Article

A Poisson Integrator for Gaussian Wavepacket Dynamics

Computing and Visualization in Science 04/2012; 9(2):45-55. DOI:10.1007/s00791-006-0019-8 pp.45-55

ABSTRACT We consider the variational approximation~of the time-dependent Schrödinger equation by Gaussians wavepackets. The corresponding finite-dimensional dynamical system inherits a Poisson (or non-canonically symplectic) structure from the Schrödinger equation by its construction via the Dirac–Frenkel–McLachlan variational principle. The variational splitting between kinetic and potential energy turns out to yield an explicit, easily implemented numerical scheme. This method is a time-reversible Poisson integrator, which also preserves the L
2 norm and linear and angular momentum. Using backward error analysis, we show long-time energy conservation for this splitting scheme. In the semi-classical limit the numerical approximations to position and momentum converge to those obtained by applying the Störmer–Verlet method to the classical limit system.

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Keywords

2 norm
 
angular momentum
 
backward error analysis
 
corresponding finite-dimensional dynamical system inherits
 
Dirac–Frenkel–McLachlan variational principle
 
Gaussians wavepackets
 
momentum converge
 
numerical approximations
 
potential energy
 
Schrödinger equation
 
splitting scheme
 
Störmer–Verlet method
 
time-dependent Schrödinger equation
 
time-reversible Poisson integrator
 
variational approximation~of
 
variational splitting