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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##### Article:Computing Semiclassical Quantum Dynamics with Hagedorn Wavepackets.
SIAM J. Scientific Computing. 01/2009; 31:3027-3041.
• ##### Chapter:Numerical Integrators for Highly Oscillatory Hamiltonian Systems: A Review
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ABSTRACT: Numerical methods for oscillatory, multi-scale Hamiltonian systems are reviewed. The construction principles are described, and the algorithmic and analytical distinction between problems with nearly constant high frequencies and with time- or state-dependent frequencies is emphasized. Trigonometric integrators for the first case and adiabatic integrators for the second case are discussed in more detail.
01/1970: pages 553-576;

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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