Quantum surface diffusion in Bohmian mechanics

S Miret-Artés

doi:10.1088/2399-6528/aae06e

Abstract

Surface diffusion of small adsorbates is analyzed in terms of the so-called intermediate scattering function and dynamic structure factor, observables in experiments using the well-known quasielastic Helium atom scattering and Helium spin echo techniques. The linear theory applied is an extension of the neutron scattering due to van Hove and considers the time evolution of the position of the adsorbates in the surface. This approach allows us to use a stochastic trajectory description following the classical, quantum and Bohmian frameworks. Three different regimes of motion are clearly identified in the diffusion process: ballistic, Brownian and intermediate which are well characterized, for the first two regimes, through the mean square displacements and Einstein relation for the diffusion constant. The Langevin formalism is used by considering Ohmic friction, moderate surface temperatures and small coverages. In the Bohmian framework, analyzed here, the starting point is the so-called Schrödinger-Langevin equation which is a nonlinear, logarithmic differential equation. By assuming a Gaussian function for the probability density, the corresponding quantum stochastic trajectories are given by a dressing scheme consisting of a classical stochastic trajectory followed by the center of the Gaussian wave packet, and issued from solving the Langevin equation (particle property), plus the time evolution of its width governed by the damped Pinney differential equation (wave property). The Bohmian velocity autocorrelation function is the same as the classical one when the initial spread rate is assumed to be zero. If not, in the diffusion regime, the Brownian-Bohmian motion shows a weak anomalous diffusion.

Highlights

Surface diffusion is one of the most elementary dynamical process occurring on surfaces and a preliminary step to more complex surface phenomena
Original content from this function and dynamic structure factor, observables in experiments using the well-known quasielastic work may be used under the terms of the Creative Helium atom scattering and Helium spin echo techniques
We are going to focus on He atoms as nondestructive probe particles used in two types of experiments, quasielastic He atom scattering (QHAS) [6] and He spin echo spectroscopy (HeSE) [7]

Summary

Introduction

Surface diffusion is one of the most elementary dynamical process occurring on surfaces and a preliminary step to more complex surface phenomena. The resulting quantum stochastic trajectories have been applied to simple systems such as the damped free particle, linear potential, and harmonic oscillator[31] and dissipative quantum tunnelling through an inverted parabolic barrier under the presence of an electric field [32] when analysing the classical-quantum transition of trajectories in the gradual decoherence process These works introduced the so-called scaled trajectories having as a particular case the Bohmian ones. Two main observable functions the so-called dynamic structure factor and intermediate scattering function are introduced and written in terms of adsorbate trajectories These trajectories are briefly presented and discussed within the general Langevin formalism starting from the so-called Caldeira-Leggett Hamiltonian in the classical and quantum frameworks, being the adsorbate coverage introduced by a collisional friction. The Brownian-Bohmian motion shows a weak anomalous diffusive behavior

General theory

Quantum Langevin equation

Applications