- #1
latentcorpse
- 1,444
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How does one obtain the formulae for the angular momentum operators in spherical polar coordinates i.e.
[itex]\hat{L_x}=i \hbar (\sin{\phi} \frac{\partial}{\partial{\theta}} + \cot{\theta} \cos{\phi} \frac{\partial}{\partial{\phi}}[/itex]
[itex]\hat{L_y}=i \hbar (-\cos{\phi}{\phi} \frac{\partial}{\partial{\theta}} + \cot{\theta} \sin{\phi} \frac{\partial}{\partial{\phi}}[/itex]
[itex]\hat{L_z}=-i \hbar \frac{\partial}{\partial{\phi}}[/itex]
[itex]\hat{L}^2=\hbar^2 \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}} \right][/itex]
?
[itex]\hat{L_x}=i \hbar (\sin{\phi} \frac{\partial}{\partial{\theta}} + \cot{\theta} \cos{\phi} \frac{\partial}{\partial{\phi}}[/itex]
[itex]\hat{L_y}=i \hbar (-\cos{\phi}{\phi} \frac{\partial}{\partial{\theta}} + \cot{\theta} \sin{\phi} \frac{\partial}{\partial{\phi}}[/itex]
[itex]\hat{L_z}=-i \hbar \frac{\partial}{\partial{\phi}}[/itex]
[itex]\hat{L}^2=\hbar^2 \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}} \right][/itex]
?