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 in units of . The quantum uncertainties Δθ and ΔL for the state are well-defined and are shown to give a lower value of the uncertainty product 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 of the state is not required to be an integer. In the case of any half-integer mean value the state constructed exhibits a remarkable critical behavior with upper and lower bounds and .