Colloquium January 21
Date: Wednesday, January 21
Time: 16:00 to 17:00
Place: 455 McBryde (Commons Room)
Speaker: Vasile Gradinaru of ETH Zürich
Title: Numerical Quantum Dynamics: from Fourier to Hagedorn wave-packets
Abstract: The time-dependent Schrödinger 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 a direct approach. We show that the Fourier method, even in a sparse grid formulation is not satisfactory for the semi-classical approximation. In the context of theoretical semi-classical models in quantum mechanics, Hagedorn discovered a family of basis functions that are particularly suitable for the approximation of the highly oscillatory solutions of the quantum molecular dynamics in many dimensions. We use these basis functions in order to construct a time-reversible, fully explicit time-stepping algorithm to approximate the solution of the semi-classical nuclear time-dependent Schrödinger equation. 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.
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