.

$$\frac{\partial^2}{\partial \theta^2}$$

$$\cot \theta \frac{\partial}{\partial \theta}$$

$$zero \frac{\partial}{\partial \phi}$$

$$zero \frac{\partial^2}{\partial \phi\partial \theta}$$

$$+ \left \{\frac{\cos^2 \theta +\sin^2 \theta}{\sin^2\theta} +1 \right \}\frac{\partial^2}{\partial \phi^2}$$ (31)

which is, ultimately

$$L^2 = -\hbar^2 \left ( \frac{1}{\sin^2\theta} \left [ \sin \t... ...\theta} + \frac{\partial^2}{\partial \phi^2} \right ] \right )$$ (32)

We see that the operator associated with Legendre's Equation (multiplied by a constant) has emerged, meaning that Legendre Polynomials and angular momentum are intimately asssociated together.