In quantum mechanics we can identify the "Angular momentum operator" $\textbf{L}$ via:
$$ \langle { \theta,\varphi } | \textbf{L} | \psi\rangle = - i (\textbf{r} \times \nabla) \psi(\theta,\varphi)$$ where $\langle \theta, \varphi | \psi \rangle = \psi(\theta,\varphi) $From this formula we can obtain: $$ \langle { \theta,\varphi } | \textbf{L}^2 | \psi\rangle = - (\textbf{r} \times \nabla)^2 \psi(\theta,\varphi) = - \Delta_2 \psi(\theta,\varphi)$$
where $\Delta_2$ is the laplacian on the 2 sphere.
Next the definition of the spherical harmonics is $\langle \theta,\phi|l m \rangle = Y_{lm} (\theta,\varphi) $. Using the fact that $\Delta_2 \psi(\theta,\varphi) = -l(l+1)$ we see that:
$$\langle { \theta,\varphi } | \textbf{L}^2 | l,m \rangle = l(l+1) Y_{lm}(\theta,\varphi)$$ which is what we expect $L^2 | l,m \rangle = l(l+1) | l,m \rangle $. I am wondering if this naive translation from kets to spherical harmonics works in more general cases. For example I am considering a coupled state:
$$ | L M l_1 l_2 \rangle = \sum_{m_1 m_2} C^{LM}_{l_1 m_1 l_2 m_2} | l_1 m_1 \rangle \otimes | l_2 m_2 \rangle $$
where the $C$ is the Clebsch gordon coefficient. Applying the $L^2$ operator on the left gives $L(L+1)$, so we can write: $$ L(L+1) \sum_{m_1 m_2} C^{LM}_{l_1 m_1 l_2 m_2} | l_1 m_1 \rangle \otimes | l_2 m_2 \rangle = \textbf{L}^2 \sum_{m_1 m_2} C^{LM}_{l_1 m_1 l_2 m_2} | l_1 m_1 \rangle \otimes | l_2 m_2 \rangle $$ Next we can multiply to the left by $\langle \theta_1,\phi_1| \otimes \langle \theta_2,\phi_2|$ on both sides to get:
$$ L(L+1) \sum_{m_1 m_2} C^{LM}_{l_1 m_1 l_2 m_2} Y_{l_1 m_1}(\Omega_1) Y_{l_2 m_2} (\Omega_2) $$
$$= - ( ( \textbf{r}_1 \times \nabla_1)^2 + (\textbf{r}_2 \times \nabla_2)^2 + 2 ( \textbf{r}_1 \times \nabla_1) \cdot ( \textbf{r}_2 \times \nabla_2) \sum_{m_1 m_2} C^{LM}_{l_1 m_1 l_2 m_2} Y_{l_1 m_1}(\Omega_1) Y_{l_2 m_2} (\Omega_2) \qquad \dagger $$
which seems like a powerful identity regarding the spherical harmonics however I did not see this identity written anywhere. Furthermore, I am confused by a comment in this post Decomposition of spherical harmonics via Clebsh-Gordan coefficients in which it is explained that $Y_{lm} (\Omega) = \langle l 0 | R(\Omega) | l m \rangle $ and not what I had written so I am not sure about the validity of $\langle \theta,\phi|l m \rangle = Y_{lm} (\Omega,\varphi)$ and in particular the use of it to derive equation $\dagger$.
