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

Abstract

An exhaustive analysis of the asymptotic time dependence of wave packets in one dimension Am.The quantum trajectory method ͑QTM͒ was recently developed to solve the hydrodynamic equations of motion in the Lagrangian, moving-with-the-fluid, picture. In this approach, trajectories are integrated for N fluid elements ͑particles͒ moving under the influence of both the force from the potential surface and from the quantum potential. In this study, distributed approximating functionals ͑DAFs͒ are used on a uniform grid to compute the necessary derivatives in the equations of motion. Transformations between the physical 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 approach 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 the transmitted density and wave function. A problem that develops when part of the wave packet reflects back into the reactant region is avoided in this study by introducing a potential ramp to sweep the reflected particles away from the barrier region.

Bohmian mechanics without pilot waves

“…However, since structured grids are used to compute all field quantities, these node difficulties are anticipated to be quite minor, in comparison with other synthetic QTM calculations that use unstructured grids -i.e., along the lines of requiring small time-step sizes to handle the large quantum forces, which should be easily accommodated using adaptive algorithms. On the other hand, it should be stated that the somewhat related algorithm of Wyatt, Kouri, and Hoffman [63] does appear to face more substantial node-related difficulties.…”

“…The first is a distributed approximating functional (DAF) approach of Wyatt, Kouri, and Hoffman [62,63], in which a new coordinate transformation is applied at each time step, in order to render the (1D) trajectory grid at that particular time uniform (i.e., regular). The transformed grid is then used to perform all spatial derivatives.…”

Computational Improvements to Quantum Wave Packet ab Initio Molecular Dynamics Using a Potential-Adapted, Time-Dependent Deterministic Sampling Technique

“…This is precisely the region where novel implementations of quantum dynamics based on Bohm's interpretation [80][81][82][83][84][85][86][87][104][105][106][107][108] have trouble on account of the fact that the "quantum potential" [) (-p 2 /2m)F…”

“…When ω 0 (R QM ) is small, the accuracy of energy and gradients is not critical, and the value of the potential in such regions is obtained though interpolation. Additional interpretations that connect the sampling function in eq 7 to the Wentzel-Kramers-Brillouin (WKB) 80 semiclassical theory and also to Bohmian mechanics [80][81][82][83][84][85][86][87][104][105][106][107][108] are discussed in Appendix A.…”

Forward−Backward Quantum Dynamics for Time Correlation Functions

“…[53][54][55][56][57][58][59][60][61][62][63][64][65][66][67] The so-called Bohmian or hydrodynamic formulation of quantum mechanics 49-52 assumes a form very similar to the semiclassical approximation, and recent work 68 has discussed the apparent similarities and fundamental differences of the two formulations. Even though the wave function (or propagator) is written as an amplitude multiplied by a phase in both theories, the semiclassical method relies on cross terms between distinct trajectories in order to account for quantum interference, while the Bohmian prescription employs a single trajectory for each point in space whose motion is governed by a quantum potential.…”