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Journal of Physics A Mathematical and General

Classical orbits and semiclassical wavepacket propagation in the Coulomb potential

I. M. Suarez Barnes, Michael Nauenberg, Mark Douglas Nockleby, Steven Tomsovic

Published May 7, 1994• 20 citations
Publisher

Abstract

A simple and elegant semiclassical theory for propagating wavepackets is presented. The development of the theory avoids the explicit use of the Green-function formalism. It involves expanding the potential to a quadratic time-dependent form and the use of multiple reference trajectories. The classical analogue in phase space predicts the accuracy of the results. The theory is applied to the calculation of the autocorrelation function in a Coulomb field in one dimension. Remarkable agreement with an exact quantum calculation is obtained in many circumstances in spite of the Coulomb singularity. All the detailed quantum behaviour, including the long-time recurrences and the spectrum, is reproduced.

DOI
10.1088/0305-4470/27/9/039

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A simple and elegant semiclassical theory for propagating wavepackets is presented. The development of the theory avoids the explicit use of the Green-function formalism. It involves expanding the potential to a quadratic time-dependent form and the use of multiple reference trajectories. The classical analogue in phase space predicts the accuracy of the results. The theory is applied to the calculation of the autocorrelation function in a Coulomb field in one dimension. Remarkable agreement with an exact quantum calculation is obtained in many circumstances in spite of the Coulomb singularity. All the detailed quantum behaviour, including the long-time recurrences and the spectrum, is reproduced.
A semiclassical method is discussed for propagating wave packets that incorporates nonlinear classical dynamical information through the use of multiple reference trajectories. We apply the theory to obtain an analytic approximation for the autocorrelation function in the Coulomb potential which is extremely accurate in many circumstances in spite of the Coulomb singularity. All the detailed quantum behavior is reproduced including the long time revivals and the spectrum. The evolving phase space structure of the classical analog reveals how well a particular case will be approximated.
Quantum wavepackets follow classical orbits and spread initially with time except for the special case when the energy levels are equally spaced, e.g. the harmonic oscillator. However, for integrable systems bound localised wavepackets can exhibit at long times quantum recurrences which do not have a classical correspondence. For the Coulomb potential the recurrence time is approximately n/3 times the Kepler period 2 pi n 3 , where n is the mean value of the principal quantum numbers of the wavepacket.
A unified semiclassical time propagator is used to calculate the semiclassical time-correlation function in three Cartesian dimensions for a particle moving in an attractive Coulomb potential. It is demonstrated that under these conditions the singularity of the potential does not cause any difficulties and the Coulomb interaction can be treated as any other nonsingular potential. Moreover, by virtue of our three-dimensional calculation, we can explain the discrepancies between previous semiclassical and quantum results obtained for the one-dimensional radial Coulomb problem.
We derive a regularized semiclassical radial propagator for the Coulomb potential, a case for which standard approaches run into well-known difficulties associated with a non-Cartesian radial coordinate and a potential singularity. Following Kleinert [Path Integrals in Quantum Mechanics, Statistics and Polymer Physics (World Scientific, Singapore, 1990)], we first perform a quantum-mechanical regularization of the propagator. The semiclassical limit is then obtained by stationary phase approximation of the resulting integrals. The semiclassical propagator so derived has the standard Van Vleck--Gutzwiller form for the radial Coulomb problem with a potential correction (Langer modification) term included. The regularized semiclassical propagator is applied to compute the autocorrelation function for a Gaussian Rydberg wave packet.
A new semiclassical approach that constructs the full semiclassical Green’s function propagation of any initial wave function directly from an ensemble of real trajectories, without root searching, is presented. Each trajectory controls a cell of initial conditions in phase space, but the cell area is not constrained by Planck’s constant. The method is shown to be accurate for rather long times in anharmonic oscillators, indicating the semiclassical time-dependent Green’s function is clearly worthy of more study. The evolution of wave functions in anharmonic potentials is examined and a spectrum from the semiclassical correlation function is calculated, comparing with exact fast Fourier transform results.
An explicit solution to the Maslov-type semiclassical theory for propagating a wave function, rather than evolving in time the Feynman kernel, is presented. It turns out that the present solution bears distinguished advantages over the semiclassical kernel, one of the most remarkable examples of which is the far less number of classical trajectories required for the propagation, basically proportional to ${P}^{2N}\ensuremath{\sim}{P}^{3N}$ for the kernel while only to PN{P}^{N}PN in our solution, where N is the dimension in configuration space and P is the number of sampling points in each dimension. As an illustrative example to show the validity of the solution, the theory is applied to the calculation of eigenvalues of the Morse oscillators, giving accurate results in a compact way.