Hydrodynamic quantum trajectories

Róbert E. Wyatt, Corey Jason Trahan

David A. Micha

Makoto Tsubota, Michikazu Kobayashi, Hiromitsu Takeuchi

Róbert E. Wyatt, Eric R. Bittner

Although the foundations of the hydrodynamical formulation of quantum mechanics were laid over 50 years ago, it has only been within the past few years that viable computational implementations have been developed. One approach to solving the hydrodynamic equations uses quantum trajectories as the computational tool. The trajectory equations of motion are described and methods for implementation are discussed, including fitting of the fields to gaussian clusters.

Corey Jason Trahan

Recently, in an effort to solve more realistic problems in quantum dynamics, much attention has been directed into numerically integrating the quantum hydrodynamic equations of motions (QHEM), as opposed to directly solving the time-dependent Schrödinger equation (TDSE). Such efforts have been provoked by the many numerical drawbacks encountered when solving the TDSE on a fixed-grid. In this dissertation, one trajectory method for integrating the QHEM is reviewed, and two novel trajectories methods are described. The first of these, the quantum trajectory method (QTM), was introduced in 1999 and has been used to solve many problems in quantum dynamics since then. However, severe numerical problems are encountered when this method is applied to problems that form wave function nodes. To get around this problem, new methods for numerically integrating the QHEM are needed. In the first novel method described, the arbitrary Lagrangian-Eulerian (ALE) method, particle trajectories are governed by a predetermined equation of motion that is user-supplied. The ALE method remedies inflation and compression problems encountered in the pure Lagrangian QTM. In the second new method discussed, the derivative propagating method (DPM), single quantum trajectories can be calculated one at a time, as opposed to the ensemble propagation of the QTM and ALE method. Using these two methods, new solutions to the QHEM are obtained where the QTM fails. In addition to solving the QHEM, the DPM is also used to solve the classical Klein-Kramers equation in this dissertation. This equation governs the Markovian phase space evolution of a system coupled to an environment such as a heat bath. This marks the first time single trajectories have been used to solve both the QHEM and the Klein-Kramers equations.