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arXiv:0907.2878

Quantum measurement theory for particle oscillations

Charis Anastopoulos, Ntina Savvidou

Published Jul 16, 2009• 0 citations
arXivHTMLPDF

Abstract

A fundamental principle of quantum theory, clearly manifested in the two-slit experiment, is that for any alternatives that cannot be distinguished by measurement physical predictions are obtained by summation of their amplitudes. In particle oscillation experiments, a particle's time of detection is not directly measured, consequently, the detection probability should involve the summation over amplitudes corresponding to different detection times. However, in contrast to the principle above, standard treatments involve summation over probabilities rather than amplitudes; this implicitly assumes the existence of a decohering mechanism. In this work, we construct the detection probabilities for particle oscillations by summation over amplitudes, corresponding to different detection times. The resulting wavelength of particle oscillations differs from the standard expression by a factor of two. Moreover, we predict a dependence of the oscillation wavelength on the threshold of the decay process used for detection.

DOI
10.48550/arxiv.0907.2878

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A fundamental principle of quantum theory, clearly manifested in the two-slit experiment, is that for any alternatives that cannot be distinguished by measurement physical predictions are obtained by summation of their amplitudes. In particle oscillation experiments, a particle's time of detection is not directly measured, consequently, the detection probability should involve the summation over amplitudes corresponding to different detection times. However, in contrast to the principle above, standard treatments involve summation over probabilities rather than amplitudes; this implicitly assumes the existence of a decohering mechanism. In this work, we construct the detection probabilities for particle oscillations by summation over amplitudes, corresponding to different detection times. The resulting wavelength of particle oscillations differs from the standard expression by a factor of two. Moreover, we predict a dependence of the oscillation wavelength on the threshold of the decay process used for detection.
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Quantum probability is very different from classical probability. Part of this difference is manifested in the generic inability of stochastic processes to describe the results of multi-time measurements of quantum mechanical systems and the fact that the complex-valued temporal correlation functions of quantum theory have no interpretation in terms of multi-time measurements. By analysing experiments involving measurements at more than one moments of time, we conclude that this inequivalence must be manifested either as a failure of the quantum logic or as the inability to define probabilities in multi-time measurements because the relative frequencies do not converge. These alternatives can be empirically distinguished as they correspond to different behaviours of the statistical data in multi-time measurements.
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.
We present a numerical implementation of the proposed Source–Detector Resonance (SDR) as a conceptual analog of a Double-Slit Interference Experiment with discrete particles. Two periodic streams of particles are emitted from two point sources at random integer multiples of a fundamental period P and corresponding frequency ω=2π/P and fly out towards a detection screen. The screen consists of a deep set of identical oscillators with eigenfrequency ω0=2π/P0. In the SDR scenario, ω≈ω0. When the particles reach the screen, they implement a periodic forcing of its oscillators at the stream’s fundamental frequency ω0. As a result, an oscillating pattern develops along the screen. The amplitude of oscillation of each oscillator saturates at a value that is determined by the balance between the periodic particle forcing and the damping of each oscillator. This is clearly proportional to the number of particles that reach a certain oscillator per unit time times the fraction of particles that reach it at its resonant frequency. The latter fraction is equal to the ratio of the Power Spectral Density (PSD) of the time series of the particles that reach the oscillator at its resonance frequency PSD(ω0) over the PSD at zero frequency PSD(0). If we further assume that each oscillator absorbs a particle and announces a detection with a probability that is proportional to the square of the ratio PSD(ω0)/PSD(0); the pattern of particle detections that develops over a thick layer of oscillators is shown to be the same as that of a Double-Slit Interference Experiment. Our result shows that when macroscopic resonant detectors interact with and detect periodic streams of discrete particles, they may create the illusion of an interference measurement, as if each discrete particle manifests a phase of its own.
A stochastic framework for quantum mechanics is presented in which the elements are measurements and amplitudes. The resulting structure provides three stochastic levels. At the bottom level are the sample points. These may be unobservable, in general. At the next level are the outcomes of measurements. The amplitude of an outcome is computed by summing the amplitudes of the sample points resulting in that outcome upon a measurement. At the highest level are the events of a measurement. The probability of an event is computed by summing the moduli squared of the amplitudes for the outcomes comprising that event. Using these guidelines, various probabilities, conditional probabilities, and expectations are defined. Superpositions of amplitude functions are investigated and superselection sectors are shown to occur in a natural way. The framework is illustrated in various mathematical models such as spin- (1)/(2) models, the two-slit experiment, and phase space quantum mechanics. Finally, a theory of discrete Feynman amplitudes is presented