For an electron in a hydrogen atom, the stationary-state wavefunctions are of the form $\psi(r, \theta, \phi)=R(r) Y_{l m}(\theta, \phi)$, where in suitable units $R$ obeys the radial equation

$$\frac{d^{2} R}{d r^{2}}+\frac{2}{r} \frac{d R}{d r}-\frac{l(l+1)}{r^{2}} R+2\left(E+\frac{1}{r}\right) R=0$$

Explain briefly how the terms in this equation arise.

This radial equation has bound-state solutions of energy $E=E_{n}$, where $E_{n}=-\frac{1}{2 n^{2}}(n=1,2,3, \ldots)$. Show that when $l=n-1$, there is a solution of the form $R(r)=r^{\alpha} e^{-r / n}$, and determine $\alpha$. Find the expectation value $\langle r\rangle$ in this state.

Determine the total degeneracy of the energy level with energy $E_{n}$.