Classical Mechanics from Stochastic Quantum Dynamics

Classical Mechanics from Stochastic Quantum Dynamics

The present study deals with the corresponding stochastic Schrödinger equation (SSE) leading to the quantum-to-classical  transition.This  work  shows  that  the  stochastic  generalisation  of  the  quantum hydrodynamic analogy (QHA) has its corresponding SSE. The SSE owns an imaginary random noise that has a finite correlation distance so that when the physical length of the problem is much smaller than it, the SSE converges to the standard Schrödinger equation. The model derives the correlation length  of  the  environmental  noise,  leaving  the  quantumpotential  energy  of  fluctuations  finite,  and shows that in non-linear (weakly bounded) systems, the term responsible of the non-local interaction in the SSE may have a finite range of efficacy maintaining its non-local effect on a finite distance. A non-linear SSE that describes the related large-scale classical dynamics is derived. The work also shows that at the edge between the quantum and the classical regime the SSE can lead to the semi-empirical  Gross-Pitaevskii  equation. The SSE  can  be  helpful  in  describing  at  larger  extent  open quantum systems where the environmental fluctuations and the classical effects are both relevant.

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