An iterative finite difference method for solving the quantum hydrodynamic equations of motion

A new stable computational method is presented for the direct numerical solution of the quantum hydrodynamic equations of motion associated with the de Broglie-Bohm formulation of quantum mechanics. The methodology is based on an iterative solution of the coupled non-linear hydrodynamic equations using a finite difference approximation for evaluating the numerical derivatives. The method is 2nd-order accurate in both space and time and exhibits exponential convergence with respect to the iteration count. The method is applied to a one-dimensional Gaussian wave packet: freely moving and scattering from an Eckart barrier. Excellent agreement is obtained between the numerical and analytic solutions for the free Gaussian wave packet. Accurate transmission probabilities are computed for the Eckart scattering problem which are in excellent agreement with two other computational methods. This method represents the first successful implementation of a stable finite difference approach for directly solving the quantum hydrodynamic equations of motion.