The Garrison-Wong quantum phase operator revisited

TL;DR

This paper revisits the Garrison-Wong quantum phase operator, providing a detailed proof of its fundamental commutation relation with the number operator and analyzing its properties and limitations.

Contribution

It offers a rigorous proof of the Heisenberg commutation relation for the Garrison-Wong phase operator and discusses its properties and the failure of Weyl relations.

Findings

Proved the Heisenberg commutation relation on a natural domain.

Analyzed the failure of Weyl commutation relations.

Discussed properties of the phase-number operator pair.

Abstract

We revisit the quantum phase operator $\Phi$ introduced by Garrison and Wong. Denoting by $N$ the number operator, we provide a detailed proof of the Heisenberg commutation relation $\Phi N - N\Phi = iI$ on the natural maximal domain $D(\Phi N) \cap D(N \Phi)$ as well as the failure of the Weyl commutation relations, and discuss some further interesting properties of this pair.