Heisenberg Uncertainty Relation for Three Canonical Observables

TL;DR

This paper extends the Heisenberg uncertainty principle to three canonical quantum observables, deriving a new triple uncertainty relation and identifying the state that saturates it, with proposals for experimental verification.

Contribution

It introduces a third observable with canonical commutation relations to position and momentum, establishing a triple uncertainty relation in quantum mechanics.

Findings

Derived the smallest possible triple uncertainty bound

Identified the squeezed state saturating the relation

Proposed quantum optical experiments for validation

Abstract

Uncertainty relations provide fundamental limits on what can be said about the properties of quantum systems. For a quantum particle, the commutation relation of position and momentum observables entails Heisenberg's uncertainty relation. A third observable is presented which satisfies canonical commutation relations with both position and momentum. The resulting triple of pairwise canonical observables gives rise to a Heisenberg-type uncertainty relation for the product of three standard deviations. We derive the smallest possible value of this bound and determine the specific squeezed state which saturates the triple uncertainty relation. Quantum optical experiments are proposed to verify our findings.