Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Physical Review Letters

Semiclassical theory of quantum propagation: The Coulomb potential

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

Published Sep 27, 1993• 43 citations
Publisher

Abstract

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.

DOI
10.1103/physrevlett.71.1961

Related papers

Classical orbits and semiclassical wavepacket propagation in the Coulomb potential

I. M. Suarez Barnes, Michael Nauenberg, Mark Douglas Nockleby +1 more

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.

Journal of Physics A Mathematical and General•May 7, 1994•20 citations

Regularized semiclassical radial propagator for the Coulomb potential

Robert S. Manning, Gregory S. Ezra

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.

Physical Review A•Aug 1, 1994•17 citations

Semiclassical dynamics with quantum trajectories: Formulation and comparison with the semiclassical initial value representation propagator

Sophya V. Garashchuk, Vitaly A. Rassolov

We present a time-dependent semiclassical method based on quantum trajectories. Quantum-mechanical effects are described via the quantum potential computed from the wave function density approximated as a linear combination of Gaussian fitting functions. The number of the fitting functions determines the accuracy of the approximate quantum potential (AQP). One Gaussian fit reproduces time-evolution of a Gaussian wave packet in a parabolic potential. The limit of the large number of fitting Gaussians and trajectories gives the full quantum-mechanical result. The method is systematically improvable from classical to fully quantum. The fitting procedure is implemented as a gradient minimization. We also compare AQP method to the widely used semiclassical propagator of Herman and Kluk by computing energy-resolved transmission probabilities for the Eckart barrier from the wave packet time-correlation functions. We find the results obtained with the Herman–Kluk propagator to be essentially equivalent to those of AQP method with a one-Gaussian density fit for several barrier widths.

The Journal of Chemical Physics•Jan 30, 2003•54 citations

Nearly real trajectories in complex semiclassical dynamics

Troy Van Voorhis, Eric J. Heller

We introduce a very general approximation to the quantum propagator that is based on the assumption that the most important contributions to the complex semiclassical propagator evolve from real classical trajectories that almost satisfy the desired boundary conditions. Our results for two systems --- the autocorrelation function for the quartic anharmonic oscillator and the photodissociation spectrum of ${\mathrm{CO}}_{2}$ --- show that these nearly real contributions yield an excellent approximation to the quantum propagator for quite long times. The approach taken here is applicable to problems with many (e.g., several hundred) degrees of freedom, and hence promises to provide an accurate and useful representation of the quantum dynamics for a wide variety of physically interesting systems.

Physical Review A•Nov 4, 2002•36 citations

Complex trajectory method in semiclassical propagation of wave packets

Marc Boiron, M. Lombardi

We propose a semiclassical wave packet propagation method relying on classical trajectories in a complex phase space. It is based on the Schrödinger wave equation and the usual expansion with respect to ℏ, except that the amplitude of the wave packet is taken into account at the very zeroth order, unlike in the usual WKB method where it is treated as a corrective or first order term. Formally, it amounts to making both the wavelength and the width of the wave packet tend to zero with ℏ. The action and consequently the classical trajectories derived are complex. This method is tested successfully in many cases, analytically or numerically, including the bounce and even the splitting of the wave packet. Our method appears to be much more accurate than the WKB method while less computationally demanding than the Van-Vleck formula. Moreover, it has a particularly interesting property: the singularities (caustics) of the usual semiclassical theories do not appear in this formalism in all cases tested.

The Journal of Chemical Physics•Mar 1, 1998•32 citations

A regular semiclassical approximation for the propagation of wave packets with complex trajectories

Fernando Parisio, Marcus A. M. de Aguiar

The semiclassical propagation of Gaussian wave packets by complex classical trajectories involves multiple contributing and noncontributing solutions interspersed by phase space caustics. Although the phase space caustics do not generally lie exactly on the relevant trajectories, they might strongly affect the semiclassical evolution depending on their proximity to them. In this paper, we derive a third-order regular semiclassical approximation which correctly accounts for the caustics and which is finite everywhere. We test the regular formula for the potential V ( x ) = 1/ x 2 , where the complex classical trajectories and phase space caustics can be computed analytically. We make a detailed analysis of the structure of the complex functions involved in the saddle point approximations and show how the changes in the steepest descent integration contour control both the contributing and noncontributing trajectories and the type of Airy function that appears in the regular approximation.

Journal of Physics A Mathematical and General•Oct 5, 2005•14 citations
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.
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.
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.
We present a time-dependent semiclassical method based on quantum trajectories. Quantum-mechanical effects are described via the quantum potential computed from the wave function density approximated as a linear combination of Gaussian fitting functions. The number of the fitting functions determines the accuracy of the approximate quantum potential (AQP). One Gaussian fit reproduces time-evolution of a Gaussian wave packet in a parabolic potential. The limit of the large number of fitting Gaussians and trajectories gives the full quantum-mechanical result. The method is systematically improvable from classical to fully quantum. The fitting procedure is implemented as a gradient minimization. We also compare AQP method to the widely used semiclassical propagator of Herman and Kluk by computing energy-resolved transmission probabilities for the Eckart barrier from the wave packet time-correlation functions. We find the results obtained with the Herman–Kluk propagator to be essentially equivalent to those of AQP method with a one-Gaussian density fit for several barrier widths.
We introduce a very general approximation to the quantum propagator that is based on the assumption that the most important contributions to the complex semiclassical propagator evolve from real classical trajectories that almost satisfy the desired boundary conditions. Our results for two systems --- the autocorrelation function for the quartic anharmonic oscillator and the photodissociation spectrum of CO2{\mathrm{CO}}_{2}CO2​ --- show that these nearly real contributions yield an excellent approximation to the quantum propagator for quite long times. The approach taken here is applicable to problems with many (e.g., several hundred) degrees of freedom, and hence promises to provide an accurate and useful representation of the quantum dynamics for a wide variety of physically interesting systems.
We propose a semiclassical wave packet propagation method relying on classical trajectories in a complex phase space. It is based on the Schrödinger wave equation and the usual expansion with respect to ℏ, except that the amplitude of the wave packet is taken into account at the very zeroth order, unlike in the usual WKB method where it is treated as a corrective or first order term. Formally, it amounts to making both the wavelength and the width of the wave packet tend to zero with ℏ. The action and consequently the classical trajectories derived are complex. This method is tested successfully in many cases, analytically or numerically, including the bounce and even the splitting of the wave packet. Our method appears to be much more accurate than the WKB method while less computationally demanding than the Van-Vleck formula. Moreover, it has a particularly interesting property: the singularities (caustics) of the usual semiclassical theories do not appear in this formalism in all cases tested.
The semiclassical propagation of Gaussian wave packets by complex classical trajectories involves multiple contributing and noncontributing solutions interspersed by phase space caustics. Although the phase space caustics do not generally lie exactly on the relevant trajectories, they might strongly affect the semiclassical evolution depending on their proximity to them. In this paper, we derive a third-order regular semiclassical approximation which correctly accounts for the caustics and which is finite everywhere. We test the regular formula for the potential V ( x ) = 1/ x 2 , where the complex classical trajectories and phase space caustics can be computed analytically. We make a detailed analysis of the structure of the complex functions involved in the saddle point approximations and show how the changes in the steepest descent integration contour control both the contributing and noncontributing trajectories and the type of Airy function that appears in the regular approximation.