Uncertainty relations in the realm of classical dynamics

TL;DR

This paper investigates the parallels between quantum and classical uncertainty relations in one-dimensional conservative systems, showing that their first and second moments of canonical observables are structurally identical.

Contribution

It demonstrates that uncertainty relations in quantum stationary states and classical ensembles share an identical structure for canonical observables.

Findings

First and second moments of observables match in quantum and classical descriptions.

Uncertainty relations have an identical structure in both quantum and classical regimes.

Classical and quantum uncertainty relations can be directly compared in conservative systems.

Abstract

It is generally believed that classical regime emerges as a limiting case of quantum theory. Exploring such quantum-classical correspondences in a more transparent manner is central to the deeper understanding of foundational aspects and has attracted a great deal of attention - starting from the early days of quantum theory. While it is often highlighted that quantum to classical transition occurs in the limit hbar tending to zero, several objections have been raised about its suitability in some physical contexts. Ehrenfest's theorem is another widely discussed classical limit - however, its inadequacy has also been pointed out in specific examples. It has been proposed that since a quantum mechanical wave function inherits an intrinsic statistical behavior, its classical limit must correspond to a classical ensemble - not an individual particle. This opens up the question "how would uncertainty relations of canonical observables compare themselves in quantum and classical realms?" In this paper we explore parallels between uncertainty relations in stationary states of quantum systems and that in the corresponding classical ensemble. We confine ourselves to one dimensional conservative systems and show, with the help of suitably defined dimensionless physical quantities, that first and second moments of the canonical observables match with each other in classical and quantum descriptions - resulting in an identical structure for uncertainty relations in both the realms.