Geometry and Dynamics of Gaussian Wave Packets and their Wigner Transforms

Abstract

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.

23 pages

References in corpus (5)

• Wave packet evolution in non-Hermitian quantum systems
• The geodesic Vlasov equation and its integrable moment closures
• The Siegel Upper Half Space is a Marsden-Weinstein Quotient: Symplectic Reduction and Gaussian Wave Packets
• Symmetry and Conservation Laws in Semiclassical Wave Packet Dynamics
• Hamiltonian approach to Ehrenfest expectation values and Gaussian quantum states

Cited by in corpus (8)

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• Hilbert geometry of the Siegel disk: The Siegel-Klein disk model
• High-order geometric integrators for the local cubic variational Gaussian wavepacket dynamics
• Approximation of Semiclassical Expectation Values by Symplectic Gaussian Wave Packet Dynamics
• The frame bundle picture of Gaussian wave packet dynamics in semiclassical mechanics
• Quantum Polar Duality and the Symplectic Camel: a Geometric Approach to Quantization