Quantization of angular momementum

[Scienc]

Can someone give me a hint on how can I prove it?

 

[studiot]

Use Spherical coordinates for the angular momentum operator

 

[math]{I^2} =  - \hbar \left[ {\frac{1}{{\sin \theta }}\frac{\partial }{{\partial \theta }}\left( {\sin \theta \frac{\partial }{{\partial \theta }}} \right) + \frac{1}{{{{\sin }^2}\theta }}\frac{{{\partial ^2}}}{{\partial {\phi ^2}}}} \right][/math]

to solve

[math]{I^2}Y\left( {\theta ,\phi } \right) = {\hbar ^2}\lambda Y\left( {\theta ,\phi } \right)[/math]

for eigenvalues.