.

We need to form L², i.e.,

$$L^2 = L_x \cdot L_x + L_y \cdot L_y + L_z \cdot L_z$$ (23)

$$L_x^2 = - \hbar^2 \left ( \sin \phi \frac{\partial}{\partial... ...\cos \phi}{\sin\theta} \frac{\partial}{\partial \phi} \right )$$ (24)

$$L_y^2 = - \hbar^2 \left ( \cos \phi \frac{\partial}{\partial... ...\sin \phi}{\sin\theta} \frac{\partial}{\partial \phi} \right )$$ (25)

$$L_z^2 = - \hbar^2 \frac{\partial \frac{\partial}{\partial \phi} }{\partial \phi}$$ (26)

For L_x² we have, upon expansion

 

$$\frac{L_x^2 }{ - \hbar^2}= \sin \phi \frac{\partial \sin \phi \fr... ...}{\partial \theta} \Rightarrow \sin^2 \phi \frac{\partial^2}{\partial \theta^2}$$

$$+ \frac{\cos \theta \cos \phi}{\sin\theta} \frac{\partial\sin \ph... ... \cot \theta \cos \phi \sin \phi\frac{\partial^2}{\partial \theta\partial \phi}$$

$$\sin \phi \frac{\partial \frac{\cos \theta \cos \phi}{\sin\theta}... ...\cos \phi \sin \phi}{\sin\theta}\frac{\partial^2}{\partial \phi\partial \theta}$$

$$+ \frac{\cos \theta \cos \phi}{\sin\theta} \frac{\partial \frac{\... ...rac{\cos^2 \theta \cos^2 \phi}{\sin^2\theta} \frac{\partial^2}{\partial \phi^2}$$ (27)

For L_y we have

 

$$\frac{L_y^2}{ - \hbar^2 }= \cos \phi \frac{\partial \cos \phi \fr... ...}}{\partial \theta} \Rightarrow \cos^2\phi \frac{\partial^2}{\partial \theta^2}$$

$$- \frac{\cos \theta \sin \phi}{\sin\theta} \frac{\partial \cos \p... ...in \phi}{\sin\theta} \cos \phi \frac{\partial^2}{\partial \theta \partial \phi}$$

$$-\cos \phi \frac{\partial\frac{\cos \theta \sin \phi}{\sin\theta}... ...s \theta \sin \phi}{\sin\theta} \frac{\partial^2}{\partial \phi\partial \theta}$$

$$+ \frac{\cos \theta \sin \phi}{\sin\theta} \frac{\partial \frac{\... ...rac{\cos^2 \theta \sin^2 \phi}{\sin^2\theta} \frac{\partial^2}{\partial \phi^2}$$ (28)

and, of course

   $$\frac{L_y^2}{ - \hbar^2 }= \frac{\partial^2}{\partial \phi^2}$$

It follows, adding Equations 27, 28 and 7, that

$$\begin{eqnarray*}\sin^2 \phi \frac{\partial^2}{\partial \theta^2} \nonumber \\ ... ...tial \phi^2} \nonumber \\ + \frac{\partial^2}{\partial \phi^2} \end{eqnarray*}$$