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Title:Theoretical methods for the study of quantum dynamics
Author(s):Chen, Jonathan E.
Director of Research:Makri, Nancy
Doctoral Committee Chair(s):Makri, Nancy
Doctoral Committee Member(s):Luthey-Schulten, Zaida A.; Bond, Stephen; Vishveshwara, Smitha
Department / Program:Chemistry
Discipline:Chemical Physics
Degree Granting Institution:University of Illinois at Urbana-Champaign
Degree:Ph.D.
Genre:Dissertation
Subject(s):quantum dynamics
correlation functions
semiclassical approximation
quantum trajectories
Bohmian trajectories
Monte Carlo
forward-backward semiclassical dynamics
forward-backward quantum dynamics
sign problem
Hamilton's law of varying action
Abstract:Several theoretical methods for the computation of quantum dynamical quantities are formulated, implemented, and applied, in the overall context of overcoming the exponential scaling property of quantum mechanics. The convergence of the approximate forward-backward semiclassical dynamics (FBSD) method has been accelerated by working with only the imaginary part of the real-time position and momentum time correlation functions. Simple manipulations allow the computationally favorable limit of a single factorization (“bead”) of the Boltzmann operator to yield greatly improved results. Numerical examples for a one-dimensional anharmonic oscillator and Lennard-Jones fluids are presented. Bohm's trajectory formulation of quantum mechanics is an exact mapping of the time-dependent Schrödinger equation onto classical fluid dynamics, plus one extra term responsible for all non-classical effects, the dreaded quantum potential. A computational procedure has been introduced to stably solve Bohm’s hydrodynamic equations of motion in the forward-backward quantum dynamics (FBQD) formulation of time correlation functions, and applied to one-dimensional bound systems at zero and low temperature. A key step in the formulation of this procedure was the use of Hamilton's law of varying action, a generalization of Hamilton’s principle of stationary action applicable to initial-value problems. This has resulted in the first reported synthetic computation of non-stationary quantum trajectories for a fully anharmonic oscillator. Success in one dimension has laid the groundwork for future extension to a rigorous quantum-classical scheme, which is briefly discussed. Other unpublished methods for obtaining time correlation functions and propagators in quantum dynamics are also discussed.
Issue Date:2010-05-19
URI:http://hdl.handle.net/2142/16094
Rights Information:Copyright 2010 Jonathan E. Chen
Date Available in IDEALS:2010-05-19
Date Deposited:May 2010


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