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arXiv:quant-ph/0703126

Quantum interference with slits

Thomas V Marcella

Published Mar 15, 2007• 1 citation
arXivHTMLPDF

Abstract

In the experiments considered here, we measure the y-component of momentum for a particle passing through a system of slits. The source-slit system is the preparation apparatus that determines the state vector. Recognizing that a system of slits is a position-measuring device allows us to ascertain that the state vector is a position state. Then, writing the state vector in momentum space provides a straightforward calculation for the probability amplitude and its corresponding probability function. Interference effects, if any, are inherent in the probability function We determine the statistical distribution of scattered particles for four different slit systems. The results are in agreement with the well-known interference patterns obtained in classical wave optics.

DOI
10.48550/arxiv.quant-ph/0703126

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In the experiments considered here, we measure the y-component of momentum for a particle passing through a system of slits. The source-slit system is the preparation apparatus that determines the state vector. Recognizing that a system of slits is a position-measuring device allows us to ascertain that the state vector is a position state. Then, writing the state vector in momentum space provides a straightforward calculation for the probability amplitude and its corresponding probability function. Interference effects, if any, are inherent in the probability function We determine the statistical distribution of scattered particles for four different slit systems. The results are in agreement with the well-known interference patterns obtained in classical wave optics.
A quantum experiment consists of a preparation procedure followed by the measurement of a specified observable. In the experiments considered here, the y -component of momentum is measured after a particle passes through a system of slits. The source–slit system is the preparation apparatus that determines the initial state |ψ⟩. Recognizing that a system of slits is a position-measuring device allows us to determine |ψ⟩. Then, a straightforward calculation gives P ( p y ) = |⟨ p y |ψ⟩| 2 the probability that a particle will pass through a system of slits with momentum p y = p sin θ. Interference effects are inherent in this probability function. We perform the calculation for four different slit systems. The method is suitable for an upper level course in quantum mechanics.
Summary form only given. We say that the quantum observables "position" and "momentum" are "complementary" because the precise knowledge of the position (momentum) implies that all possible outcomes of measuring the momentum (position) are equally probable. In two-slit experiments, the common understanding is that the position-momentum uncertainty relation makes it impossible to determine which slit the photon (or electron) passes through without at the same time disturbing the object enough to destroy the interference pattern. In 1982, Scully et al. suggested a way around this position-momentum uncertainty obstacle and proposed a quantum eraser to obtain which-path or particle-like information without scattering or otherwise introducing large uncontrolled phase factors to disturb the interference. Since 1982, quantum has been claimed in several experiments; however, the original proposed ideas have not been fully demonstrated. We demonstrate them here.
The Young’s single- and double-slit experiments are explained using six-component quantum and one-component quasi-classical photon wave functions in the coordinate representation, which are modeled by different mechanisms of photon diffraction on slits or simultaneous emission of two photons by (two) independent sources. This explanation is compared with the well-known statement about the interference of a particle with itself when passing through both slits simultaneously in the Young’s experiment on the basis of the postulated Feynman amplitudes. It is argued that these amplitudes cannot be wave functions, in the literal sense, describing some states of quantum particles. In all cases, under the assumption of the linear polarization of photons, a pronounced interference pattern has been obtained, which suggests that the coordinate wave function of a photon can explain the phenomena of the one- and two-photon interference. This conclusion, in turn, makes it possible to associate the coordinate wave function of a photon (two photons, in the entangled state) measured in some recent experiments with the quasi-classical wave function of a photon (two photons), although it does not take into account its (their) polarization properties.
Whether the loss of visibility due to measurement in two-slit interference patterns can always be attributed to momentum transfer depends upon how one defines the last term. The momentum transfer to a quantum particle can be determined unambiguously if it was initially in a momentum eigenstate, but that is not the state relevant to a two-slit experiment. A sensible answer for the two-slit problem was obtained by Wiseman et al. [Phys. Rev. A 56, 944 (1997)] using the formalism of the Wigner function. Here I show that a more general answer can be obtained using the Bohmian formulation of quantum mechanics, in which particles have a definite position and momentum at all times. By following all possible trajectories of the particle it is possible to calculate the probability distribution ${P}_{\mathrm{total}}^{B}{(p}^{\ensuremath{'}})$ for it to receive a momentum transfer ${p}^{\ensuremath{'}}$ as a result of the measurement. Furthermore, the 1-norm of this distribution obeys the relation $〈|{p}^{\ensuremath{'}}|{〉}_{\mathrm{total}}^{B}d>~2\ensuremath{\Elzxh}(1\ensuremath{-}V)/\ensuremath{\pi},$ where VVV is the visibility of the interference pattern and ddd is the slit separation. This confirms that the momentum transfer in a welcher Weg (which path) experiment is in accord with the uncertainty principle. Like the Wignerian analysis, the Bohmian analysis clearly distinguishes between a local momentum transfer (which occurs even if the particle is localized at a single slit) and a nonlocal momentum transfer (which occurs only if the particle is delocalized, at both slits). In the Bohmian analysis the former occurs at the time of the measurement, while the latter occurs after the measurement, and is a consequence of Bohm's ``quantum potential.''
The particle- interference-experiments, with the photons and electrons have clearly shown that they are always detected as localized „particles‟; and the density of their detections exhibit a pattern similar to the interference of waves. This experimental-observation is now explained here, making the current, „probabilistic interpretation‟, a past history. The „particle ‟ is first mathematically characterized as an „impulse function ‟ in space; and then Fourier-transformed into the wave-number-domain, as a wide band of constituent-waves as shown in the fig. 1&2. The „particle ‟ is explained as an „event ‟ of constructive-superimposition of a wide band of „pre-quantum-waves‟. This „event ‟ occurs in space at regular distances; and at regular instances in time. We know that even the purest monochromatic laser source has a „line-width ‟ of a few Giga-Hertz, or at least a few kilo-Hertz. So in the particle-interference-experiments, they are these wide bands of waves which pass through both the slits, even in the case of single photon states, and interfere physically. This physical interference of the wide bands of pre-quantum-waves determines the position and time of emergence of the „particle‟. Thus, form the theoretical view-point, there are only the waves, and waves alone; and the „particle ‟ is a specific situation of constructive-superimposition of all the spectral-components of pre-quantum-waves; so there is no wave-particle-duality.
Since the wavefunction of a photon only describes the probability of photon detection in time and space, it is impossible to derive uniquely defined trajectories describing the path taken by the photon between emission and detection. However, it is possible to test whether a particular set of trajectories is consistent with the statistics observed at different times for photons in the same initial state. Recently, I have shown that quantum interference effects between position and momentum can result in a violation of inequalities associated with motion along straight lines. Here, I present a more detailed analysis on the origin of the effect and its relation with other experimentally observable aspects of quantum statistics such as weak measurements and quantum tomography. It is shown that the interference pattern between a quantum state component of well-defined position and a quantum state component of well-defined momentum describes a modified causality relation between the positions detected at different times. The phase of the interference pattern is identified with the classical action of particle motion and the relation between uncertainty and causality is considered. The specific case of single photon wavefunctions is used to explain the possibilities and limitations of control at the ultimate quantum limit.