Symplectic semiclassical wave packet dynamics

The paper gives a symplectic-geometric account of semiclassical Gaussian wave packet dynamics. We employ geometric techniques to “strip away” the symplectic structure behind the time-dependent Schrödinger equation and incorporate it into semiclassical wave packet dynamics. We show that the Gaussian wave packet dynamics of Heller is a Hamiltonian system with respect to the symplectic structure, apply the theory of symplectic reduction and reconstruction to the dynamics, and discuss dynamic and geometric phases in semiclassical mechanics. We also propose an asymptotic approximation of the potential term that provides a practical semiclassical correction term to the approximation by Heller. A simple harmonic oscillator example is worked out to illustrate the results, along with the canonical and action–angle coordinates for the system. Finally, we look into the geometry behind the Hagedorn parametrization of Gaussian wave packet dynamics.