An arbitrary Lagrangian-Eulerian approach to solving the quantum hydrodynamic equations of motion: Equidistribution with "smart" springs

Recently, the quantum trajectory method (QTM) has been utilized in solving several quantum mechanical wave packet scattering problems including barrier transmission and electronic nonadiabatic dynamics. By propagating the real-valued action and amplitude functions in the Lagrangian frame, only a fraction of the grid points needed for Eulerian fixed-grid methods are used while still obtaining accurate solutions. Difficulties arise, however, near wave function nodes and in regions of sharp oscillatory features, and because of this many quantum mechanical problems have not yet been amenable to solution with the QTM. This study proposes a hybrid of both the Lagrangian and Eulerian techniques in what is termed the arbitrary Lagrangian-Eulerian method (ALE). In the ALE method, an additional equation of motion governing the momentum of the grid points is coupled into the quantum hydrodynamic equations. These new "quasi-" Bohmian trajectories can be dynamically adapted to the emergent features of the time evolving hydrodynamic fields and are non-Lagrangian. In this study it is shown that the ALE method applied to an uphill ramp potential that was previously unsolvable by the current Lagrangian QTM not only yields stable transmission probabilities with accuracies comparable to that of a high resolution Eulerian method, but does so with a small number of grid points and for extremely long propagation times. To determine the grid point positions at each new time, an equidistribution method is used that is constructed similar to the stiffness matrix of a classical spring system in equilibrium. Each "smart" spring is dependent on a local function M(x) called the monitor function which can sense gradients or curvatures of the fields surrounding its position. To constrain grid points from having zero separation and possible overlap, a new system of equations is derived that includes a minimum separation parameter which prevents this from occurring. (C) 2003 American Institute of Physics.