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 $\theta$ and the corresponding angular momentum operator $L=-id\mathrm{/}d\theta$. The quantum uncertainties $\mathrm{\Delta }\theta$ and $\mathrm{\Delta }L$ for the state are well-defined and are, e.g., shown to give a lower value of the uncertainty product $\mathrm{\Delta }\theta \mathrm{\Delta }L$ 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 $\mathrm{\Delta }\theta <\sqrt{{\pi }^2\mathrm{/}3-2}$ and $\mathrm{\Delta }L>1\mathrm{/}2$.