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Numerical evaluation of coherent-state path integrals with application to time-dependent problems

Joachim Stolze, Bernd Burghardt

Published Jan 1, 1999• 0 citations
Publisher

Abstract

We study the application of the coherent-state path integral as a numerical tool for wave-packet propagation. The numerical evaluation of path integrals is reduced to a matrix-vector multiplication scheme. Together with a split-operator technique we apply our method to a time-dependent double-well potential.
DOI
10.17877/de290r-2663

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The numerical evaluation of coherent-state path integrals for quantum dynamical problems is discussed for one-dimensional examples. To propagate an initial state, we use the normal and antinormal ordered coherent-state path integrals combined with a split-operator technique dividing the Hamiltonian into harmonic and anharmonic parts. For numerical purposes integrations must be approximated by quadrature formulae. This leads to a matrix multiplication scheme which is systematically tested for the double-well and Morse potentials. The method is accurate for propagation times much longer than the natural time scale of the system, and it allows for short as well as long time steps without loss of stability.
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We study the application of the semiclassical coherent-state path-integral technique to the problem of scattering. A numerical method based on initial-value trajectories and their stability information is proposed to solve the root search for the complex trajectories that determine the semiclassical propagator. Very good agreement between the numerically exact quantum and semiclassical results for the correlation function can be achieved for scattering from one-dimensional barrier as well as well-type potentials by using just one single complex trajectory at each time.
Numerical solutions of the quantum time-dependent integro-differential Schrödinger equation in a coherent state Husimi representation are investigated. Discretization leads to propagation on a grid of nonorthogonal coherent states without the need to invert an overlap matrix, with the further advantage of a sparse Hamiltonian matrix. Applications are made to the evolution of a Gaussian wave packet in a Morse potential. Propagation on a static rectangular grid is fast and accurate. Results are also presented for a moving rectangular grid, guided at its center by a mean classical path, and for a classically guided moving grid of individual coherent states taken from a Monte Carlo ensemble.