On quantum states for angular position and angular momentum of light

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 = − i d / d θ in units of ℏ = 1 . The quantum uncertainties Δ θ and Δ L for the state are well-defined and are shown to give a lower value of the uncertainty product Δ θ Δ L 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 Δ θ = π 2 / 3 − 2 and Δ L = 1 / 2 .