skip to main content
OSTI.GOV title logo U.S. Department of Energy
Office of Scientific and Technical Information

Title: Using semiclassical trajectories for the time-evolution of interacting quantum-mechanical systems

Abstract

We have developed a method that recasts the time-propagation of dynamic, mutually interacting quantum-mechanical wavefunctions principally as the time-evolution of many classical particles. Our approach utilizes an approximation of Feynman path integrals, known as the semiclassical method, to reduce the path integral to only the 'classical' paths connecting the wavefunction at one time step to the next. In exchange for simplifying the path sampling, each classical path's contribution gains a determinant term dependent on the path and its environment. Like virtual particles in quantum field theory, 'virtual classical particles' are said to follow these classical paths. Pushing these virtual classical particles provides the necessary data to evolve quantum wavefunctions in time. Particle-based techniques implemented on parallel computers can then be used to propagate quantum systems using this alternative method.

Authors:
 [1];  [2];  [2]
  1. Dauger Research, Inc., P.O. Box 3074, Huntington Beach, CA 92605 (United States) and UCLA Plasma Physics Group, Department of Physics and Astronomy, University of California, Los Angeles, CA 90095 (United States). E-mail: d@daugerresearch.com
  2. UCLA Plasma Physics Group, Department of Physics and Astronomy, University of California, Los Angeles, CA 90095 (United States)
Publication Date:
OSTI Identifier:
20687261
Resource Type:
Journal Article
Resource Relation:
Journal Name: Journal of Computational Physics; Journal Volume: 209; Journal Issue: 2; Other Information: DOI: 10.1016/j.jcp.2005.03.028; PII: S0021-9991(05)00187-7; Copyright (c) 2005 Elsevier Science B.V., Amsterdam, The Netherlands, All rights reserved; Country of input: International Atomic Energy Agency (IAEA)
Country of Publication:
United States
Language:
English
Subject:
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; FEYNMAN PATH INTEGRAL; LAGRANGIAN FUNCTION; MATHEMATICAL EVOLUTION; QUANTUM FIELD THEORY; QUANTUM MECHANICS; SEMICLASSICAL APPROXIMATION; TRAJECTORIES; VIRTUAL PARTICLES; WAVE FUNCTIONS

Citation Formats

Dauger, D.E., Decyk, V.K., and Dawson, J.M. Using semiclassical trajectories for the time-evolution of interacting quantum-mechanical systems. United States: N. p., 2005. Web. doi:10.1016/j.jcp.2005.03.028.
Dauger, D.E., Decyk, V.K., & Dawson, J.M. Using semiclassical trajectories for the time-evolution of interacting quantum-mechanical systems. United States. doi:10.1016/j.jcp.2005.03.028.
Dauger, D.E., Decyk, V.K., and Dawson, J.M. Tue . "Using semiclassical trajectories for the time-evolution of interacting quantum-mechanical systems". United States. doi:10.1016/j.jcp.2005.03.028.
@article{osti_20687261,
title = {Using semiclassical trajectories for the time-evolution of interacting quantum-mechanical systems},
author = {Dauger, D.E. and Decyk, V.K. and Dawson, J.M.},
abstractNote = {We have developed a method that recasts the time-propagation of dynamic, mutually interacting quantum-mechanical wavefunctions principally as the time-evolution of many classical particles. Our approach utilizes an approximation of Feynman path integrals, known as the semiclassical method, to reduce the path integral to only the 'classical' paths connecting the wavefunction at one time step to the next. In exchange for simplifying the path sampling, each classical path's contribution gains a determinant term dependent on the path and its environment. Like virtual particles in quantum field theory, 'virtual classical particles' are said to follow these classical paths. Pushing these virtual classical particles provides the necessary data to evolve quantum wavefunctions in time. Particle-based techniques implemented on parallel computers can then be used to propagate quantum systems using this alternative method.},
doi = {10.1016/j.jcp.2005.03.028},
journal = {Journal of Computational Physics},
number = 2,
volume = 209,
place = {United States},
year = {Tue Nov 01 00:00:00 EST 2005},
month = {Tue Nov 01 00:00:00 EST 2005}
}
  • A closed-form exact solution is presented for the time evolution operator of a nonrelativistic hydrogen atom driven by a circularly polarized monochromatic light beam. The solution has the form U(t,0) = exp(-itF)exp(-itG), where F and G are time independent operators. All temporal effects of the oscillating field are included. Extension to other systems is easily made. A generalization of the rotating wave approximation to multilevel systems is also presented. (auth)
  • Semiclassical approximation of real-time quantum effects is analyzed with the aid of the semiclassical initial value representation (SC-IVR) and Wigner distribution functions. We utilize these two ingredients to propose a new version of the semiclassical correlation function that contains, in principle, all quantum-mechanical effects. The advantage of this formulation is that it allows for a stepwise approximation specifically for real-time quantum effects based on a gradual inclusion of more degrees of freedom into the integral responsible for interference. From numerical calculations, this procedure does not seem to depend significantly on the choice coordinates if all degrees of freedom are coupled.more » This freedom from the coordinate choice removes possible ambiguities in applying this method. Several example cases are presented to demonstrate the usefulness of this approach. (c) 2000 American Institute of Physics.« less
  • No abstract prepared.
  • Stripping and excitation cross sections are calculated, using a time-dependent discrete-variable approach, for collisions of protons with energies from 0.5 keV to 2 MeV with He{sup +} initially in the 1s, 2s, 2p, 3s, 3p, and 3d states. This quantum-semiclassical approach takes trajectory curvature into account. The spatial and temporal convergence properties of the method are analyzed for the ground and higher states. The results are in good agreement with existing accurate calculations and experimental cross sections, available for E{sub p}{>=}3.73 keV. Results are also obtained for lower-energy collisions where the cross sections are still significant but trajectory curvature ismore » important.« less
  • The time independent semiclassical treatment of barrier tunneling has been understood for a very long time. Several semiclassical approaches to time dependent tunneling through barriers have also been presented. These typically involve trajectories for which the position variable is a complex function of time. In this paper, a method is presented that uses only real valued trajectories, thus avoiding the complications that can arise when complex trajectories are employed. This is accomplished by expressing the time dependent wave packet as an integration over momentum. The action function in the exponent in this expression is expanded to second order in themore » momentum. The expansion is around the momentum, p{sub 0{sup *}}, at which the derivative of the real part of the action is zero. The resulting Gaussian integral is then taken. The stationary phase approximation requires that the derivative of the full action is zero at the expansion point, and this leads to a complex initial momentum and complex tunneling trajectories. The “pseudo-stationary phase” approximation employed in this work results in real values for the initial momentum and real valued trajectories. The transmission probabilities obtained are found to be in good agreement with exact quantum results.« less