Numerical implementation of quantum fluid dynamics: a working example

In the last years new interest has been addressed to the fluid dynamical formulation of quantum mechanics of Madelung-De Broglie-Bohm. Various attempts to implement numerically this formulation have been presented recently concerning photodissociation and scattering, sharing the use of the lagrangian and semi-lagrangian method to discretize the spatial independent variables. The possibility of using moving numerical grids suited for wave packet propagation, in general computationally cheaper than fixed grids, is attractive but difficult when QFD is applied to molecular scattering. This aspect is discussed in the paper, and a particular solution of the problem is proposed. The result for transmission probability through the Eckart barrier as a function of momentum is obtained by Fourier transforming the final wave packet; the agreement with analytical results is good on a large range, the computational time is reasonable and the algorithm used is relatively simple in comparison to other ones proposed for solving this problem.