Quantum Dynamics with Trajectories: Introduction to Quantum Hydrodynamics

David A. Micha

Jeremy B. Maddox

Daisuke Matsumoto, K. Hayashi, Tomonori Ida +3 more

Abstract Wyatt and coworkers introduced a new approach (Wyatt, Quantum Dynamics with Trajectories Introduction to Quantum Hydrodynamics, Springer, New York, 2005; Lopreore and Wyatt, Phys Rev Lett, 1999, 82, 5190; Wyatt et al., J Chem Phys, 2001, 114, 5113; Lopreore and Wyatt, J Chem Phys, 2002, 116, 1228; Trahan and Wyatt, J Chem Phys, 2003, 118, 4784; Trahan et al., J Chem Phys, 2005, 122, 164104; Babyuk and Wyatt, J Chem Phys, 2006, 125, 64112) which is based on quantum hydrodynamics to the quantum wavepacket dynamics and applied to several systems including tunneling and nonadiabatic transition. This approach is expected to overcome some disadvantages of the conventional wavepacket dynamics, such as order‐N problem and use of absorbing potential. We applied this method to two‐dimensional model of quantum tunneling system and compared the new approach with the conventional wavepacket simulation since the validity of this method to multidimensional systems is not well known. In the simulation, we adopted two different model potentials (Model I and Model II) and calculated time dependence of the probability density and the tunneling probability in both models, where we showed explicitly that the results of the quantum hydrodynamics are identical with those of the conventional method. Using the moving grid and the regridding algorithm in the quantum hydrodynamics, we can effectively avoid order‐N problem and use of the absorbing potential. In addition to these computational advantages, the flux vector is easily obtained by the quantum hydrodynamics and these vectors field helps us to intuitively understand detailed tunneling dynamics. © 2007 Wiley Periodicals, Inc. Int J Quantum Chem, 2007

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.