University of Illinois Urbana-Champaign

Theoretical methods for the study of quantum dynamics

Chen, Jonathan E.

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https://hdl.handle.net/2142/16094

Description

Title
Theoretical methods for the study of quantum dynamics
Author(s)
Chen, Jonathan E.
Issue Date
2010-05-19T18:34:42Z
Director of Research (if dissertation) or Advisor (if thesis)
Makri, Nancy
Doctoral Committee Chair(s)
Makri, Nancy
Committee Member(s)
  • Luthey-Schulten, Zaida A.
  • Bond, Stephen
  • Vishveshwara, Smitha
Department of Study
Chemistry
Discipline
Chemical Physics
Degree Granting Institution
University of Illinois at Urbana-Champaign
Degree Name
Ph.D.
Degree Level
Dissertation
Date of Ingest
2010-05-19T18:34:42Z
Keyword(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.
Graduation Semester
2010-5
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http://hdl.handle.net/2142/16094
Copyright and License Information
Copyright 2010 Jonathan E. Chen

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