Development and Numerical Analysis of "Black-Box" Counterpropagating Wave Algorithm for Exact Quantum Scattering Calculations

Abstract

] a bipolar counter-propagating wave decomposition, Ψ = Ψ+ + Ψ−, was presented for stationary bound states Ψ of the one-dimensional Schrödinger equation, such that the components Ψ± approach their semiclassical WKB analogs in the large action limit. The corresponding bipolar quantum trajectories are classical-like and well-behaved, even when Ψ has many nodes, or is wildly oscillatory. In this paper, the earlier results are used to construct a universal "black-box" algorithm, numerically robust, stable and efficient, for computing accurate scattering quantities of any quantum dynamical system in one degree of freedom.

Reconciling Semiclassical and Bohmian Mechanics: IV. Multisurface Dynamics

“…This yields somewhat complicated results, analogous to Ref. 4 Eq. (12), which are excluded here for the sake of brevity.…”

“…3 and Ref. 4 [basically, constructing a convective term to get rid of first-order spatial derivatives of the wavefunction in the hydrodynamic frame, and introducing an explicit time dependence via ∂Ψ ± /∂t = −(i/h)EΨ ± ], we are then led to the following coupled hydrodynamic (Lagrangian) time evolution equations:…”

Bohmian mechanics without pilot waves

“…However, one drawback that still remains in the present approach is that the density of grid points (if not their spatial extent) may still need to be high, in order to accommodate interference oscillations. This latter difficulty might be ameliorated in a bipolar treatment [49,53,58], as may be explored in the future.…”

Efficient quantum trajectory representation of wavefunctions evolving in imaginary time

“…(5). The bipolar wavefunction representation has the capability of reproducing QM interference via superposition of multiple subwavepackets, 34,35 but is more complicated in concept and in practice (especially in high dimensions).…”