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American Journal of Physics

Computer modeling of resonance scattering in the time domain

Rodney C. Greenhow

Published Mar 1, 1994• 0 citations
Publisher

Abstract

Microcomputer generation of motion pictures illustrating solutions of Schrödinger’s time-dependent equation is described. The technique is used to model the resonance scattering of a Gaussian wave packet by a spherical step-potential well. The formation of transient resonance states is observed, and the decay times correlated with the widths of the resonances.

DOI
10.1119/1.17604

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Microcomputer generation of motion pictures illustrating solutions of Schrödinger’s time-dependent equation is described. The technique is used to model the resonance scattering of a Gaussian wave packet by a spherical step-potential well. The formation of transient resonance states is observed, and the decay times correlated with the widths of the resonances.
Computer integration of Schrödinger’s time-independent wave equation with graphical display of solutions provides a valuable teaching aid, and is much used in the study of bound states. Much less attention has been paid to continuum solutions although these provide a richer set of phenomena. Programs are described that integrate the one-dimensional equation for potential steps and barriers, and model impedance matching, barrier penetration, and field emission. The radial equation for spherically symmetric potentials is also integrated and graphical demonstrations of scattering phase shifts, a simple model of the Ramsauer effect, and resonance scattering are provided. The programs have been used by undergraduates to investigate resonance scattering from spherical step and Woods–Saxon potentials, and a new power law relating resonance cross-section breadths and energies has been established.
Using an analytical model potential which contains resonant and bound states, we show that the decay of the resonances can be simulated by Lindblad operators. For that purpose, the unitary time evolution of an initial Gaussian wave packet in the model potential is compared with the non-unitary time evolution, obtained by solving the Lindblad equation, of the same wave packet in a potential which coincides with the model potential in the region of interest but does not contain resonances. In the latter case, dissipative effects are accounted for by Lindblad operators which lead to phenomenological friction and diffusion constants in the equations of motion. We suggest how those constants can be determined in a non-heuristic way, being directly connected to the width of the resonance in the model potential which we calculate using the complex rotation method.
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.
A method for the solution of the time-dependent Schrödinger equation in two dimensions is presented which is based on the hydrodynamic analogy to quantum mechanics. The continuum introduced in that analogy is approximated by a finite number of particles, whose trajectories are computed. The procedure is applied here to the dynamics of a Gaussian wave packet on a two-dimensional quadratic potential surface containing a saddle point. Comparison with the known analytical solution indicates that the method is capable of accurate results with substantially less computer time required than in methods previously employed. This particular check problem was chosen because of its close relationship to the wave-packet dynamics of chemical reaction rates.