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Journal of Jianghan University

The Origin of the Statistical Character of the Quantum Mechanics

Huang Bao-xing

Published Jan 1, 2002• 0 citations
OpenAlex

Abstract

It is discussed the distributions of the wavefunction of the quantum mechanics in the classical limit, the single particle system and the ensemble of the classical mechanics. It proved that the wavefunction of the quantum mechanics can only describe an ensemble and can't describe the motion of the single particle. It is the origin of the statistical character of the quantum mechanics.

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It is discussed the distributions of the wavefunction of the quantum mechanics in the classical limit, the single particle system and the ensemble of the classical mechanics. It proved that the wavefunction of the quantum mechanics can only describe an ensemble and can't describe the motion of the single particle. It is the origin of the statistical character of the quantum mechanics.
Because of the use of different limiting procedures, there are two conflict conclusions on wave function. One is that a wave function can describe a single particle and the other is that it can only describe an ensemble in the classical limit. In this paper the limiting procedures have been compared. We put the synthesized limit n→∞, →0 but keep E(n, ) to be the actual measured value of energy. Under this limit condition, we can not only prove that a wave function can only describe an ensemble, but also can make clear the dynamical behavior of the particles of the ensemble. Calculations show that the reason why quantum mechanics cannot describe a particle is not because of the motion equation but because of the definition of state for the particle.
We show that the quantum wavefunction, interpreted as the probability density of finding a single non-localized quantum particle, which evolves according to classical laws of motion, is an intermediate description of a material quantum particle between the quantum and classical realms. Accordingly, classical and quantum mechanics should not be treated separately, a unified description in terms of the Wigner distribution function being possible. Although defined on classical phase space coordinates, the Wigner distribution function accommodates the nonlocalization property of quantum systems, and leads to both the Schrodinger equation for the quantum wavefunction and to the definition of position and momentum operators.
It is shown that the wave function describes the state of the statistical ensemble E[S] of individual particles, or the statistical average particle . This result follows from the fact that in the classical limit h=0 the Schroedinger equation turns to the dynamic equations for the statistical ensemble of classical particles. The statement that the wave function describes the state of an individual particle is incompatible with the quantum mechanics formalism. It is shown that the statistical interpretation of quantum mechanics is a corollary of the fact, that the QM formalism is the technique of the statistical ensemble description, restricted by constraints of the dynamic equation linearity.
It is proved the mathematical theorem, that the wave function describes the statistical ensemble of particles, but not a single particle. Supposition, that the wave function describes a single particle appears to be incompatible with formalism of quantum mechanics. One discusses the reasons, why this very simple statement has not been proved mathematically for many years. The reason lies in application of the trial and error methods for construction of the quantum mechanics. Application of this method as the main tool of investigation during eighty years generated ”fitting mentality ” of all microwold researchers. In this paper we prove a very important theorem, which claims that the wave function may not describe an individual quantum particle. It describes always a statistical ensemble of quantum particles. We shall show, that the action AS for the Schrödinger particle SS (the dynamic system described by the Schrödinger equation) turns into the action AE[Scl] for the statistical ensemble E [Scl] of free classical particles Scl, when the quantum constant � → 0. Such a transition is possible only in the case, when the wave function ψ describes a statistical ensemble of quantum particles, but not a single particle. For the free Schrödinger particle SS the action has the form
It is shown that Schrödinger's equation may be derived from three postulates. The first is a kind of statistical metamorphosis of classical mechanics, a set of two relations which are obtained from the canonical equations of particle mechanics by replacing all observables by statistical averages. The second is a local conservation law of probability with a probability current which takes the form of a gradient. The third is a principle of maximal disorder as realized by the requirement of minimal Fisher information. The rule for calculating expectation values is obtained from a fourth postulate, the requirement of energy conservation in the mean. The fact that all these basic relations of quantum theory may be derived from premises which are statistical in character is interpreted as a strong argument in favor of the statistical interpretation of quantum mechanics. The structures of quantum theory and classical statistical theories are compared, and some fundamental differences are identified.
The Bohmian formulation of quantum mechanics is used in order to describe the measurement process in an intuitive way without a reduction postulate in the framework of a deterministic single system theory. Thereby the motion of the hidden classical particle is chaotic during almost all nontrivial measurement processes. For the correct reproduction of experimental results, it is further essential that the distribution function P(x)P(x)P(x) of the results of a position measurement is identical with ∣Ψ∣2|Ψ|^2∣Ψ∣2 of the wavefunction ΨΨΨ of the single system under consideration. It is shown that this feature is not an additional assumption, but can be derived strictly from the chaotic motion of a single system during a sequence of measurements, providing a completely deterministic picture of the statistical features of quantum mechanics.