For an electron of mass $m$ in a hydrogen atom, the time-independent Schrödinger equation may be written as

$$-\frac{\hbar^{2}}{2 m r^{2}} \frac{\partial}{\partial r}\left(r^{2} \frac{\partial \psi}{\partial r}\right)+\frac{1}{2 m r^{2}} L^{2} \psi-\frac{e^{2}}{4 \pi \epsilon_{0} r} \psi=E \psi$$

Consider normalised energy eigenstates of the form

$$\psi_{l m}(r, \theta, \phi)=R(r) Y_{l m}(\theta, \phi)$$

where $Y_{l m}$ are orbital angular momentum eigenstates:

$$L^{2} Y_{l m}=\hbar^{2} l(l+1) Y_{l m}, \quad L_{3} Y_{l m}=\hbar m Y_{l m}$$

where $l=1,2, \ldots$ and $m=0, \pm 1, \pm 2, \ldots \pm l$. The $Y_{l m}$ functions are normalised with $\int_{\theta=0}^{\pi} \int_{\phi=0}^{2 \pi}\left|Y_{l m}\right|^{2} \sin \theta d \theta d \phi=1 .$

(i) Write down the resulting equation satisfied by $R(r)$, for fixed $l$. Show that it has solutions of the form

$$R(r)=A r^{l} \exp \left(-\frac{r}{a(l+1)}\right)$$

where $a$ is a constant which you should determine. Show that

$$E=-\frac{e^{2}}{D \pi \epsilon_{0} a}$$

where $D$ is an integer which you should find (in terms of $l$ ). Also, show that

$$|A|^{2}=\frac{2^{2 l+3}}{a^{F} G !(l+1)^{H}},$$

where $F, G$ and $H$ are integers that you should find in terms of $l$.

(ii) Given the radius of the proton $r_{p} \ll a$, show that the probability of the electron being found within the proton is approximately

$$\frac{2^{2 l+3}}{C}\left(\frac{r_{p}}{a}\right)^{2 l+3}\left[1+\mathcal{O}\left(\frac{r_{p}}{a}\right)\right]$$

finding the integer $C$ in terms of $l$.

[You may assume that $\int_{0}^{\infty} t^{l} e^{-t} d t=l !$.]