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arXiv:0902.3880

On the Uncertainty Relations in Stochastic Mechanics

D. A. Trifonov, Blagovest A. Nikolov, Ivaïlo M. Mladenov

Published Feb 23, 2009• 2 citations
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Abstract

It is shown that the Bohm equations for the phase $S$ and squared modulus $ρ$ of the quantum mechanical wave function can be derived from the classical ensemble equations admiting an aditional momentum $p_s$ of the form proportional to the osmotic velocity in the Nelson stochastic mechanics and using the variational principle with appropriate change of variables. The possibility to treat grad$S$ and $p_s$ as two parts of the momentum of quantum ensemble particles is considered from the view point of uncertainty relations of Robertson - Schroedinger type on the examples of the stochastic image of quantum mechanical canonical coherent and squeezed states.

DOI
10.48550/arxiv.0902.3880

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It is shown that the Bohm equations for the phase SSS and squared modulus ρρρ of the quantum mechanical wave function can be derived from the classical ensemble equations admiting an aditional momentum psp_sps​ of the form proportional to the osmotic velocity in the Nelson stochastic mechanics and using the variational principle with appropriate change of variables. The possibility to treat gradSSS and psp_sps​ as two parts of the momentum of quantum ensemble particles is considered from the view point of uncertainty relations of Robertson - Schroedinger type on the examples of the stochastic image of quantum mechanical canonical coherent and squeezed states.
We derive the equations of quantum mechanics and quantum thermodynamics from the assumption that a quantum system can be described by an underlying classical system of particles. Each component ϕjϕ_jϕj​ of the wave vector is understood as a stochastic complex variable whose real and imaginary parts are proportional to the coordinate and momentum associated to a degree of freedom of the underlying classical system. From the classical stochastic equations of motion, we derive a general equation for the covariance matrix of the wave vector which turns out to be of the Lindblad type. When the noise changes only the phase of ϕjϕ_jϕj​, the Schrödinger and the quantum Liouville equation are obtained. The component ψjψ_jψj​ of the wave vector obeying the Schrödinger equation is related to stochastic wave vector by ∣ψj∣2=⟨∣ϕj∣2⟩|ψ_j|^2=\langle|ϕ_j|^2\rangle∣ψj​∣2=⟨∣ϕj​∣2⟩.
Abstract We derive the equations of quantum mechanics and quantum thermodynamics from the assumption that a quantum system can be described by an underlying classical system of particles. Each component φ j of the wave vector is understood as a stochastic complex variable whose real and imaginary parts are proportional to the coordinate and momentum associated with a degree of freedom of the underlying classical system. From the classical stochastic equations of motion, we derive a general equation for the covariance matrix of the wave vector, which turns out to be of the Lindblad type. When the noise changes only the phase of φ j , the Schrödinger and the quantum Liouville equations are obtained. The component ψ j of the wave vector obeying the Schrödinger equation is related to the stochastic wave vector by | ψ j | 2 = ⟨ | ϕ j | 2 ⟩ .
Among the famous formulations of quantum mechanics, the stochastic picture developed since the middle of the last century remains one of the less known ones. It is possible to describe quantum mechanical systems with kinetic equations of motion in configuration space based on conservative diffusion processes. This leads to the representation of physical observables through stochastic processes instead of self-adjoint operators. The mathematical foundations of this approach were laid by Edward Nelson in 1966. It allows a different perspective on quantum phenomena without necessarily using the wave-function. This article recaps the development of stochastic mechanics with a focus on variational and extremal principles. Furthermore, based on recent developments of optimal control theory, the derivation of generalized canonical equations of motion for quantum systems within the stochastic picture are discussed. These so-called quantum Hamilton equations add another layer to the different formalisms from classical mechanics that find their counterpart in quantum mechanics.