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 A

Direct trajectory method for semiclassical wave functions

Shuangbo Yang, Michael E. Kellman

Published Jul 13, 2000• 7 citations
Publisher

Abstract

This paper reports a method to build a semiclassical wave function corresponding to an invariant torus satisfying Einstein-Brillouin-Keller quantization conditions. Instead of calculating the stability matrix of the trajectory at each step, as in the standard method of Keller [Ann. Phys. (N.Y.) 4, 180 (1958)] or the modification of Maslov and Fedoriuk [Semiclassical Approximations in Quantum Mechanics (Reidel, Dordrecht, 1981)], we use the actual density of the trajectory, calculated by running the trajectory and counting passages through cells in coordinate space. The method is tested for a system of coupled Morse oscillators, and found to be comparable in accuracy to the standard method. It may be more useful than the standard method for testing ideas for semiclassical quantization in the chaotic regime.

DOI
10.1103/physreva.62.022105

Related papers

Addendum to “Direct trajectory method for semiclassical wave functions”

Shuangbo Yang, Michael E. Kellman

For two kinetically coupled Morse oscillators, we present details of the method of calculation in Ref. 1 of the semiclassical wave function for an Einstein-Brillouin-Keller quantizing torus, using a single trajectory and its stability matrix. This is the ``standard'' method, contrasted in Ref. 1 with our new ``direct'' method, which does not require the calculation of the stability matrix.

Physical Review A•Feb 5, 2002•11 citations

Semiclassical wave function near a strong resonance

Shuangbo Yang, Michael E. Kellman

A method for Einstein-Brillouin-Keller (EBK) semiclassical wave functions is applied to the case where the quantizing torus is close to a chain of resonance islands, and standard EBK wave functions are distorted by the resonance islands. The method tested here is a modification of a recent ``direct trajectory'' implementation of the standard EBK method. The phase factor of the semiclassical wave function is obtained exactly as by Yang and Kellman [Phys. Rev. A 62, 022105 (2000); 65, 034103 (2002)], by running a single trajectory on the quantizing torus. However, rather than obtaining the amplitude factor from this trajectory, the method here uses an ensemble of trajectories uniformly distributed in a region enclosing an area \ensuremath{\Elzxh} around the torus. The agreement between the resulting semiclassical wave function and the quantum wave function is excellent.

Physical Review A•Jun 5, 2002•3 citations

Semiclassical quantization via adiabatic invariance of classical action variables

Tasko P. Grozdanov, Subhash Saini, Howard S. Taylor

The semiclassical method based on the Einstein-Brillouin-Keller quantization of invariant phase-space tori and the hypothesis of adiabatic invariance of classical action variables is applied in the calculation of highly excited energy levels of two-dimensional coupled oscillator systems. The results obtained are in good agreement with quantum-mechanical calculations even in the regions where the corresponding classical dynamics becomes apparently irregular. General limitations of the method are discussed.

Physical Review A•Jan 1, 1986•39 citations

Semiclassical Quantization in Many Dimensions

Roy D. Williams

We examine the semiclassical limit of the quantum energy spectrum in many dimensions: by means of a WKB-like ansatz leading to Einstein-Brillouin-Keller (EBK) quantization, by means of a path integral, hence associating a bound state with a particular classical periodic trajectory, and by the Birkhoff-Gustavson (BG) transformation to action-angle variables. We extend the EBK method to many-fermion systems using coherent states; and apply both EBK using surfaces of section, and the BG transformation to an SU(3) schematic nuclear shell model. We describe a new algorithm for finding periodic trajectories of a Lagrangian system with polynomial potential. It is applied to the Henon-Heiles system with good results, and these trajectories are used to quantize the system. The EBK and BG methods have some success, while periodic trajectory quantization fails. We discuss possible reasons for this failure and future approaches to these problems.

