Time-dependent wave packet propagation using quantum hydrodynamics

Brian K. Kendrick

Dmytro Babyuk, Róbert E. Wyatt

A conceptually simple approach, the covering function method (CFM), is developed to cope with the node problem in the hydrodynamic formulation of quantum mechanics. As nodes begin to form in a scattering wave packet (detected by a monitor function), a nodeless covering wave function is added to it yielding a total function that is also nodeless. Both local and global choices for the covering function are described. The total and covering functions are then propagated separately in the hydrodynamic picture. At a later time, the actual wave function is recovered from the two propagated functions. The results obtained for Eckart barrier scattering in one dimension are in excellent agreement with exact results, even for very long propagation times t=1.2 ps. The capability of the CFM is also demonstrated for multidimensional propagation of a vibrationally excited wave packet.

J. Broeckhove, Luc Lathouwers, Piet Van Leuven

Brian K. Kendrick

The quantum hydrodynamic equations associated with the de Broglie–Bohm formulation of quantum mechanics are solved using a meshless method based on a moving least squares approach. An arbitrary Lagrangian–Eulerian frame of reference is used which significantly improves the accuracy and stability of the method when compared to an approach based on a purely Lagrangian frame of reference. A regridding algorithm is implemented which adds and deletes points when necessary in order to maintain accurate and stable calculations. It is shown that unitarity in the time evolution of the quantum wave packet is significantly improved by propagating using averaged fields. As nodes in the reflected wave packet start to form, the quantum potential and force become very large and numerical instabilities occur. By introducing artificial viscosity into the equations of motion, these instabilities can be avoided and the stable propagation of the wave packet for very long times becomes possible. Results are presented for the scattering of a wave packet from a repulsive Eckart barrier.

Chia‐Chun Chou

Claude M. Dion, A. Hashemloo, G. Rahali