Phase states and phase operators for the quantum harmonic oscillator

How does the classical notion of ‘‘phase’’ apply to a quantum harmonic oscillator H=1/2(q${\mathrm{^}}^2$+p${\mathrm{^}}^2$), [q^,p^]=iħ, which cannot have sharp position and momentum? A quantum state ρ^ can be assigned a definite classical phase only if it is a large-amplitude localized state. Our only demand, therefore, on a (Hermitian) phase operator φ^ is that the phase distribution P(Φ)= Tr{δ(φ^-Φ)ρ^} attribute the correct sharp phase to any such ‘‘classical phase’’ state. This requires that the Weyl symbol [φ^${]}_{\mathit{w}}$(q,p) of φ^ tend to θ mod2π as r→∞, where θ=${\tan }^{\mathrm{-}1}$(p/q) and r=(${\mathit{q}}^2$+${\mathit{p}}^2$ ${)}^{1/2}$. There are infinitely many such phase operators. Each is expressible as φ^=[${\tan }^{\mathrm{-}1}$(p^/q^)${]}_{\mathrm{\Omega }}$, where Ω specifies an ordering rule for q^ and p^. The commutator -i[H^,φ^]=1-2π[δ(${\tan }^{\mathrm{-}1}$p^/q^)${]}_{\mathrm{\Omega }}$ corresponds to the Poisson bracket {H,${\mathrm{\phi }}_{\mathrm{cl}}$ ${\mathrm{\}}}_{\mathrm{}\mathrm{PB}}$=1-2πδ(θ) for the single-valued classical phase ${\mathrm{\phi }}_{\mathrm{cl}}$=θ mod2π. Phase states Γ^(Φ) are defined by the condition that their Weyl symbols [Γ^(Φ)${]}_{\mathit{w}}$(r,θ)→δ(θ-Φ) as r→∞. If moreover ${\mathrm{\int }}_0^{2\mathrm{\pi }}$dΦΓ^(Φ)=1^, then Γ^(Φ) is a phase probability operator measure (POM). In particular, δ(φ^-Φ) is a phase POM.