Seguramente estoy cometiendo un error de cálculo trivial pero no lo encuentro.
Déjalo: $$L(r,\theta,\phi)= \frac{\hbar}{i} r \left( \frac{1}{r\sin\theta} \frac{\partial}{\partial \phi} \hat{\phi} - \frac{1}{r}\frac{\partial }{\partial \theta} \hat{\theta} \right)$$ $$L^2=L\cdot L= -\hbar^2 r^2 \left( \frac{1}{r\sin\theta} \frac{\partial}{\partial \phi} \hat{\phi} - \frac{1}{r}\frac{\partial }{\partial \theta} \hat{\theta} \right)^2=$$ $$= -\hbar^2 r^2 \left( \frac{1}{r\sin\theta} \frac{\partial}{\partial \phi} \hat{\phi} - \frac{1}{r}\frac{\partial }{\partial \theta} \hat{\theta} \right) \cdot \left( \frac{1}{r\sin\theta} \frac{\partial}{\partial \phi} \hat{\phi} - \frac{1}{r}\frac{\partial }{\partial \theta} \hat{\theta} \right)=$$ $$-\hbar^2 \left( \frac{1}{\sin^2(\theta)} \frac{\partial^2}{\partial \phi^2} + \frac{\partial^2}{\partial \theta^2} \right)$$
Pero sé que el resultado correcto es: $$L^2=-\hbar^2 \left( \frac{1}{\sin^2(\theta)} \frac{\partial^2}{\partial \phi^2} + \frac{\partial^2}{\partial \theta^2} + \cot\theta \frac{\partial}{\partial \theta} \right)$$
