On the Uncertainty Relations in Stochastic Mechanics

TL;DR

This paper derives Bohmian quantum equations from classical stochastic mechanics and explores their implications for uncertainty relations in quantum states like coherent and squeezed states.

Contribution

It demonstrates how Bohm equations can be obtained from classical ensemble equations with stochastic elements and examines the uncertainty relations in this framework.

Findings

Bohm equations derived from classical stochastic mechanics.

Uncertainty relations analyzed for coherent and squeezed states.

Stochastic interpretation offers insights into quantum uncertainties.

Abstract

It is shown that the Bohm equations for the phase $S$ and squared modulus $\rho$ of the quantum mechanical wave function can be derived from the classical ensemble equations admiting an aditional momentum $p_s$ of the form proportional to the osmotic velocity in the Nelson stochastic mechanics and using the variational principle with appropriate change of variables. The possibility to treat grad$S$ and $p_s$ as two parts of the momentum of quantum ensemble particles is considered from the view point of uncertainty relations of Robertson - Schroedinger type on the examples of the stochastic image of quantum mechanical canonical coherent and squeezed states.