Past exam of the quantum physics course of the University of Oxford / 2016 / 8 / c / Solution

There is no allowed spherical harmonic with $m=l+1$, so $L_{+}{Y}_l^l=0$. The equation $-i\hslash {\partial }_{\varphi }Y=\hslash lY$ first gives $Y=f(\theta )e^{il\varphi }$. Combining the differential expressions for $L_x,L_y$ yieldsIts annihilation of the highest-weight state becomesIntegrating $f^{\prime }/f=l\cot \theta$ gives $f=C{\sin }^l\theta$, with the constant solution included for $l=0$. HenceThe phase $e^{il\varphi }$ is single valued for integral orbital quantum number $l$; normalization fixes only $|C|$, leaving a conventional constant phase.