•Oct 31, 2019•0 citations

Multidimensional wave functions from classical trajectories

Michael J. Davis, Eric J. Heller

A new technique is developed to generate semiclassical wave functions. The method uses only information already available from a standard semiclassical quantization of a system. Linear superpositions of Gaussian coherent states that lie along quantizing classical trajectories are used, with phases given by the action integrals plus a Maslov-type correction. Wave functions generated in this way suffer from none of the problems with caustics that primitive semiclassical wave functions encounter. The semiclassical wave functions are convenient for subsequent use in applications, e.g., molecular spectra. By generating wave functions for several simple systems, we show that under most circumstances these wave functions are very accurate approximations to the true quantum states.

The Journal of Chemical Physics•Oct 15, 1981•98 citations

Cellular dynamics: A new semiclassical approach to time-dependent quantum mechanics

Eric J. Heller

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.

The Journal of Chemical Physics•Feb 15, 1991•393 citations
This paper reports a method to build a semiclassical wave function corresponding to an invariant torus satisfying Einstein-Brillouin-Keller quantization conditions. Instead of calculating the stability matrix of the trajectory at each step, as in the standard method of Keller [Ann. Phys. (N.Y.) 4, 180 (1958)] or the modification of Maslov and Fedoriuk [Semiclassical Approximations in Quantum Mechanics (Reidel, Dordrecht, 1981)], we use the actual density of the trajectory, calculated by running the trajectory and counting passages through cells in coordinate space. The method is tested for a system of coupled Morse oscillators, and found to be comparable in accuracy to the standard method. It may be more useful than the standard method for testing ideas for semiclassical quantization in the chaotic regime.
A method for Einstein-Brillouin-Keller (EBK) semiclassical wave functions is applied to the case where the quantizing torus is close to a chain of resonance islands, and standard EBK wave functions are distorted by the resonance islands. The method tested here is a modification of a recent ``direct trajectory'' implementation of the standard EBK method. The phase factor of the semiclassical wave function is obtained exactly as by Yang and Kellman [Phys. Rev. A 62, 022105 (2000); 65, 034103 (2002)], by running a single trajectory on the quantizing torus. However, rather than obtaining the amplitude factor from this trajectory, the method here uses an ensemble of trajectories uniformly distributed in a region enclosing an area \ensuremath{\Elzxh} around the torus. The agreement between the resulting semiclassical wave function and the quantum wave function is excellent.
The semiclassical method based on the Einstein-Brillouin-Keller quantization of invariant phase-space tori and the hypothesis of adiabatic invariance of classical action variables is applied in the calculation of highly excited energy levels of two-dimensional coupled oscillator systems. The results obtained are in good agreement with quantum-mechanical calculations even in the regions where the corresponding classical dynamics becomes apparently irregular. General limitations of the method are discussed.
We examine the semiclassical limit of the quantum energy spectrum in many dimensions: by means of a WKB-like ansatz leading to Einstein-Brillouin-Keller (EBK) quantization, by means of a path integral, hence associating a bound state with a particular classical periodic trajectory, and by the Birkhoff-Gustavson (BG) transformation to action-angle variables. We extend the EBK method to many-fermion systems using coherent states; and apply both EBK using surfaces of section, and the BG transformation to an SU(3) schematic nuclear shell model. We describe a new algorithm for finding periodic trajectories of a Lagrangian system with polynomial potential. It is applied to the Henon-Heiles system with good results, and these trajectories are used to quantize the system. The EBK and BG methods have some success, while periodic trajectory quantization fails. We discuss possible reasons for this failure and future approaches to these problems.
A new technique is developed to generate semiclassical wave functions. The method uses only information already available from a standard semiclassical quantization of a system. Linear superpositions of Gaussian coherent states that lie along quantizing classical trajectories are used, with phases given by the action integrals plus a Maslov-type correction. Wave functions generated in this way suffer from none of the problems with caustics that primitive semiclassical wave functions encounter. The semiclassical wave functions are convenient for subsequent use in applications, e.g., molecular spectra. By generating wave functions for several simple systems, we show that under most circumstances these wave functions are very accurate approximations to the true quantum states.
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.