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Physics Essays

Fuzziness in Quantum Mechanics

Alex Granik, H. John Caulfield

Published Sep 1, 1996• 8 citations
Publisher

Abstract

It is shown that quantum mechanics can be regarded as what one might call a "fuzzy" mechanics whose underlying logic is the fuzzy one, in contradistinction to the classical "crisp" logic.Therefore classical mechanics can be viewed as a crisp limit of a "fuzzy" quantum mechanics.Based on these considerations it is possible to arrive at the Schroedinger equation directly from the Hamilton-Jacobi equation.The link between these equations is based on the fact that a unique ( "crisp") trajectory of a classical particle emerges out of a continuum of possible paths collapsing to a single trajectory according to the principle of least action.This can be interpreted as a consequence of an assumption that a quantum "particle" "resides" in every path of the continuum of paths which collapse to a single( unique) trajectory of an observed classical motion.A wave function then is treated as a function describing a deterministic entity having a fuzzy character.As a consequence of such an interpretation, the complimentarity principle and wave-particle duality can be abandoned in favor of a fuzzy deterministic microoobject.

DOI
10.4006/1.3029260

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Fuzziness in Quantum Mechanics

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It is shown that quantum mechanics can be regarded as what one might call a "fuzzy" mechanics whose underlying logic is the fuzzy one, in contradistinction to the classical "crisp" logic. Therefore classical mechanics can be viewed as a crisp limit of a "fuzzy" quantum mechanics. Based on these considerations it is possible to arrive at the Schroedinger equation directly from the Hamilton-Jacobi equation. The link between these equations is based on the fact that a unique ("crisp") trajectory of a classical particle emerges out of a continuum of possible paths collapsing to a single trajectory according to the principle of least action. This can be interpreted as a consequence of an assumption that a quantum "particle" "resides" in every path of the continuum of paths which collapse to a single(unique) trajectory of an observed classical motion. A wave function then is treated as a function describing a deterministic entity having a fuzzy character. As a consequence of such an interpretation, the complimentarity principle and wave-particle duality can be abandoned in favor of a fuzzy deterministic microoobject.

