Geometry and Dynamics of Gaussian Wave Packets and Their Wigner Transforms




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Amer Inst Physics


We find a relationship between the dynamics of the Gaussian wave packet and the dynamics of the corresponding Gaussian Wigner function from the Hamiltonian/symplectic point of view. The main result states that the momentum map corresponding to the natural action of the symplectic group on the Siegel upper half space yields the covariance matrix of the corresponding Gaussian Wigner function. This fact, combined with Kostant's coadjoint orbit covering theorem, establishes a symplectic/Poisson-geometric connection between the two dynamics. The Hamiltonian formulation naturally gives rise to corrections to the potential terms in the dynamics of both the wave packet and the Wigner function, thereby resulting in slightly different sets of equations from the conventional classical ones. We numerically investigate the effect of the correction term and demonstrate that it improves the accuracy of the dynamics as an approximation to the dynamics of expectation values of observables.



Quantum theory, Correlation (Statistics), Hamiltonian mechanics, Wave mechanics, Coherent states, Differential geometry, Integral transforms, Riemannian manifolds, Vector fields

Leverhulme Trust Research Project Grant No. 2014-112 and by London Mathematical Society Grant No. 31320


CC BY 4.0 (Attribution), ©2017 The Authors