- #1
bjnartowt
- 284
- 3
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,
[tex]{\left\langle {{\bf{\hat n}}|\ell ,\ell } \right\rangle = Y_\ell ^\ell (\theta ,\phi ) = {C_\ell }{e^{{\bf{i}}\ell \phi }}{{(\sin \theta )}^\ell }}[/tex]
...and lowering from there, by applying this lowering operator...
[tex]{\left( {{\bf{i}}{\textstyle{\partial \over {\partial \theta }}} + \cot \theta {\textstyle{\partial \over {\partial \phi }}}} \right)^n}[/tex]
..."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?