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The uncertainty relation between angle and orbital angular momentum had notbeen formulated in a similar form as the uncertainty relation between positionand linear momentum because the angle variable is not represented by a quantummechanical self-adjoint operator. Instead of the angle variable operator, weintroduce the complex position operator $ \hat{Z} = \hat{x}+i \hat{y} $ andinterpret the order parameter $ \mu = \langle \hat{Z} \rangle / \sqrt{ \langle\hat{Z}^\dagger \hat{Z} \rangle} $ as a measure of certainty of the angledistribution. We prove the relation between the uncertainty of angular momentumand the angle order parameter. We prove also its generalizations and discussexperimental methods for testing these relations.

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