Chapter 6.6, Problem 6.19P

(a)

To determine

Show that $\left[{\widehat{L}}_z,{\widehat{V}}_{\pm }\right]=\pm h{\widehat{V}}_{+}$, and $\left[{\widehat{L}}^2,{\widehat{V}}_{\pm }\right]=2h^2{\widehat{V}}_{\pm }\pm 2h{\widehat{V}}_{\pm }{\widehat{L}}_z\mp 2h{\widehat{V}}_z{\widehat{L}}_{\pm }$.

(b)

To determine

Show that, if $\psi$ is an eigen state of ${\widehat{L}}^2$ and ${\widehat{L}}_z$ with eigenvalues $\mathcal{l}\left(\mathcal{l}+1\right){\hslash }^2$ and $\mathcal{l}\hslash$ respectively, then either $\widehat{V}+\psi$ is zero or $\widehat{V}+\psi$ is also an eigen state of ${\widehat{L}}^2$ and ${\widehat{L}}_z$ with eigenvalues $\left(\mathcal{l}+1\right)\left(\mathcal{l}+2\right){\hslash }^2$ and $\left(\mathcal{l}+1\right)\hslash$ respectively.