Hagedorn Wavepackets for the Time Dependent Schroedinger Equation

Vasile Gradinaru, SAM, ETH Zurich, Switzerland

Monday, November 3
at 16.15, HG E 1.1

The time-dependent Schroedinger equation in many space dimensions governs the quantum molecular dynamics. The standard discretisation strategies suffer of the curse of dimensionality. The usual rescue is to give up the linearity of the equation in order to reduce the dimensionality of the problem. On the contrary, we choose the direct approach by Hagedorn-wave-packets.
Discovered in the context of theoretical semi-classical models in quantum mechanics, these generalisations of the Hermite-functions have not been put to use in computational algorithms so far. We build a time-reversible, fully explicit time-stepping algorithm to approximate the solution of the Hagedorn wave-packet dynamics. The algorithm is based on a splitting between the kinetic and potential part of the Hamiltonian operator, as well as on a splitting of the potential into its local quadratic approximation and the remainder. The algorithm is robust in the semi-classical limit and allows the treatment of multi-particle problems by thinning out the basis according to a hyperbolic cross approximation, and of high-dimensional problems by Hartree-type approximations in a moving coordinate frame.