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Proceedings of the Samahang Pisika ng Pilipinas

How the knowledge of the particle's path dictates the dynamics of the quantum time-of-arrival operator

Denny Lane B. Sombillo, Eric A. Galapon

Published Jun 7, 2017• 0 citations
OpenAlex

Abstract

The two-slit experiment can produce two distinct patterns on the screen based on our knowledge of the particle's path. A bright spot on the screen corresponds to a point where the particle appeared. In the theory of quantum time-of-arrival the appearance of particle is due to the time-or-arrival measurement. In this work, we demonstrate how the two distinct pattern on the screen be interpreted using the theory of quantum time-of-arrival. It turns out that our knowledge of the particle's path determine the set of time-of-arrival operator's eigenfunctions with distinct dynamics.

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The two-slit experiment can produce two distinct patterns on the screen based on our knowledge of the particle's path. A bright spot on the screen corresponds to a point where the particle appeared. In the theory of quantum time-of-arrival the appearance of particle is due to the time-or-arrival measurement. In this work, we demonstrate how the two distinct pattern on the screen be interpreted using the theory of quantum time-of-arrival. It turns out that our knowledge of the particle's path determine the set of time-of-arrival operator's eigenfunctions with distinct dynamics.
The unexpected connection is unravelled between the collapse of the wave function on the appearance of particle and the quantum time-of-arrival problem in one dimension. To do so, a theory of quantum first time of arrival is developed in the interacting case for arbitrary arrival point in one dimension based on a self-adjoint and canonical coarse graining of a time-of-arrival operator that derives the classical time-of-arrival observable. The appearance of particle in quantum mechanics is then considered in the light of this theory. It is found that the appearance of particle arises as a combination of the collapse of the initial wave function into one of the eigenfunctions of the time-of-arrival operator, followed by a unitary Schrödinger evolution of the eigenfunction.
In the standard Copenhagen interpretation of Quantum Mechanics, the norm of the Schrödinger wave function is interpreted as the probability density of finding the particle at a certain small volume in space at a given time. This means that particle positions are very fuzzy and cannot be known until a position measurement is made. In the famous two-slit experiment, particles are emitted from a known source position and then are detected at definite screen positions, but the path any particle took to get to that detected position on the screen, in particular, which slit the particle went through to get to the detector, cannot be known. Such travel path questions are even regarded as illegitimate, since between position measurements the particle is said to behave not as a particle but as a wave, which is spread out through all available space without any particular defined position and without any defined path. It is only at the source point and at the points of detection that the particle can be said to have a position in the standard Copenhagen interpretation of Quantum Mechanics. This paper presents an alternative interpretation, and experiments to verify or disprove this interpretation are also presented.
Abstract Despite the astonishing successes of quantum mechanics, due to some fundamental problems such as the measurement problem and quantum arrival time problem, the predictions of the theory are in some cases not quite clear and unique. Especially, there are various predictions for the joint spatiotemporal distribution of particle detection events on a screen, which are derived from different formulations and interpretations of the quantum theory. Although the differences are typically small, we show that these predictions can be experimentally distinguished by a proposed unconventional double-slit configuration, which is realizable using present-day single-atom interferometry. This experiment would enrich our understanding of the foundations of quantum mechanics.
3.1 Interference of a Single Photon The kinds of study originally conducted by Thomas Young - in particular, the celebrated "double slit" experiment that we briefly examined in Chapter 1 (see Fig. 1) - provide a capacity to reveal far more than the wave nature of light, important as that is. True enough, the observed characteristic fringe pattern, when monochromatic light passes through two closely spaced slits onto a screen, appears to be simply explicable on the basis of waves from each slit undergoing constructive and destructive interference at different positions on the screen. Yet, astonishingly, these patterns of interference are still exhibited when individual photons of a low-intensity input beam reach the screen one at a time, showing that it cannot in fact be a matter of different photons having to interfere with each other. Indeed, Dirac enunciated the general principle that photons can never directly interact with each other. The double slit and other such experiments indicate that each photon interacts in a delocalized, wave-like manner with both slits at the same time. Successive experiments build interference patterns that conform to an exact mathematical distribution formula, but the position at which an individual photon is detected in any particular instance cannot be predicted; this is a quintessential illustration of quantum uncertainty. Using the path integral approach of quantum theory, the phenomenon is more accurately represented by introducing a summation over all possible routes between the source and the screen on which the pattern appears, signifying an entire "lifetime" for each photon involved in the experiment. This relates to the fact that the actual trajectory is unknown without an observation, and thus we only obtain the probabilities of the photon striking the screen at different positions and times; all trajectories (potentialities) must be considered in a superposition. Each route is considered equally probable, and the variation in phase produces quantum interference. The differences in phase of each trajectory, at varying positions on the screen, generates the fringe pattern. An intriguing variation on this experiment involves "marking" the photons so that the actual route of each photon to the screen is known. Remarkably, in such a case, the fringe patterns do not appear because quantum interference cannot occur. A consequence of the observation of the photon route is an effect that, in other quantum mechanical connections, is known as a collapse of the wavefunction. However, it is possible for the fringe patterns to reappear when information on the route of the photons is lost; this is known as a quantum eraser.
This paper aims to understand how sequential, separate and independent impacts of particles on the screen of a double-slit experiment produce an interference pattern when both slits are open and how no such pattern is produced when detectors are placed at the slits. These strange phenomena can be explained by conjecturing the existence of a certain " thickness " of time. This conjecture is falsifiable, fits the observed results and maximizes simplicity for explaining other quantum phenomena without contradicting the established principles and laws of quantum mechanics.