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Quantum wave packet dynamics with trajectories: Implementation with adaptive Lagrangian grids
The quantum trajectory method was recently developed to solve the hydrodynamic equations of motion in the Lagrangian, moving-with-the-fluid, picture. In this approach, trajectories are integrated for ...

Integrating the quantum Hamilton–Jacobi equations by wavefront expansion and phase space analysis

J. Chem. Phys. 113, 8888 (2000); doi:10.1063/1.1319987

Issue Date: 22 November 2000

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Eric R. Bittner
Department of Chemistry, University of Houston, Houston, Texas 77204

Robert E. Wyatt
Department of Chemistry and Biochemistry, University of Texas at Austin, Austin, Texas 78712
In this paper we report upon our computational methodology for numerically integrating the quantum Hamilton–Jacobi equations using hydrodynamic trajectories. Our method builds upon the moving least squares method developed by Lopreore and Wyatt [Phys. Rev. Lett. 82, 5190 (1999)] in which Lagrangian fluid elements representing probability volume elements of the wave function evolve under Newtonian equations of motion which include a nonlocal quantum force. This quantum force, which depends upon the third derivative of the quantum density, rho, can vary rapidly in x and become singular in the presence of nodal points. Here, we present a new approach for performing quantum trajectory calculations which does not involve calculating the quantum force directly, but uses the wavefront to calculate the velocity field using mv = [bold del]S, where S/[h-bar] is the argument of the wave function psi. Additional numerical stability is gained by performing local gauge transformations to remove oscillatory components of the wave function. Finally, we use a dynamical Rayleigh–Ritz approach to derive ancillary equations-of-motion for the spatial derivatives of rho, S, and v. The methodologies described herein dramatically improve the long time stability and accuracy of the quantum trajectory approach even in the presence of nodes. The method is applied to both barrier crossing and tunneling systems. We also compare our results to semiclassical based descriptions of barrier tunneling. ©2000 American Institute of Physics.
History: Received 13 July 2000; accepted 1 September 2000
Permalink: http://link.aip.org/link/?JCPSA6/113/8888/1
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EDITORIALLY RELATED

  1. Quantum wave packet dynamics with trajectories: Implementation with adaptive Lagrangian grids
    Robert E. Wyatt et al.
    J. Chem. Phys. 113, 8898 (2000)

KEYWORDS and PACS

Keywords
PACS
  • 03.65.Ge
    Quantum mechanics, field theories, and special relativity Quantum mechanics Solutions of wave equations: bound states
  • 03.65.Sq
    Quantum mechanics, field theories, and special relativity Quantum mechanics Semiclassical theories and applications
  • YEAR: 2000

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PUBLICATION DATA

ISSN:
0021-9606 (print)   1089-7690 (online)
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REFERENCES (27)
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