Quantum wave packet dynamics with trajectories: Implementation with distributed approximating functionals

The quantum trajectory method (QTM) was recently developed to solve the hyd rodynamic equations of motion in the Lagrangian, moving-with-the-fluid, pic ture. In this approach, trajectories are integrated for N fluid elements (p articles) moving under the influence of both the force from the potential s urface and from the quantum potential. In this study, distributed approxima ting functionals (DAFs) are used on a uniform grid to compute the necessary derivatives in the equations of motion. Transformations between the physic al grid where the particle coordinates are defined and the uniform grid are handled through a Jacobian, which is also computed using DAFs. A difficult problem associated with computing derivatives on finite grids is the edge problem. This is handled effectively by using DAFs within a least squares a pproach to extrapolate from the known function region into the neighboring regions. The QTM-DAF is then applied to wave packet transmission through a one-dimensional Eckart potential. Emphasis is placed upon computation of th e transmitted density and wave function. A problem that develops when part of the wave packet reflects back into the reactant region is avoided in thi s study by introducing a potential ramp to sweep the reflected particles aw ay from the barrier region. (C) 2000 American Institute of Physics. [S0021- 9606(00)00224-5].