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Theoretical and computational study of time dependent scattering on a 2D surface

M Sohn

Published Mar 20, 2020• 0 citations
Publisher

Abstract

The quantum mechanical treatment of the elastic scattering of atoms from a crystal surface provides valuable information, such as surface properties and gas-surface interaction potentials. However, since it is based on the stationary state solution, it does not provide the details of the scattering process in the neighborhood of the surface, especially when atoms are physically adsorbed. In this thesis, the time evolution of the scattering process is treated in 2D with a model potential, V(x, z) = -|g|δ(z) + λδ(z)cos(2πx/a), using the Gaussian wave packet approach. The focus is on the case where the Gaussian wave packet makes a transition into a selective adsorption state because it can provide information on the probability density of selectively adsorbed particles as well as the details of the scattering process in the neighborhood of the surface. The obtained Gaussian wave packet solution shows a transition into a selective adsorption state. However, the probability density of selectively adsorbed particles cannot be accurately determined because the Gaussian wave packet constructed from the Born approximate time-independent wave function does not conserve the total probability density.

DOI
10.34917/1348560

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The quantum mechanical treatment of the elastic scattering of atoms from a crystal surface provides valuable information, such as surface properties and gas-surface interaction potentials. However, since it is based on the stationary state solution, it does not provide the details of the scattering process in the neighborhood of the surface, especially when atoms are physically adsorbed. In this thesis, the time evolution of the scattering process is treated in 2D with a model potential, V(x, z) = -|g|δ(z) + λδ(z)cos(2πx/a), using the Gaussian wave packet approach. The focus is on the case where the Gaussian wave packet makes a transition into a selective adsorption state because it can provide information on the probability density of selectively adsorbed particles as well as the details of the scattering process in the neighborhood of the surface. The obtained Gaussian wave packet solution shows a transition into a selective adsorption state. However, the probability density of selectively adsorbed particles cannot be accurately determined because the Gaussian wave packet constructed from the Born approximate time-independent wave function does not conserve the total probability density.
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.
We develop methodology for performing time dependent quantum mechanical calculations by representing the wave function as a sum of Gaussian wave packets (GWP) each characterized by a set of parameters such as width, position, momentum, and phase. The problem of computing the time evolution of the wave function is thus reduced to that of finding the time evolution of the parameters in the Gaussians. This parameter motion is determined by minimizing the error made by replacing the exact wave function in the time dependent Schrödinger equation with its Gaussian representation approximant. This leads to first order differential equations for the time dependence of the parameters, and those describing the packet position and the momentum of each packet have some resemblance with the classical equations of motion. The paper studies numerically the strategy needed to achieve the best GWP representation of time dependent processes. The issues discussed are the representation of the initial wave function, the numerical stability and the solution of the differential equations giving the evolution of the parameters, and the analysis of the final wave function. Extensive comparisons are made with an approximate method which assumes that the Gaussians are independent and their width is smaller than the length scale over which the potential changes. This approximation greatly simplifies the calculations and has the advantage of a greater resemblance to classical mechanics, thus being more intuitive. We find, however, that its range of applications is limited to problems involving localized degrees of freedom that participate in the dynamic process for a very short time. Finally we give particular attention to the notion that the GWP representation of the wave function reduces the dynamics of one quantum degree of freedom to that of a set of pseudoparticles (each represented by one packet) moving according to a pseudoclassical (i.e., classical-like) mechanics whose ‘‘phase space’’ is described by a position and momentum as well as a complex phase and width.
A wave-packet formalism is presented for obtaining the differential cross section for the elastic scattering of a single particle by a potential. The method avoids difficulties inherent in the more common formulations such as interference between incident and scattered waves, box normalization, and density of final states. The time sequence of events in the scattering process is pictured in an intuitively appealing manner. The initial form of the wave packet is specified explicitly, and the time development of the wave packet is determined from the time-translation operator. The asymptotic form of the wave packet is investigated after a sufficiently long time lapse, and the cross section is obtained from a calculation of particle flux densities.
The scattering process of wave packets is often described recovering the simple monochromatic scheme supposing that the particle comes from −∞ with a sharp momentum distribution, thus allowing for approximations usually used in common calculations of physical quantities involved in the process. In this work, we study the scattering process of a Gaussian wave packet impinging on the step potential, finding out that its dynamics depends on the initial conditions of the incident particle, and in particular on the wave packet origin x0. We propose a semi-classical approximated model to describe the dynamics of the scattering wave packet, also defining a characteristic time interval tf as the time required for the formation of transmitted packet beyond the step. Through a comparison with the numerical solution of the SchrÖdinger equation, our model explains the tf as a function of the origin x0 and spread of the incident wave packet in coordinate space, giving rise to such dependence for a finite distance scattering processes.
Atom-laser interaction is at the heart of the vibrant field of quantum optics. The dynamical properties of a moving atom submitted to a laser radiation are strongly \ninfluenced by the position- and time-dependence of the latter. The full physical state of the atom must include information about the centre-of-mass motion and the internal structure of the atom. It is challenging to obtain a complete description of the full state of the atom. However, there exists an alternative approach to the dynamics of an atom in the presence of a laser, which is based on the concept of matter-wave absorption. \n \nIn this thesis, we theoretically study the nonrelativistic one-dimensional motion of an electrically neutral quantum particle in the presence of a thin time-dependent absorber, representing the laser radiation. Our aim is to better understand the precise connection between time-dependent matter-wave absorption and the interaction between an atom and a localised time-dependent laser. \n \nOur analysis is based on two different approaches to the problem. The first one describes the moving atom by a two-level system, and represents the laser radiation \nby an off-diagonal δ-potential. The second model treats the atom as a structureless particle, and describes the laser by a time-dependent absorbing barrier. While the \nformer model can be derived from first principles, the treatment of the absorbing barrier in the latter model lacks a rigorous quantum mechanical justification. The \nmain outcome of our work is to provide a solid physical ground for the absorber model, by explicitly connecting it to the δ-potential model. We have thus extended \nthe range of theoretical tools useful for investigating the effects of time-dependent laser radiation on quantum matter.