,5] have been suggested in the literature for calculating the arrival time distribution such as those based on axiomatic approaches, trajectory models of quantum mechanics, attempts to define and calculate the arrival time distribution using the consistent histories approach, and attempts of constructing the time of arrival operator, etc. Thus there is an inherent nonuniqueness within the formalism of quantum mechanics for calculating time distributions such as the arrival time distribution.

Quantum interference in the time-of-flight distribution

“…In one dimension, the current density J(x, t) tells us the rate at which probability is flowing past the point x. So, interpreting the one dimensional continuity equation in terms of the flow of physical probability, the Born interpretation for the squared modulus of the wave function and its time derivative suggest that the arrival time distribution of the particles reaching a detector located at x = X can be calculated [17,18,19,20,21,22,23,24,25] using the probability current density J(x, t). It should also be noted that J(x, t) can be negative, hence one needs to take the modulus sign in order to use the above definition.…”

“…We ignore this small spin-dependent contribution here in our present discussion, as the estimated magnitude of the spin-dependent current is roughly 10 5 to 10 6 times smaller than the Schrödinger current. It was emphasized that the probability current density approach not only provides an unambiguous definition of arrival time at the quantum mechanical level [17,18,19,20,21,22,23,24,25], but also adresses the issue of obtaining the proper classical limit of the TOF of massive quantum particles [24,25]. Now, to keep the discussion concise, here we restrict ourselves to the case of onedimensional motion, but our resulting conclusion does not depend on three-dimensional extension which will be straightforward as discussed later at the end of this Section.…”

“…So, the need for a quantum analysis of TOF distribution is not merely a conceptual but a practical issue, asking how to predict the TOF distribution using only classical and semiclassical ingredients in a purely quantum scenario like this (interference in the TOF distribution for quantum particles). Now, in spite of the emphasis of quantum theory on the observable concept, there is no commonly accepted recipe to incorporate time observables and their probability distributions in the quantum formalism, and there is considerable difficulty and debate over the issue of defining time (for example, tunneling time, decay time, arrival time) as an observable [12,13,14,15,16,17,18,19,20,21,22,23,24,25]. Even for the simplest case of arrival time problem there is no unique way to calculate the probability distribution in the quantum formalism [17].…”

“…Despite this, many researchers have evidently not been discouraged from seeking an expression for the arrival time distribution (or the quantum TOF distribution) within a consistent theoretical framework. Several logically consistent schemes for the treatment of the arrival time distribution have been formulated, such as those based on axiomatic approaches [14], operator constructions [15], measurement based approaches [16,19], trajectory models [21] and probability current density approach [17,18,19,20,21,22,23,24,25]. We will use here probability current density approach to calculate the quantum TOF distribution which is logically consistent and also physically motivating.…”

“…Second, we have just mentioned that there is an inherent nonuniqueness within the formalism of quantum mechanics for calculating the TOF or arrival time distribution. It remains an open question as to what extent these different quantum mechanical approaches [12,13,14,15,16,17,18,19,20,21,22,23,24,25] for calculating the time distributions can be tested or empirically discriminated. Our proposal of measuring matter-wave interference in TOF distribution has the potential to empirically resolve ambiguities inherent in the theoretical formulations of the quantum TOF distribution.…”