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arXiv:quant-ph/0111094

Two Slits Interference Is Compatible with Particles' Trajectories

Vladimir V. Kisil

Published Nov 19, 2001• 6 citations
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Abstract

We propose a simple numerical experiment of two slits interference of particles. It disproves the popular belief that such an interference is incompatible with a knowledge which slit each particle came through or, more generally, ``quantum particles could not have trajectories''. Our model is an illustration to the contextual interpretation of quantum probabilities.

DOI
10.48550/arxiv.quant-ph/0111094

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•Sep 1, 2014•0 citations
We propose a simple numerical experiment of two slits interference of particles. It disproves the popular belief that such an interference is incompatible with a knowledge which slit each particle came through or, more generally, ``quantum particles could not have trajectories''. Our model is an illustration to the contextual interpretation of quantum probabilities.
We consider a classical analogue of the well known quantum two-slit experiment. Charged particles are scattered on flat screen with two slits and hit the second screen. We show that the probability distribution on the second screen when both slits are open is not given by the sum of distributions for each slit separately, but has an extra interference term that is given with the quantum rule of the addition of probabilistic alternatives. We show that the proposed classical model has a context dependence and could be adequately described with contextual formalism.
A number of assertions have recently been made that in two-particle interference devices, Bohmian trajectories may not reproduce exactly all statistical predictions of quantum mechanics. Specifically, let two identical bosons go through identical slits arranged symmetrically with respect to a plane, with wave functions transforming into each other under the symmetry operation. Then it is claimed that pairs of particles following Bohmian trajectories and registered on a screen opposite the slits may have observable spatial distribution properties not predicted by quantum mechanics. We point out that alleged discrepancies in fact originate from a specific sampling assumption on Bohmian particles that violates their general probability distribution.
The causal quantum mechanics (i.e. Bohmian or de Broglie-Bohm or Bohm-de Broglie quantum mechanics) has made possible to calculate the trajectories of electrons in a typical double-slit experiment [C. Philippidis et al., Il Nuovo Cimento, 52 B, 15-28 (1979)]. The trajectories do not correspond to an uniform movement but to an accelerated one. The acceleration is caused by the quantum potential. From the quantum theoretical point of view, there is a probability for the electron to emit photons, with a certain emission power, during its movement from the slits to the screen. We find a quantum general formula for the emission power of photons, valid independently of the interpretation. Then, according to the Copenhagen interpretation, this formula gives a strictly zero value for the emission power because the electron moves as a free particle after it leaves the slit and before reach the screen. Then, there is no emission of photons. In the case of the causal interpretation, the emission power results, for a concrete real experiment, in a very tiny but not a zero value, driven by the square of the quantum potential gradient. We give an idea of the type of spectrum that could be measured. A brief idea of a possible experimental arrangement in order to detect this effect, is given.
The present investigation is about the quantum mechanics of multislit interference experiments. One of the cornerstones of our understanding of quantum mechanics is provided by the analysis of this type of experiments. Beginning with Bohr and Einstein, the discussion revolved mostly around single-slit diffraction and double-slit interference, probing the complementarity of path determination and the appearance of interference fringes, and contrasting the familiar with the quantum. Here, we are concerned with developing a systematic understanding and description of multislit interference experiments, i.e. setups in which a plurality of slits is illuminated. We provide a characterisation of a number of relevant observables, discussing those that are compatible and may be measured jointly, and also incompatible observables which cannot be measured jointly but instead display quantum uncertainty. \nWe begin with a discussion of a particular modification of the classic double- slit interference experiment which highlights the realisation of specific position and momentum observables which are jointly measurable. Although there are technical results regarding the coexistence of specific position and momentum observables, it may be surprising that ubiquitous experimental setups provide a preparation of such quantum states. We proceed with a discussion of the particular character of the incompatibility of certain measurements by building on an initial heuristic argument provided by Aharonov, Pendleton and Petersen. We prove, extend and discuss a formulation of un- certainty suitable for the context of multislit experiments. We conclude with a comparison of this formulation of uncertainty with an alternative uncertainty formulation developed by Uffink and Hilgevoord. Although these two uncertainty formulations are very different technically, we demonstrate that the same tradeoff is expressed independently.