Uncertainty relation between angle and orbital angular momentum: interference effect in electron vortex beams

The uncertainty relation between angle and orbital angular momentum had not been formulated in a similar form as the uncertainty relation between position and linear momentum because the angle variable is not represented by a quantum mechanical self-adjoint operator. Instead of the angle variable operator, we introduce the complex position operator $\hat{Z} = \hat{x}+i \hat{y}$ and interpret the order parameter $\mu = \langle \hat{Z} \rangle / \sqrt{ \langle \hat{Z}^\dagger \hat{Z} \rangle}$ as a measure of certainty of the angle distribution. We prove the relation between the uncertainty of angular momentum and the angle order parameter. We prove also its generalizations and discuss experimental methods for testing these relations.