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

The Journal of Chemical Physics

Time dependent quantum propagation in phase space

Dmitrii V. Shalashilin, Mark Sheard Child

Published Dec 8, 2000• 113 citations
Publisher

Abstract

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.

DOI
10.1063/1.1322075

Related papers

Quantum dynamics using a discretized coherent state representation: An adaptive phase space method

L. Mauritz Andersson

We introduce a discretized coherent state representation (DCSR) for quantum dynamics. Expansion of a wave function in the nonorthogonal slightly overcomplete set is made with an identity operator computed using an iterative refinement method. Calculating the inverse of the overlap matrix is not necessary. The result is an accurate and efficient representation, where you only put basis functions in the region of phase space where the wave function is nonvanishing. Compared to traditional spatial grid methods, fewer grid points are needed. The DCSR can be viewed as an application of the Weyl–Heisenberg frame and extends it into a useful computational method. A scheme for fully quantum mechanical propagation is constructed and applied to the realistic problem of highly excited vibration in the heavy diatomic molecule Rb2. Compared to split-operator propagation in a conventional spatial grid, an order of magnitude longer time steps can be taken and fewer grid points are needed. The computational effort scales linearly with the number of basis functions. Nonreflecting boundary conditions are a natural property of the representation and is illustrated in a model of predissociation.

The Journal of Chemical Physics•Jul 15, 2001•33 citations

A strategy for time dependent quantum mechanical calculations using a Gaussian wave packet representation of the wave function

Shin‐ichi Sawada, Robert Heather, Bret Jackson +1 more

We develop methodology for performing time dependent quantum mechanical calculations by representing the wave function as a sum of Gaussian wave packets (GWP) each characterized by a set of parameters such as width, position, momentum, and phase. The problem of computing the time evolution of the wave function is thus reduced to that of finding the time evolution of the parameters in the Gaussians. This parameter motion is determined by minimizing the error made by replacing the exact wave function in the time dependent Schrödinger equation with its Gaussian representation approximant. This leads to first order differential equations for the time dependence of the parameters, and those describing the packet position and the momentum of each packet have some resemblance with the classical equations of motion. The paper studies numerically the strategy needed to achieve the best GWP representation of time dependent processes. The issues discussed are the representation of the initial wave function, the numerical stability and the solution of the differential equations giving the evolution of the parameters, and the analysis of the final wave function. Extensive comparisons are made with an approximate method which assumes that the Gaussians are independent and their width is smaller than the length scale over which the potential changes. This approximation greatly simplifies the calculations and has the advantage of a greater resemblance to classical mechanics, thus being more intuitive. We find, however, that its range of applications is limited to problems involving localized degrees of freedom that participate in the dynamic process for a very short time. Finally we give particular attention to the notion that the GWP representation of the wave function reduces the dynamics of one quantum degree of freedom to that of a set of pseudoparticles (each represented by one packet) moving according to a pseudoclassical (i.e., classical-like) mechanics whose ‘‘phase space’’ is described by a position and momentum as well as a complex phase and width.

The Journal of Chemical Physics•Sep 15, 1985•205 citations

Description of tunneling with the help of coupled frozen Gaussians

Dmitrii V. Shalashilin, Mark Sheard Child

A previous initial value coherent state representation is applied to Gaussian wave packet propagation on one-dimensional and two-dimensional double well potentials. The method uses a basis of trajectory guided frozen Gaussian coherent states, sampled from a Monte Carlo ensemble, for numerical solutions of the quantum time dependent integro-differential Schrödinger equation. Accurate tunneling splittings are obtained. Comparisons are also made with corresponding results obtained by the Herman–Kluk semiclassical initial value representation wave packet method.

The Journal of Chemical Physics•Jun 1, 2001•53 citations

Quantum mechanics in phase space: New approaches to the correspondence principle

Go. Torres-Vega, John H. Frederick

We present a time evolution equation that provides a novel basis for the treatment of quantum systems in phase space and for the investigation of the quantum-classical correspondence. Through the use of a generalized Husimi transform, we obtain a phase space representation of the time-dependent Schrödinger equation directly from the coordinate representation. Such an equation governs the time evolution of densities such as the Husimi density entirely in phase space, without recourse to a coordinate or momentum representation. As an application of the phase-space Schrödinger equation, we compute the eigenfunctions of the harmonic oscillator in phase space, relate these to the Husimi transform of coordinate representation eigenstates, and investigate the coherent state, its time evolution, and classical limit (ℏ→0) for the probability density generated by this state. Finally, we discuss our results as they relate to the quantum-classical correspondence, and quasiclassical trajectory simulations of quantum dynamics.

The Journal of Chemical Physics•Dec 15, 1990•111 citations

Multiscale quantum propagation using compact-support wavelets in space and time

Haixiang Wang, Ramiro Acevedo, Heather Mollé +3 more

Orthogonal compact-support Daubechies wavelets are employed as bases for both space and time variables in the solution of the time-dependent Schrodinger equation. Initial value conditions are enforced using special early-time wavelets analogous to edge wavelets used in boundary-value problems. It is shown that the quantum equations may be solved directly and accurately in the discrete wavelet representation, an important finding for the eventual goal of highly adaptive multiresolution Schrodinger equation solvers. While the temporal part of the basis is not sharp in either time or frequency, the Chebyshev method used for pure time-domain propagations is adapted to use in the mixed domain and is able to take advantage of Hamiltonian matrix sparseness. The orthogonal separation into different time scales is determined theoretically to persist throughout the evolution and is demonstrated numerically in a partially adaptive treatment of scattering from an asymmetric Eckart barrier.

