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International Journal of Theoretical Physics

A statistical analysis of the two-slit experiment: Or some remarks on quantum probability

Domenico Costantini

Published Dec 1, 1993• 3 citations
Publisher
DOI
10.1007/bf00673004

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Abstract This report examines the characteristics of the Heisenberg–Kennard uncertainty product in the double-slit experiment with an asymmetric slits configuration. For completeness, the symmetric double-slit experiment is also revisited. We employ the Budiyono–Rohrlich statistical model of quantum mechanics to derive explicit expressions for the related product based on the Gaussian position probability function. We analyze the effects of varying the slit widths and the separation distance. We observe that changing the ratio of the two slits’ widths and their separation can reversibly switch between the relative-phase-dependent coherent uncertainty products, providing a mechanism for tuning the associated uncertainty product through the geometric configuration of the slits. The theoretically predicted geometry-induced switching phenomenon, driven by the nonzero expectation value of position and the differing slopes of the position probability function, as represented by the Fisher information term, may enable applications in precision which-way-related experiments and weak-measurement protocols. We also perform a geometric analysis based on a Greenberger–Yasin duality-like relation to investigate the characteristics of the transition from coherent to decoherent conditions.
We start with the discussion on misapplication of classical probability theory by Feynman in his analysis of the two slit experiment (by following the critical argumentation of Koopman, Ballentine, and the author of this paper). The seed of Feynman's conclusion on the impossibility to apply the classical probabilistic description for the two slit experiment is treatment of conditional probabilities corresponding to different experimental contexts as unconditional ones. Then we move to the Bell type inequalities. Bell applied classical probability theory in the same manner as Feynman and, as can be expected, he also obtained the impossibility statement. In contrast to Feynman, he formulated his no-go statement not in the probabilistic terms, but by appealing to nonlocality. This note can be considered as a part of the author's attempts for getting rid off nonlocality from quantum physics.
We give a mathematical golden mean distribution based probabilistic confirmation of a recent spectacular experiment with light. The experiment in question is a three-slit variant of the well known two-slit set up of quantum mechanics. The outcome of the sophisticated experiment revealed the looped path of light on the quantum scale and consequently the Peano-Hilbert geometry of spacetime, ergo its fractal-Cantorian nature. The mathematics used here on the other hand is the remarkably simple and insightful golden mean probability distribution known from a famous paradox known in social sciences as the voter paradox.