Semiclassical Dynamics Based on Quantum Trajectories

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We present a trajectory-based method that incorporates quantum effects in the context of Hamiltonian dynamics. It is based on propagation of trajectories in the presence of quantum potential within the hydrodynamic formulation of the Schrödinger equation. The quantum potential is derived from the density approximated as a linear combination of gaussian functions. One-gaussian fit gives exact result for parabolic potentials, as do successful semiclassical methods. The limit of the large number of fitting gaussians and trajectories gives the full quantum–mechanical result. The method is systematically improvable from classical to fully quantum, as demonstrated on a transmission through the Eckart barrier.