 Angular momentum expectation

For a normalized periodic angular wavefunction, $⟨L_z⟩=-i\hslash {\int }_0^{2\pi }{\psi }^{\ast }{\partial }_{\varphi }\psi d\varphi$. A real periodic wavefunction with matching boundary values has zero mean, even though nonzero angular momenta can have positive measurement probabilities.