Chapter 6, Problem 6.34P

(a)

To determine

To show that ${\widehat{M}}_z{\psi }_{n\mathcal{l}\mathcal{l}}=-\frac{1}{\sqrt{2}}\frac{1}{\sqrt{l+1}}{c}_{n\mathcal{l}}{\psi }_{n\left(\mathcal{l}+1\right)\mathcal{l}}$.

(b)

To determine

To show that ${\widehat{M}}_{-}{\widehat{M}}_{+}{\psi }_{n\mathcal{l}\mathcal{l}}={\left(\frac{e^2}{4\pi {\epsilon }_0}\right)}^2\left[1-{\left(\frac{l+1}{n}\right)}^2\right]{\psi }_{n\mathcal{l}\mathcal{l}}-{\widehat{M}}_Z^2{\psi }_{n\mathcal{l}\mathcal{l}}$.

(c)

To determine

The constant $c_{n\mathcal{l}}$, and to show that $c_{n\mathcal{l}}$ is non zero unless $\mathcal{l}=n-1$.