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

arXiv:physics/9812013 · Physical Review A

Semiclassical time-dependent propagation in three dimensions for a Coulomb potential

Gerd van de Sand, Jan Michael Rost

Published Mar 1, 1999• 19 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physreva.59.r1723

Related papers

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

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

Semiclassical theory of quantum propagation: The Coulomb potential

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

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.

Physical Review Letters•Sep 27, 1993•43 citations

Three-dimensional non-approximate Coulomb interaction between two trapped quantum particles

Nicolás Z. Lizama, Sebastian Camilo Carrasco, Jose Antonio Rogan +1 more

The two trapped quantum particles interacting problem is generalized to three dimensions, and the exact Coulomb potential is used. The system is solved by expanding the wavefunction in terms of the isotropic harmonic oscillator eigenfunctions and Hydrogen atom eigenfunctions independently, showing that each one results in a prime approximation for different domains of the normalized coupling constant of the relative interactions, suggesting that the combination of the basis is enough to build a well-suited base for the non-approximate problem. The results are compared to previous works that use a model of approximate finite-rage soft-core interaction model of the problem to give insights into the many-body states of strongly correlated ultracold bosons and fermions. We conclude that the proposed three-dimensional approach facilitates the distinction between bosons and fermions while the solutions given by the expansions better define the behavior of the particles for repulsive potentials. In addition, we discuss the substantial differences between our work and the previous approximate model.

Scientific Reports•Oct 24, 2023•1 citation

Some Quantum-Mechanical Relations in case of singular operators and potentials

Anzor Khelashvili, Teimuraz Nadareishvili

Elaboration of some fundamental relations in three dimensional quantum mechanics is considered in connection to taking into account the restriction on domains in radial distance. In such cases the boundary behavior of the radial wave function and singularity of operators at the origin of coordinates contribute to these relations. In particular, we derive the relation between the average value of operator time derivative and the time derivative of mean value of this operator, which usually is considered as the same by definition. The deviation from known result is deduced manifested by extra term, which depends on boundary behavior mentioned above. The general form for this extra term participates to the hypervirial like theorems. As a particular case, the virial theorem for a Coulomb potential is considered and correction to the Kramers sum rule is derived. Moreover the corrected Ehrenfest theorem is deduced also and its consistency with real physical picture is demonstrated.

•Jun 2, 2018•0 citations

Exact propagators for moving hard-wall potentials

M. G. E. da Luz, Bin Kang Cheng

The semiclassical approximation has been used to evaluate the exact propagators for two one-dimensional quantum systems: (a) a free particle interacting with one hard-wall potential moving with constant velocity, and (b) a free particle inside a rigid box with one wall moving uniformly in time. The results derived from these propagators are in agreement with those of the corresponding Schrodinger equations.

Journal of Physics A Mathematical and General•Sep 7, 1992•20 citations
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 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.
The two trapped quantum particles interacting problem is generalized to three dimensions, and the exact Coulomb potential is used. The system is solved by expanding the wavefunction in terms of the isotropic harmonic oscillator eigenfunctions and Hydrogen atom eigenfunctions independently, showing that each one results in a prime approximation for different domains of the normalized coupling constant of the relative interactions, suggesting that the combination of the basis is enough to build a well-suited base for the non-approximate problem. The results are compared to previous works that use a model of approximate finite-rage soft-core interaction model of the problem to give insights into the many-body states of strongly correlated ultracold bosons and fermions. We conclude that the proposed three-dimensional approach facilitates the distinction between bosons and fermions while the solutions given by the expansions better define the behavior of the particles for repulsive potentials. In addition, we discuss the substantial differences between our work and the previous approximate model.
Elaboration of some fundamental relations in three dimensional quantum mechanics is considered in connection to taking into account the restriction on domains in radial distance. In such cases the boundary behavior of the radial wave function and singularity of operators at the origin of coordinates contribute to these relations. In particular, we derive the relation between the average value of operator time derivative and the time derivative of mean value of this operator, which usually is considered as the same by definition. The deviation from known result is deduced manifested by extra term, which depends on boundary behavior mentioned above. The general form for this extra term participates to the hypervirial like theorems. As a particular case, the virial theorem for a Coulomb potential is considered and correction to the Kramers sum rule is derived. Moreover the corrected Ehrenfest theorem is deduced also and its consistency with real physical picture is demonstrated.