A new method for wave packet dynamics: Derivative propagation along quantum trajectories

Abstract

A new method is proposed for computing the time evolution of quantum mechanical wave packets. Equations of motion for the real-valued functions C and S in the complex action S=C(r,t)+iS(r,t)/ℏ, with ψ(r,t)=exp(S), involve gradients and curvatures of C and S. In previous implementations of the hydrodynamic formulation, various time-consuming fitting techniques of limited accuracy were used to evaluate these derivatives around each fluid element in an evolving ensemble. In this study, equations of motion are developed for the spatial derivatives themselves and a small set of these are integrated along quantum trajectories concurrently with the equations for C and S. Significantly, quantum effects can be included at various orders of approximation, no spatial fitting is involved, there are no basis set expansions, and single quantum trajectories (rather than correlated ensembles) may be propagated, one at a time. Excellent results are obtained when the derivative propagation method is applied to anharmonic potentials involving barrier transmission.