A new hybrid moving grid approach to wave packet dynamics is described. Exterior points within a nonrigid frame follow Lagrangian trajectories obtained by solving the hydrodynamic equations of motion. Internal grid points within one or more open windows follow non-Lagrangian adaptive paths. Within these windows, problems encountered with quantum trajectories near wave function nodes are circumvented by directly solving the moving path transform of the Schrödinger equation. Excellent results are obtained for evolution of the density in a double well potential even though multiple ripples develop in the density.
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→∞, the limiting relative motion becomes linear and preserves Gaussianity. With the area scale set to
ℏ/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
ℏ 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.