Quantum fluid dynamics in the Lagrangian representation and applications to photodissociation problems

Abstract

This paper considers the practical utility of quantum fluid dynamics ͑QFD͒ whereby the time-dependent Schrödinger's equation is transformed to observing the dynamics of an equivalent ''gas continuum.'' The density and velocity of this equivalent gas continuum are respectively the probability density and the gradient of the phase of the wave function. The numerical implementation of the QFD equations is carried out within the Lagrangian approach, which transforms the solution of Schrödinger's equation into following the trajectories of a set of mass points, i.e., subparticles, obtained by discretization of the continuum equations. The quantum dynamics of the subparticles which arise in the present formalism through numerical discretization are coupled by the density and the quantum potential. Numerical illustrations are performed for photodissociation of NOCl and NO 2 treated as two-dimensional models. The dissociation cross sections ͑ ͒ are evaluated in the dramatically short CPU times of 33 s for NOCl and 40 s for NO 2 on a Pentium-200 MHz PC machine. The computational efficiency comes from a combination of ͑a͒ the QFD representation dealing with the near monotonic amplitude and phase as dependent variables, ͑b͒ the Lagrangian description concentrating the computation effort at all times into regions of highest probability as an optimal adaptive grid, and ͑c͒ the use of an explicit time integrator whereby the computational effort grows only linearly with the number of discrete points.

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

“…If done appropriately, such a decomposition can lead to very important ramifications for quantum trajectory methods (QTMs) 5,6,7,8,9,10,11,12,13,14 -i.e., trajectorybased numerical techniques for performing exact quantum dynamics calculations, based on Bohmian mechanics 15,16,17,18,19,20 -due to nonlinearity of the Bohmian equations of motion. In particular, the earlier series of articles has culminated in a set of trajectorybased time-dependent methods for computing stationary scattering quantities (the theoretical underpinning of all chemical reactions) in one degree of freedom (DOF).…”

Quantum tunneling dynamics using hydrodynamic trajectories

“…We have found that the convergence of the least-squares procedure is greatly improved by using a logarithmic form of the density 15,17,19 ,…”

“…While there are a number of papers which have computed quantum trajectories having obtained the wave function, relatively little work has been done in developing computational methods based upon the description which does not rely upon first constructing the wave function. [10][11][12][13][14][15] However, Wyatt and co-workers have recently described a mesh-less finite-element method for integrating the de Broglie-Bohm equations using Lagrangian hydrodynamic trajectories. [16][17][18] Similar methods are widely used in computational fluid dynamics (CFD) to simulate fluid flow dynamics in porous media (such as an oil reservoir) and other systems with complex topologies.…”

Flux continuity and probability conservation in complexified Bohmian mechanics

“…More recently, innova- * Electronic address: Email: Bill.Poirier@ttu.edu tions spearheaded by members of the chemical physics community have led to the use of quantum trajectory methods (QTMs) as a "synthetic" tool, i.e. to solve the time-dependent Schrödinger equation (TDSE) itself [10,11,12].…”