Chapter 4.3, Problem 4.24P

(a)

To determine

Derive expression $L_{+}{L}_{-}=-{\hslash }^2\left(\frac{{\partial }^2}{\partial {\theta }^2}+\cot \theta \frac{\partial }{\partial \theta }+{\cot }^2\theta \frac{{\partial }^2}{\partial {\varphi }^2}+i\frac{\partial }{\partial \varphi }\right)$.

(b)

To determine

Derive expression $L^2=-{\hslash }^2\left(\frac{1}{\sin \theta }\frac{\partial }{\partial \theta }\left(\sin \theta \frac{\partial }{\partial \theta }\right)+\frac{1}{{\sin }^2\theta }\frac{{\partial }^2}{{\partial }^2\varphi }\right)$.