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Physical Review A

Brownian motion of a quantum particle

Deepak Kumar

Published Mar 1, 1984• 21 citations
Publisher

Abstract

Proposed here is an exactly soluble model for the field-free motion of a quantum particle interacting with a heat bath. The heat-bath interaction includes a fluctuating random force and a frictional force of Langevin type. The momentum and position distributions obtained in the model have a close relation to the corresponding solutions of the Fokker-Planck equation of a free classical particle.

DOI
10.1103/physreva.29.1571

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Proposed here is an exactly soluble model for the field-free motion of a quantum particle interacting with a heat bath. The heat-bath interaction includes a fluctuating random force and a frictional force of Langevin type. The momentum and position distributions obtained in the model have a close relation to the corresponding solutions of the Fokker-Planck equation of a free classical particle.
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In a paper with the same title as the above, Manoliu and Kittel calculated the high temperature ensemble average of the nonequal time product of the displacement and random force in the classical Brownian motion problem. Here, using a generalized quantum Langevin equation (GLE), we generalize these considerations to the quantum case and to the case of an arbitrary heat bath (such as, e.g., the blackbody radiation heat bath). Also, we consider an arbitrary temperature and we generalize beyond Brownian motion to the case of a harmonically bound particle in a heat bath. In addition, we point out that correlations of the type considered are widely used in many applications in a variety of different areas of physics and we review some recent applications of the GLE.
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.
The Brownian motion of a particle coupled to a heat bath can be derived from an exactly solvable many-body system. This motion is governed by the Langevin equation in Newtonian mechanics, and by an operator equation formally identical to the Langevin equation when the system composed of the particle plus the heat bath is quantized. To get this formal similarity between the classical and the quantal equations of motion of the particle, the classical dissipative system is formulated in terms of a modified Hamilton–Jacobi equation and is then quantized using the Schrödinger method. The result is identical with the Schrödinger–Langevin equation that has been obtained by quantizing the entire system and then isolating the motion of the particle. The non-linear wave equation describing the motion of a particle subject to conservative and time-dependent forces as well as frictional forces has been applied to the problems of motion of a wave-packet, and of the scattering and trapping of heavy-ions.
The macroscopic description of a quantum particle with passive dissipation and moving in an arbitrary external potential is formulated in terms of the generalized Langevin equation. The coupling with the heat bath corresponds to two terms: a mean force characterized by a memory function \ensuremath{\mu}(t) and an operator-valued random force. Explicit expressions are given for the correlation and commutator of the random force. The random force is never Markovian. It is shown that \ensuremath{\mu}\ifmmode \tilde{}\else \~{}\fi{}(z), the Fourier transform of the memory function, must be a positive real function, analytic in the upper half-plane and with Re[\ensuremath{\mu}\ifmmode \tilde{}\else \~{}\fi{}(\ensuremath{\omega}+i0+{0}^{+}0+)] a positive distribution on the real axis. This form is then derived for the independent-oscillator model of a heat bath. It is shown that the most general quantum Langevin equation can be realized by this simple model. A critical comparison is made with a number of other models that have appeared in the literature.
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.