Bohmian mechanics and Fisher information for $q$-deformed Schr\"odinger equation

TL;DR

This paper explores Bohmian mechanics within a $q$-deformed Schr"odinger framework, deriving a deformed Fisher information and Cramér-Rao bound, and applies it to a particle in a potential well to reveal the algebraic structure's role.

Contribution

It introduces a $q$-algebra approach to Bohmian mechanics, deriving a deformed Fisher information and Cramér-Rao bound, and demonstrates their preservation in stationary states.

Findings

Deformed Fisher information functional is derived from the $q$-algebra.

A deformed Cramér-Rao bound is established and preserved in stationary states.

The $q$-algebraic structure influences the quantum formalism and bounds.

Abstract

We discuss the Bohmian mechanics by means of the deformed Schr\"odinger equation for position dependent mass, in the context of a $q$-algebra inspired by nonextensive statistics. A deduction of the Bohmian quantum formalism is performed by means of a deformed Fisher information functional, from which a deformed Cram\'er-Rao bound is derived. Lagrangian and Hamiltonian formulations, inherited by the $q$-algebra, are also developed. Then, we illustrate the results with a particle confined in an infinite square potential well. The preservation of the deformed Cram\'er-Rao bound for the stationary states shows the role played by the $q$-algebraic structure.