Phase of the quantum oscillator

TL;DR

This paper explores the quantum phase of oscillators, emphasizing the importance of conjugate operators, extending Hilbert spaces, and proposing a new phase uncertainty measure with implications for number-phase relations.

Contribution

It introduces a new phase uncertainty definition suitable for periodic systems and connects oscillator Hilbert spaces with gauge-invariant rotor spaces.

Findings

Proposes a phase uncertainty measure based on periodicity.

Establishes a number-phase uncertainty relation.

Identifies oscillator Hilbert space with a gauge-invariant rotor space.

Abstract

Requirements of a conjugate operator are emphasized, especially in its role in uncertainty relations.It is argued that in many contexts it is necessary to extend the Hilbert space in order to define a conjugate operator as in gauge theories. Example of a particle in a box is analysed. This is closely related to the quantum oscillator through cosine states of Susskind and Glogower.It is used to justify London's phase wave functions albeit as part of a larger Hilbert space. A new definition phase uncertainty neccessiated by periodicity is proposed.It is close to the usual r.m.s. definition.Corresponding number- phase uncertainty relation is obtained and its implications are discussed. Hilbert space of an oscillator is identified with the Hilbert space of a planar rotor with a $Z_2$ gauge invariance.This is used to construct states analogous to the cosine and sine states and to illustrate unitary equivalence of Hilbert spaces.