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arXiv:1902.05930 · Fluctuation and Noise Letters

Brownian Motion and Quantum Mechanics

Roumen Tsekov

Published Nov 19, 2019• 6 citations
arXivHTMLPDF

Abstract

A theoretical parallel between the classical Brownian motion and quantum mechanics is explored via two stochastic sources. It is shown that, in contrast to the classical Langevin force, quantum mechanics is driven by turbulent velocity fluctuations with diffusive behavior. In the case of simultaneous action of the thermal and quantum noises, the quantum Brownian motion is described as well.

DOI
10.1142/s0219477520500170

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A theoretical parallel between the classical Brownian motion and quantum mechanics is explored via two stochastic sources. It is shown that, in contrast to the classical Langevin force, quantum mechanics is driven by turbulent velocity fluctuations with diffusive behavior. In the case of simultaneous action of the thermal and quantum noises, the quantum Brownian motion is described as well.
We give a new description of quantum Brownian motion in terms of stochastic pure states. The corresponding path integral propagator allows us to establish a direct connection to the classical Langevin equation, in the Schrodinger picture. We show that in the quantum domain, one is naturally led to consider two stochastic processes driving the Brownian dynamics, one of them representing thermal fluctuations, as in the classical case. The second process reflects growing entanglement between the Brownian particle and its environment, and is therefore a truly quantum noise process. Technically, our result rests on the representation of the full propagator of the Brownian particle and its environment in a coherent state basis. For open system dynamics that may be described by a master equation, such stochastic schemes have already been proven to be efficient Monte Carlo methods. Here, we give a new stochastic scheme that covers low temperatures and strong friction, the latter limit being the case Einstein originally investigated.
Introduction The apparent generality of the title to this chapter is appropriately reduced by sufficiently precisely defining the terms used. We have in mind the natural extension to quantum mechanical settings of the ideas usually covered in a discussion of stochastic processes in classical mechanical settings. In classical mechanical settings, the fundamental process is Brownian motion, which describes the influence of external fluctuating forces on a small particle. The mathematical apparatus developed for the study of Brownian motion has been generalized and applied to irreversible thermodynamics (Onsager and Machlup, 1953), fluctuating hydrodynamics (Fox and Uhlenbeck, 1970), light scattering from fluids and mixtures (Berne and Pecora, 1976) and chemical reactions (McQuarrie and Keizer, 1981). In each case, the stochastic process involves fluctuations generated by thermal motions. We are not considering the stochastic trajectories (chaos) of classically deterministic motion in non-linear systems (Lichtenberg and Lieberman, 1983), which is an entirely separate context for stochastic processes. Our goal in this chapter is to study the effects of thermal fluctuations in quantum mechanical settings. Therefore, we exclude a discussion of interpreting quantum mechanics as a classical stochastic motion (Nelson, 1966), an idea which should have been put to rest by Grabert, Hänggi and Talkner (1979). We also exclude a discussion of quantum chaos, which is currently a very active area of research (Zaslavsky, 1981). Just as Brownian motion provides the paradigm for stochastic processes in classical mechanical settings, the Anderson–Kubo oscillator (Anderson, 1954; Kubo, 1954; Kubo, Toda and Nashitsume, 1985) is the paradigm for stochastic processes in quantum mechanical settings.
In the frames of classical mechanics the generalized Langevin equation is derived for an arbitrary mechanical subsystem coupled to the harmonic bath of a solid. A time-acting temperature operator is introduced for the quantum Klein-Kramers and Smoluchowski equations, accounting for the effect of the quantum thermal bath oscillators. The model of Brownian emitters is theoretically studied and the relevant evolutionary equations for the probability density are derived. The Schrodinger equation is explained via collisions of the target point particles with the quantum force carriers, transmitting the fundamental interactions between the point particles. Thus, electrons and other point particles are no waves and the wavy chapter of quantum mechanics originated for the force carriers. A stochastic Lorentz-Langevin equation is proposed to describe the underlaying Brownian-like motion of the point particles in quantum mechanics. Considering the Brownian dynamics in the frames of the Bohmian mechanics, the density functional Bohm-Langevin equation is proposed, and the relevant Smoluchowski-Bohm equation is derived. A nonlinear master equation is proposed by proper quantization of the classical Klein-Kramers equation. Its equilibrium solution in the exact canonical Gibbs density operator, while the well-known Caldeira-Leggett equation is simply a linearization at high temperature. In the case of a free quantum Brownian particles, a new law for the spreading of the wave packet it discovered, which represents the quantum generalization of the classical Einstein law of Brownian motion. A new projector operator is proposed for the collapse of the wave function of a quantum particle moving in a classical environment. Its application results in dissipative Schrodinger equations, as well as in a new form of dissipative Liouville equation in classical mechanics.
A simple scheme of unified description of different physical phenomena by using the Langevin type equations is reviewed. Within this approach much attention is being paid to the study of Brownian and quantum motions. Stochastic equations with a white noise term give all characteristics of the Brownian motion. Some generalization of the Langevin type equations allows us to obtain nonlinear equations of particles' motion, which are formally equivalent to the Schrödinger equation. Thus, we establish Nelson's stochastic mechanics on the basis of the Langevin equation.
A study of the non-dissipative Brownian motion in vacuum is presented. The noise source associated to the stochastic process assumed in this work is vacuum fluctuations of some quantum field capable of interact with a massive particle. For the associated Fokker-Planck equation we do a perturbation theory which has terms presenting dynamics similar to that satisfied by the probability amplitude in Quantum Theory even though the physical interpretation be classical. In particular classical observables present expected values that coincide with those calculated using Quantum Mechanics.
Abstract We propose a Langevin equation to describe the quantum Brownian motion of bounded particles based on a distinctive formulation concerning both the fluctuation and dissipation forces. The fluctuation force is similar to that employed in the classical case. It is a white noise with a variance proportional to the temperature. The dissipation force is not restricted to be proportional to the velocity and is determined in a way as to guarantee that the stationary state is given by a density operator of the Gibbs canonical type. To this end we derived an equation that gives the time evolution of the density operator, which turns out to be a quantum Fokker–Planck–Kramers equation. The approach is applied to the harmonic oscillator in which case the dissipation force is found to be non Hermitian and proportional to the velocity and position.