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Physical review. B, Solid state

Rate theory for solids. V. Quantum Brownian-motion model

Jerome H. Weiner, R. E. Forman

Published Jul 15, 1974• 25 citations
Publisher

Abstract

A model for interstitial diffusion is studied in which the impurity atom is regarded as a Brownian particle constrained to move in a single direction and subject to a periodic potential. A quantum mechanical theory is developed for the rate at which atoms leave a given potential well. The theory makes use of an ensemble of minimum-uncertainty wave packets to describe thermal equilibrium in the well and computes the tunneling probability through the potential barrier on the basis of the Schr\"odinger-Langevin equation in order to incorporate the effect of an interacting heat bath. The result of the interactions is to reduce the departure rate at a given temperature level, but the qualitative nature of the rate expression remains unchanged and predicts Arrhenius plots with straight-line behavior at high-temperature levels and curvature due to tunneling at low-temperature levels. The dynamics of the wave packet after it leaves a well is studied by computer solution of the Schr\"odinger-Langevin equation and it is found that the persistence of motion decreases with increasing strength of interaction with the heat bath.

DOI
10.1103/physrevb.10.325

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A model for interstitial diffusion is studied in which the impurity atom is regarded as a Brownian particle constrained to move in a single direction and subject to a periodic potential. A quantum mechanical theory is developed for the rate at which atoms leave a given potential well. The theory makes use of an ensemble of minimum-uncertainty wave packets to describe thermal equilibrium in the well and computes the tunneling probability through the potential barrier on the basis of the Schr"odinger-Langevin equation in order to incorporate the effect of an interacting heat bath. The result of the interactions is to reduce the departure rate at a given temperature level, but the qualitative nature of the rate expression remains unchanged and predicts Arrhenius plots with straight-line behavior at high-temperature levels and curvature due to tunneling at low-temperature levels. The dynamics of the wave packet after it leaves a well is studied by computer solution of the Schr"odinger-Langevin equation and it is found that the persistence of motion decreases with increasing strength of interaction with the heat bath.
The effects of tunneling are included in a one-dimensional rate theory which utilizes the concept of an ensemble of appropriate minimum-uncertainty wave packets (coherent states). An approximate tunneling formula for such wave packets is developed for a piecewise quadratic potential representing a well followed by a barrier; it is based on an exact solution for a Gaussian wave packet moving on a parabolic barrier. A new method for the numerical solution of the time-dependent Schr\"odinger equation is presented and used to check this approximate tunneling formula. The addition of tunneling to the effect of quantum statistics gives a rate formula which approximates the classical result at high-temperature levels and approaches a constant rate determined by tunneling at very low-temperature levels.
A theory for rate processes in solids is presented which includes the effects of quantum statistics, tunneling, and the NNN-dimensional character of the problem. It utilizes the concept of an ensemble of appropriate minimum-uncertainty wave packets to describe the conditions of thermal equilibrium in an NNN-dimensional potential well. An exact solution is found for the motion of Gaussian wave packets on a second-degree NNN-dimensional potential surface. This solution is valid also when the initial principal directions of the wave packet differ from those of the potential surface so that the problem is not separable. It is used to compute the tunneling probability for a Gaussian wave packet on a second-degree NNN-dimensional potential surface containing a saddle point. This probability depends only on the wave-packet characteristics in the saddle-point direction, and this facilitates its incorporation in the rate expression. Since the initial characteristics of the wave packets of the ensemble are determined by the potential-well character, and their subsequent tunneling probability is determined by the saddle-point character, the theory includes the effect of a difference in orientation between the two sets of principal directions. For an artificial two-dimensional problem formulated specifically to study this aspect of the process, an orientation difference is found to produce large effects only at low temperatures where tunneling is important.
Quantum Brownian motion in a periodic cosine potential is studied and a simple estimate of the tunneling effect is obtained in the frames of a semiclassical quasi-equilibrium approach. It is shown that the latter is applicable for heavy particles but electrons cannot be described properly since the quantum effects dominate over the thermal ones. The electron purely quantum diffusion is investigated at zero temperature and demonstrates that electrons do not obey the classical Einstein law of Brownian motion in the fi eld of periodic potentials, since the dispersion of the Gaussian wave packet increases logarithmically in time. Diffusion in the fi eld of periodic potentials is important for particle transport via ordered structures, typical in condensed matter. The underlying mechanism is Brownian motion. If the diffusing particles are quantum ones, e.g. electrons, protons, light atoms, etc., their migration is described via the theory of quantum Brownian motion [1, 2], which is signifi cantly affected by the tunneling effect [3]. Hereafter an alternative approach to the problem of quantum Brownian motion in periodic potentials is developed based on the Bohm quantum potential [4]. In contrast to the WKB theory valid in vacuum the present paper describes tunneling in a dissipative environment. The starting point is a nonlinear quantum Smoluchowski-like equation [5, 6] 0 [ ( ) ] / t x x b x V Q d b β ∂ ρ = ∂ ρ∂ + β + ∂ ρ β ∫ (1)
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.
We have presented a simple approach to quantum theory of Brownian motion and barrier crossing dynamics. Based on an initial coherent state representation of bath oscillators and an equilibrium canonical distribution of quantum-mechanical mean values of their co-ordinates and momenta we have derived a c number generalized quantum Langevin equation. The approach allows us to implement the method of classical non-Markovian Brownian motion to realize an exact generalized non-Markovian quantum Kramers' equation. The equation is valid for arbitrary temperature and friction. We have solved this equation in the spatial diffusion-limited regime to derive quantum Kramers' rate of barrier crossing and analyze its variation as a function of the temperature and friction. While almost all the earlier theories rest on quasiprobability distribution functions (e.g., Wigner function) and path integral methods, the present work is based on true probability distribution functions and is independent of path integral techniques. The theory is a natural extension of the classical theory to quantum domain and provides a unified description of thermally activated processes and tunneling.
An idealized model for the quantum tunneling of a particle in interaction with a heat bath is studied by computer simulation. The model may be motivated by consideration of a bound hydrogen–oxygen pair in a metal lattice; the H atom is treated quantum mechanically, the O atom is treated classically, and a double-well potential couples the two. In order to represent the O atom interactions with the rest of the lattice as heat bath, its motion is described by a classical Langevin equation. Computer simulation of the model shows that, with all other parameters held fixed, tunneling through low energy barriers is governed by energy splitting, while for high energy barriers the transmission approach agrees well with simulation results.