The Schr\"odinger equation and its generalized analogs derived by the quantum hydrodynamic representation

TL;DR

This paper investigates how quantum hydrodynamic analogies relate to Schr"odinger equations, especially under fluctuations, revealing limitations and generalizations of the mathematical correspondence between these frameworks.

Contribution

It demonstrates the conditions under which Schr"odinger equations can be derived from quantum hydrodynamics and introduces generalized forms applicable to stochastic and classical limits.

Findings

Ensemble solutions of QHA are broader than Schr"odinger solutions under fluctuations.

Mathematical correspondence between QHA and Schr"odinger equations breaks down with fluctuations.

Generalized Schr"odinger-like equations can describe stochastic and classical limits.

Abstract

The derivation of the Schr\"odinger-like equations from the system of equations of the quantum hydrodynamic analogy (QHA) is analyzed in presence of fluctuations. If in absence of fluctuation each QHA solution can be tracked back to the equivalent one of the Schr\"odinger problem, in presence of fluctuations the work shows that the ensemble of solution of the QHA equations is wider than the Schr\"odinger one and the tracking-back procedure from the QHA problem to the original Schr\"odinger equation is not generally possible. The break-down of the mathematical correspondence is inspected and correlated to the characteristics of the states. Finally, the paper shows that if the form of Schr\"odinger equation is generalized, the QHA stochastic case as well as its classical limit can be translated into a Schr\"odinger-like representation.