The Journal of Chemical Physics•Oct 12, 2004•7 citations

Hamilton‐jacobi dynamics for the solution of time dependent quantum problems I. Formalism and wave packet propagation in one dimension

Ersin Yurtsever, Jürgen Brickmann

Abstract Two methods for the numerical integration of the time‐dependent Schrödinger equation with given initial conditions (initial wave packet) are presented. The first method (method A) is based on the Schrödinger representation of the quantum‐dynamical system while the second one (method B) is based upon the intermediate representation. In both cases the quantum dynamical equation is transformed into a system of Hamilton‐Jacobi type equations of motion as occurring in multi particle classical dynamics, i.e. standard molecular dynamics techniques can be applied for the integration. The dynamics of a minimum uncertainty Gaussian wave packet in a strongly anharmonic oscillator is taken as an example.

Berichte der Bunsengesellschaft für physikalische Chemie•Apr 1, 1994•4 citations
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.
We introduce a discretized coherent state representation (DCSR) for quantum dynamics. Expansion of a wave function in the nonorthogonal slightly overcomplete set is made with an identity operator computed using an iterative refinement method. Calculating the inverse of the overlap matrix is not necessary. The result is an accurate and efficient representation, where you only put basis functions in the region of phase space where the wave function is nonvanishing. Compared to traditional spatial grid methods, fewer grid points are needed. The DCSR can be viewed as an application of the Weyl–Heisenberg frame and extends it into a useful computational method. A scheme for fully quantum mechanical propagation is constructed and applied to the realistic problem of highly excited vibration in the heavy diatomic molecule Rb2. Compared to split-operator propagation in a conventional spatial grid, an order of magnitude longer time steps can be taken and fewer grid points are needed. The computational effort scales linearly with the number of basis functions. Nonreflecting boundary conditions are a natural property of the representation and is illustrated in a model of predissociation.
We develop methodology for performing time dependent quantum mechanical calculations by representing the wave function as a sum of Gaussian wave packets (GWP) each characterized by a set of parameters such as width, position, momentum, and phase. The problem of computing the time evolution of the wave function is thus reduced to that of finding the time evolution of the parameters in the Gaussians. This parameter motion is determined by minimizing the error made by replacing the exact wave function in the time dependent Schrödinger equation with its Gaussian representation approximant. This leads to first order differential equations for the time dependence of the parameters, and those describing the packet position and the momentum of each packet have some resemblance with the classical equations of motion. The paper studies numerically the strategy needed to achieve the best GWP representation of time dependent processes. The issues discussed are the representation of the initial wave function, the numerical stability and the solution of the differential equations giving the evolution of the parameters, and the analysis of the final wave function. Extensive comparisons are made with an approximate method which assumes that the Gaussians are independent and their width is smaller than the length scale over which the potential changes. This approximation greatly simplifies the calculations and has the advantage of a greater resemblance to classical mechanics, thus being more intuitive. We find, however, that its range of applications is limited to problems involving localized degrees of freedom that participate in the dynamic process for a very short time. Finally we give particular attention to the notion that the GWP representation of the wave function reduces the dynamics of one quantum degree of freedom to that of a set of pseudoparticles (each represented by one packet) moving according to a pseudoclassical (i.e., classical-like) mechanics whose ‘‘phase space’’ is described by a position and momentum as well as a complex phase and width.
A previous initial value coherent state representation is applied to Gaussian wave packet propagation on one-dimensional and two-dimensional double well potentials. The method uses a basis of trajectory guided frozen Gaussian coherent states, sampled from a Monte Carlo ensemble, for numerical solutions of the quantum time dependent integro-differential Schrödinger equation. Accurate tunneling splittings are obtained. Comparisons are also made with corresponding results obtained by the Herman–Kluk semiclassical initial value representation wave packet method.
We present a time evolution equation that provides a novel basis for the treatment of quantum systems in phase space and for the investigation of the quantum-classical correspondence. Through the use of a generalized Husimi transform, we obtain a phase space representation of the time-dependent Schrödinger equation directly from the coordinate representation. Such an equation governs the time evolution of densities such as the Husimi density entirely in phase space, without recourse to a coordinate or momentum representation. As an application of the phase-space Schrödinger equation, we compute the eigenfunctions of the harmonic oscillator in phase space, relate these to the Husimi transform of coordinate representation eigenstates, and investigate the coherent state, its time evolution, and classical limit (ℏ→0) for the probability density generated by this state. Finally, we discuss our results as they relate to the quantum-classical correspondence, and quasiclassical trajectory simulations of quantum dynamics.
Orthogonal compact-support Daubechies wavelets are employed as bases for both space and time variables in the solution of the time-dependent Schrodinger equation. Initial value conditions are enforced using special early-time wavelets analogous to edge wavelets used in boundary-value problems. It is shown that the quantum equations may be solved directly and accurately in the discrete wavelet representation, an important finding for the eventual goal of highly adaptive multiresolution Schrodinger equation solvers. While the temporal part of the basis is not sharp in either time or frequency, the Chebyshev method used for pure time-domain propagations is adapted to use in the mixed domain and is able to take advantage of Hamiltonian matrix sparseness. The orthogonal separation into different time scales is determined theoretically to persist throughout the evolution and is demonstrated numerically in a partially adaptive treatment of scattering from an asymmetric Eckart barrier.
Abstract Two methods for the numerical integration of the time‐dependent Schrödinger equation with given initial conditions (initial wave packet) are presented. The first method (method A) is based on the Schrödinger representation of the quantum‐dynamical system while the second one (method B) is based upon the intermediate representation. In both cases the quantum dynamical equation is transformed into a system of Hamilton‐Jacobi type equations of motion as occurring in multi particle classical dynamics, i.e. standard molecular dynamics techniques can be applied for the integration. The dynamics of a minimum uncertainty Gaussian wave packet in a strongly anharmonic oscillator is taken as an example.