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 quantum potential 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 semiempirical
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.