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Zeitschrift für Naturforschung A

Fuzzy Transitions from Quantum to Classical Mechanics and New Phenomena of Mesoscopic Objects

J. P. Hsu

Published Feb 1, 1997• 0 citations
Publisher

Abstract

Abstract A new "phase invariant" equation of motion for both microscopic and macroscopic objects is proposed. It reduces to the probabilistic wave equation for small masses and the deterministic classical equation for large masses. The motions of mesoscopic objects and fuzzy transitions between quantum and classical mechanics are discussed on the basis of the generalized equation. Experimental tests of new predictions are discussed.

DOI
10.1515/zna-1997-1-209

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Abstract A new "phase invariant" equation of motion for both microscopic and macroscopic objects is proposed. It reduces to the probabilistic wave equation for small masses and the deterministic classical equation for large masses. The motions of mesoscopic objects and fuzzy transitions between quantum and classical mechanics are discussed on the basis of the generalized equation. Experimental tests of new predictions are discussed.
In a generalized quantum-mechanical framework with fuzzy dynamical variables, there exists a small radical length R${10}^{\mathrm{\ensuremath{-}}17}$ cm, which is related to the physical impossibility of unlimited accuracy in measuring a particle's position. It sheds light on the problem of a detailed and gradual transition, as the mass increases, from the probabilistic behavior of microscopic objects (mcR\ensuremath{\ll}\ensuremath{\Elzxh}) to the deterministic behavior of macroscopic objects (mcR\ensuremath{\gg}\ensuremath{\Elzxh}), a problem unaddressable within the framework of ordinary quantum mechanics. A generalized equation of motion for a particle with an arbitrary mass m is postulated. It possesses a new local gauge symmetry and reduces to the Schr\"odinger and the Hamilton-Jacobi equations in the limits of small m and large m, respectively. For the double-slit experiment, the theory predicts that, as the mass increases with the momentum fixed, the interference pattern will have roughly half the usual intensity and two bright spots when mcR/\ensuremath{\Elzxh}\ensuremath{\sim}1.
"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
In this paper, I review one of the most important and interesting parts of my new book “Fuzzy Systems to Quantum Mechanics” (see Ref. 1). Several conclusions in this part are worth introducing here. First of all, the motion of a mass point in classic mechanics has also waviness and the wave function of the motion of a mass point is composed of wave functions of countably infinite microscopic particles. Secondly, based on the waviness of the motion of a mass point we surely know the new conclusion described as the wave-mass-point dualism in classic mechanics. And thirdly, by using the closed relation between the wave-mass-point dualism in classic mechanics and the wave-particle dualism in quantum mechanics, unified theory of classic mechanics and quantum mechanics is naturally formed.
An explicit model allowing a unified description of microscopic and macroscopic systems is exhibited. First, a modified quantum dynamics for the description of macroscopic objects is constructed and it is shown that it forbids the occurrence of linear superpositions of states localized in far-away spatial regions and induces an evolution agreeing with classical mechanics. This dynamics also allows a description of the evolution in terms of trajectories. To set up a unified description of all physical phenomena, a modification of the dynamics, with respect to the standard Hamiltonian one, is then postulated also for microscopic systems. It is shown that one can consistently deduce from it the previously considered dynamics for the center of mass of macroscopic systems. Choosing in an appropriate way the parameters of the so-obtained model one can show that both the standard quantum theory for microscopic objects and the classical behavior for macroscopic objects can all be derived in a consistent way. In the case of a macroscopic system one can obtain, by means of appropriate approximations, a description of the evolution in terms of a phase-space density distribution obeying a Fokker-Planck diffusion equation. The model also provides the basis for a conceptually appealing description of quantum measurement.
There exists a paradigm in which Quantum Mechanics is an exclusively developed theory to explain phenomena on a microscopic scale. As the Planck's constant is extremely small, h∼10−34J.sh\sim10^{-34}{J.s}h∼10−34J.s, and as in the relation of de Broglie the wavelength is inversely proportional to the momentum; for a mesoscopic or macroscopic object the Broglie wavelength is very small, and consequently the undulatory behavior of this object is undetectable. In this paper we show that with a particle oscillating around its classical trajectory, the action is an integer multiple of a quantum of action, S=nhoS = nh_{o}S=nho​. The quantum of action, hoh_{o}ho​, which plays a role equivalent to Planck's constant, is a free parameter that must be determined and depends on the physical system considered. For a mesoscopic and macroscopic system: ho≫hh_{o}\gg hho​≫h, this allows us to describe these systems with the formalism of quantum mechanics.
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.