How Are Angular Momentum Operators Calculated in Spherical Polar Coordinates?

[latentcorpse]

How does one obtain the formulae for the angular momentum operators in spherical polar coordinates i.e.

\hat{L_x}=i \hbar (\sin{\phi} \frac{\partial}{\partial{\theta}} + \cot{\theta} \cos{\phi} \frac{\partial}{\partial{\phi}}
\hat{L_y}=i \hbar (-\cos{\phi}{\phi} \frac{\partial}{\partial{\theta}} + \cot{\theta} \sin{\phi} \frac{\partial}{\partial{\phi}}
\hat{L_z}=-i \hbar \frac{\partial}{\partial{\phi}}
\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]

?

 

[malawi_glenn]

Use the del-operator in spherical coordinates.

 

[latentcorpse]

so \hat{L}=\hat{r} \times \hat{p}

where \hat{p}=-i \hbar \nabla

do i use \nabla as the gradient operator in spherical polars?

also do i write r in terms of x,y,z or in terms of r,theta,phi?

 

[malawi_glenn]

have you even tried?

 

[latentcorpse]

yes. my notes run through it for the Cartesian case and subsitute r=(x,y,z) and \nabla=(\partial_x,\partial_y,\partial_z).
regardelss of which of the combinations i of r and del iuse above i can't get the right answer.

surely if i sub in for del in spherical polars I'm actually computing \hat{L_r},\hat{L_\theta},\hat{L_\phi} rather than \hat{L_x},\hat{L_y},\hat{L_z}, no?

 

[malawi_glenn]

http://simons.hec.utah.edu/TheoryPage/BookPDF/Appendix G-Angular Mom .pdf

and

http://simons.hec.utah.edu/TheoryPage/BookPDF/Appendix A-Mathematics Rev.pdf