Half-ring sine-squared angular state

The normalized periodic state $\psi =\sqrt{8/(3\pi )}{\sin }^2\varphi$ on $0<\varphi <\pi$, zero on the remaining semicircle, has $⟨L_z⟩=0$ and $\mathrm{Var}(L_z)=4{\hslash }^2/3$. Projection on $e^{i\varphi }/\sqrt{2\pi }$ gives probability $64/(27{\pi }^2)$ for $L_z=\hslash$. Its first derivative is continuous at the joins, so integration by parts has no boundary contribution.