Angular momentum powering operator L[-] - applying n times

[bjnartowt]

Homework Statement

I want to generate the m-th spherical harmonic from the spherical harmonic with all "l" of the total angular momentum in the z-direction,

$${\left\langle {{\bf{\hat n}}|\ell ,\ell } \right\rangle = Y_\ell ^\ell (\theta ,\phi ) = {C_\ell }{e^{{\bf{i}}\ell \phi }}{{(\sin \theta )}^\ell }}$$

...and lowering from there, by applying this lowering operator...

$${\left( {{\bf{i}}{\textstyle{\partial \over {\partial \theta }}} + \cot \theta {\textstyle{\partial \over {\partial \phi }}}} \right)^n}$$

..."n" times, as you can see. My author, Sakurai, claims this is done in many "elementary" books on QM. What would be the first step to handling this "n" iterated operator-expansion?

Homework Equations

The Attempt at a Solution

 

[nickjer]

I don't believe it is intended to be expanded. The elementary part he is describing is that you are supposed to use the lowering operator iteratively to solve for each harmonic. It is very cumbersome but not too technical.