On Quantum States for angular Position and Angular Momentum of Light

In the present paper we construct a properly defined quantum state 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. The quantum uncertainties Δθ and ΔL for the state are well-defined and are, e.g., shown to give a lower value of the uncertainty product ΔθΔL than the minimal uncertainty states of Ref.\cite{Padgett_2004}. The mean value < L > of the state 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< sqrt{π²/3 -2} and ΔL > 1/2.