Variational Gaussian Wave-Packet Dynamics from Constrained Classical Trajectory Bundles

Abstract

Variational Gaussian wave-packet dynamics is usually obtained by restricting quantum evolution to Gaussian wave packets. We instead construct finite-$N$ dynamics for a labeled bundle of classical trajectories under a constraint fixing the symplectic-area scale of its covariance. At finite $N$, the bundle itself is the dynamical state: it may be non-Gaussian, and the potential force on each trajectory is evaluated from the full potential without a local harmonic approximation. For sequences whose initial empirical distributions approach a smooth Gaussian density as $N \to \infty$, the limiting relative motion becomes linear and preserves Gaussianity. With the area scale set to $\hbar/2$, the limiting centroid and width equations coincide with those of the Gaussian time-dependent variational principle, and the limiting phase-space density coincides with the corresponding Gaussian Wigner density for matched initial data. Here $N$ counts trajectories, not orders in a Moyal or $\hbar$ expansion. This phase-space correspondence reveals two complementary constructions of the same Gaussian dynamics: a variational reduction of quantum evolution and the Gaussian large-$N$ limit of a constrained classical trajectory bundle.