/sci/ - Science & Math

Trying to prove the orthogonality of the spherical harmonics but I'm getting stuck at this integration by parts. I've already prove that, for two harmonics, [math]Y_{l}^{m}(\Omega)[/math] and [math]Y_{l^{\prime}}^{m^{\prime}}(\Omega)[/math], the inner product between these two is proportional to [math]\delta_{mm^{\prime}}[/math], but now I'm trying to prove that it's also proportional to [math]\delta_{ll^{\prime}}[/math]. I was able to prove this so far by representing the harmonics as state vectors and taking the inner product thusly: [math]\langle l^{\prime}, m|l,m \rangle[/math], but I also want to accomplish this in the position representation of the harmonics.

So far I have:
[math]\langle Y_{l^{\prime}}^{m}(\Omega), Y_{l}^{m}(\Omega) \rangle = \int_{\Omega^2}L_{+}^{l^{\prime} - m}Y_{l^{\prime}}^{l^{\prime} *}(\Omega)L_{-}^{l-m}Y_{l}^{l}(\Omega)d\Omega[/math]
where the angular momentum ladder operators are defined as such: [math]L_{-}=e^{-i\phi}\left ( -\frac{\partial }{\partial \theta} + i \cot{\theta}\frac{\partial }{\partial \phi} \right )[/math] and [math]L_{+}=e^{i\phi}\left ( \frac{\partial }{\partial \theta} + i \cot{\theta}\frac{\partial }{\partial \phi} \right )[/math]

My strategy here is to get [math]L_{+}[/math] and [math]L_{-}[/math] next to each other akin to [math]\langle l^{\prime},l^{\prime} | L_{+}^{l^{\prime}-m} L_{-}^{l-m} | l,l \rangle[/math] through IBP but I'm not sure what to select for my u and dv. Also, [math]\Omega = (\theta, \phi)[/math], [math]d\Omega = \sin{\theta}d\theta d\phi[/math], and [math]\Omega^2[/math] is the surface of the unit sphere.