Integrating the quantum Hamilton-Jacobi equations by wavefront expansion and phase space analysis

In this paper we report upon our computational methodology for numerically integrating the quantum Hamilton-Jacobi equations using hydrodynamic trajec tories. Our method builds upon the moving least squares method developed by Lopreore and Wyatt [Phys. Rev. Lett. 82, 5190 (1999)] in which Lagrangian fluid elements representing probability volume elements of the wave functio n evolve under Newtonian equations of motion which include a nonlocal quant um force. This quantum force, which depends upon the third derivative of th e quantum density, rho, can vary rapidly in x and become singular in the pr esence of nodal points. Here, we present a new approach for performing quan tum trajectory calculations which does not involve calculating the quantum force directly, but uses the wavefront to calculate the velocity field usin g mv=delS, where S/(h) over bar is the argument of the wave function psi. A dditional numerical stability is gained by performing local gauge transform ations to remove oscillatory components of the wave function. Finally, we u se a dynamical Rayleigh-Ritz approach to derive ancillary equations-of-moti on for the spatial derivatives of rho, S, and upsilon. The methodologies de scribed herein dramatically improve the long time stability and accuracy of the quantum trajectory approach even in the presence of nodes. The method is applied to both barrier crossing and tunneling systems. We also compare our results to semiclassical based descriptions of barrier tunneling. (C) 2 000 American Institute of Physics. [S0021-9606(00)01144-2].