Hydrodynamic quantum trajectories

In this work we invoke the de Broglie-Bohm interpretation of quantum mechanics as a foundation for developing a series of computationally efficient strategies that numerically describe the time evolution of a quantum mechanical system in terms of a finite ensemble of hydrodynamic trajectories. The de Broglie-Bohm formulation is expanded to incorporate the dynamics of a tagged quantum mechanical system immersed in a dissipative heat bath. This extension involves a hydrodynamic reformulation of the Caldeira-Leggett master equation for the reduced quantum density matrix and provides a novel interpretation of the interplay between energy relaxation, quantum coherence, and the classical limits of quantum mechanics. Using the Wigner phase space representation, we formulate several hierarchies of equations motion for the hydrodynamic moments and cumulants of the quantum density matrix. We show that in the presence of a dissipative medium, the hierarchy of density cumulants can be truncated to low order. This semiclassical approximation is based upon the rapid decay of quantum coherence information in systems at high temperature and can be extended to the low-temperature regime by introducing an ad hoc quantum coherence length-scale. Finally, we present a completely different methodology, based upon Bayesian probability theory, for simulating the ground state properties of a multi-dimensional system by estimating the quantum distribution function and quantum force associated with a statistical ensemble of hydrodynamic quantum trajectories.