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It is shown that quantum mechanics can be regarded as what one might call a "fuzzy" mechanics whose underlying logic is the fuzzy one, in contradistinction to the classical "crisp" logic.Therefore classical mechanics can be viewed as a crisp limit of a "fuzzy" quantum mechanics.Based on these considerations it is possible to arrive at the Schroedinger equation directly from the Hamilton-Jacobi equation.The link between these equations is based on the fact that a unique ( "crisp") trajectory of a classical particle emerges out of a continuum of possible paths collapsing to a single trajectory according to the principle of least action.This can be interpreted as a consequence of an assumption that a quantum "particle" "resides" in every path of the continuum of paths which collapse to a single( unique) trajectory of an observed classical motion.A wave function then is treated as a function describing a deterministic entity having a fuzzy character.As a consequence of such an interpretation, the complimentarity principle and wave-particle duality can be abandoned in favor of a fuzzy deterministic microoobject.
It is shown that quantum mechanics can be regarded as what one might call a "fuzzy" mechanics whose underlying logic is the fuzzy one, in contradistinction to the classical "crisp" logic. Therefore classical mechanics can be viewed as a crisp limit of a "fuzzy" quantum mechanics. Based on these considerations it is possible to arrive at the Schroedinger equation directly from the Hamilton-Jacobi equation. The link between these equations is based on the fact that a unique ("crisp") trajectory of a classical particle emerges out of a continuum of possible paths collapsing to a single trajectory according to the principle of least action. This can be interpreted as a consequence of an assumption that a quantum "particle" "resides" in every path of the continuum of paths which collapse to a single(unique) trajectory of an observed classical motion. A wave function then is treated as a function describing a deterministic entity having a fuzzy character. As a consequence of such an interpretation, the complimentarity principle and wave-particle duality can be abandoned in favor of a fuzzy deterministic microoobject.
The fussy "roots" of quantum mechanics are traced directly to the Hamilton-Jacobi equation of classical mechanics. It is shown that the Shroedinger equation can be derived from the Hamilton-Jacobi equation. A deep underlying unity of both equations lies in the fact that a unique trajectory of a classical particle is "selected" out of many-continuum paths according to the Principle of least action. We can say that a classical particle has a membership in every path of the above set, which collapses to the winning single trajectory of a real motion. At the same time it can also be said that a quantum mechanical "particle" has different degrees of membership in a set of many-continuum paths where all of them contribute to the dynamics of the quantum mechanical particle. This allows one to provide an interpretation of the wave function as a parameter describing deterministic entity endowed by a fuzzy character. As a logical consequence of such an interpretation the complimentarity principle and the wave-particle duality concept can be abandoned in favor of a fuzzy deterministic microobject. This idea leads to a possibility of a quantum mechanical computer based on the fuzzy logic.
Some recent experiments claim to show that any model in which a quantum state represents mere information about an underlying physical reality of the system must make predictions which contradict those of quantum theory. The present work undertakes to investigate the issue of reality, treading a more fundamental route from the Hamilton-Jacobi equation of classical mechanics to the Schrodinger equation of quantum mechanics. Motivation for this is a similar approach from the eikonal equation in geometrical optics to the wave equation in electromagnetic theory. We rewrite the classical Hamilton-Jacobi equation as a wave equation and seek to generalise de Broglie's wave particle duality by demanding that both particle and light waves have the freedom of being described by any square-integrable function. This generalisation, which allows superposition also for matter wave functions, helps us to obtain the Schrodinger equation, whose solution can be seen to be as much objective as the classical mechanics wave function. Several other equations which one writes in quantum mechanics, including the eigenvalue equations for observables, series expansion of energy states in terms of eigenstates of observables other than energy, etc., can be written in the classical case too. Absence of any collapse of the wave function, entanglement, etc. in the classical realm have their origin in the nonlinearity of the classical wave equation. These considerations indicate that many of the puzzles in quantum mechanics are present also in classical mechanics in a dormant form, which fact shall help to demystify quantum mechanics to a great extent.
"This unique compendium represents important action of fuzzy systems to quantum mechanics. From fuzzy sets to fuzzy systems, it also gives clear descriptions on the development on fuzzy logic, where the most important result is the probability presentation of fuzzy systems. The important conclusions on fuzzy systems are used in the study of quantum mechanics, which is a very new idea. Eight important conclusions are obtained. The author has proved that mass-point motions in classical mechanics must have waves, which means that any mass-point motion in classical mechanics has wave mass-point dualism as well as any microscopic particle motion must have wave-particle dualism. Based on this conclusion, it has been proven that classical mechanics and quantum mechanics are unified."--Publisher's website
The Copenhagen interpretation of quantum mechanics assumes the existence of the classical deterministic Newtonian world. We argue that in fact the Newton determinism in classical world does not hold and in classical mechanics there is fundamental and irreducible randomness. The classical Newtonian trajectory does not have a direct physical meaning since arbitrary real numbers are not observable. There are classical uncertainty relations, i.e. the uncertainty (errors of observation) in the determination of coordinate and momentum is always positive (non zero). A "functional" formulation of classical mechanics was suggested. The fundamental equation of the microscopic dynamics in the functional approach is not the Newton equation but the Liouville equation for the distribution function of the single particle. Solutions of the Liouville equation have the property of delocalization which accounts for irreversibility. The Newton equation in this approach appears as an approximate equation describing the dynamics of the average values of the position and momenta for not too long time intervals. Corrections to the Newton trajectories are computed. An interpretation of quantum mechanics is attempted in which both classical and quantum mechanics contain fundamental randomness. Instead of an ensemble of events one introduces an ensemble of observers.
We present an explicit physical example of an experimental situation on a physical entity that gives rise to a fuzzy set. The fuzziness in the example is due to ∞uctuations of the experimental apparatus, and not to an indeterminacy about the states of the physical entity, and is described by a varying parameter †. For zero value of the parameter (no ∞uctuations), the example reduces to a classical mechanics situation, and the corresponding fuzzy set is a quasi-crisp set. For maximal value (maximal ∞uctuations), the example gives rise to a quantum fuzzy set, more precisely a spin- model. In between, we have a continuum of fuzzy situations, neither classical, nor quantum. We believe that the example can make us understand the nature of the quantum mechanical fuzziness and probability, and how these are related to the classical situation.