Half-ring sine-squared angular state
= Half-ring sine-squared angular state
The normalized periodic state $\psi=\sqrt{8/(3\pi)}\sin^2\phi$ on $0<\phi<\pi$, zero on the remaining semicircle, has $\langle L_z\rangle=0$ and $\operatorname{Var}(L_z)=4\hbar^2/3$. Projection on $e^{i\phi}/\sqrt{2\pi}$ gives probability $64/(27\pi^2)$ for $L_z=\hbar$. Its first derivative is continuous at the joins, so integration by parts has no boundary contribution.