On quantum states for angular position and angular momentum of light

Abstract

In the present paper we construct a properly defined quantum state ψ(θ) expressed in terms of elliptic Jacobi theta functions for the self-adjoint observables angular position θ and the corresponding angular momentum operator L=−id/dθ in units of ℏ=1 . The quantum uncertainties Δθ and ΔL for the state are well-defined and are shown to give a lower value of the uncertainty product ΔθΔL in contrast to the so called minimal uncertainty states as discussed in Franke-Arnold et al (2004 New J. Phys. 6 103-1-8). The mean value ⟨L⟩ of the state ψ(θ) is not required to be an integer. In the case of any half-integer mean value ⟨L⟩ the state constructed exhibits a remarkable critical behavior with upper and lower bounds Δθ=π2/3−2 and ΔL=1/2 .