= Solution
For the two inline operator labels, \b[(a)] $L_z$ has <eigenvalue> $m_\ell\hbar$, and \b[(b)] $L^2$ has <eigenvalue> $\ell(\ell+1)\hbar^2$, with $\ell=0,1,\ldots$ and $m_\ell=-\ell,\ldots,\ell$. The subscript distinguishes the <magnetic quantum number> from the <electron> <mass>.
Put $b=a_0(\ell+1)$ and $\kappa=e^2/(4\pi\epsilon_0)$. For the proposed radial shape,
$$
\frac1{r^2R}\frac{d}{dr}\left(r^2\frac{dR}{dr}\right)
=\frac{\ell(\ell+1)}{r^2}-\frac{2(\ell+1)}{br}+\frac1{b^2}.
$$
The angular-momentum term cancels the inverse-square part. The inverse-radius coefficient vanishes if $\hbar^2(\ell+1)/(mb)=\kappa$. Therefore
$$
\boxed{a_0=\frac{4\pi\epsilon_0\hbar^2}{me^2}},\qquad
\boxed{E=-\frac{\hbar^2}{2ma_0^2(\ell+1)^2}}.
$$
This <nodeless Coulomb radial eigenstate> has <principal quantum number> $n=\ell+1$, not the complete set of hydrogen radial states. Since the <spherical harmonic> is normalized, radial normalization requires
$$
1=|A|^2(2\ell+2)!\left(\frac b2\right)^{2\ell+3},\qquad
\boxed{A=\frac{(2/b)^{\ell+3/2}}{\sqrt{(2\ell+2)!}}}
$$
up to an arbitrary <phase>.
For the <hydrogen ground-state nuclear-volume probability>, use $R=2a_0^{-3/2}e^{-r/a_0}$. With $u=r_p/a_0\ll1$,
$$
P(r<r_p)=1-e^{-2u}(1+2u+2u^2)
\simeq\boxed{\frac43\left(\frac{r_p}{a_0}\right)^3}.
$$
No numerical <proton> radius is supplied or required. A sharp <proton> boundary is an approximation to a charge distribution, and the point-Coulomb potential changes inside an extended <proton>. Finite <proton> <mass>, relativistic and electromagnetic corrections also limit the ideal <wavefunction>'s precision.
Classically the <kinetic energy> must satisfy $E-V(r)\geq0$. For $E=-\kappa/(2a_0)$ this demands \b[$r\leq2a_0$]. The <hydrogen ground-state probability beyond the classical turning point> is
$$
P(r>2a_0)=\int_{2a_0}^\infty\frac{4r^2}{a_0^3}e^{-2r/a_0}\,dr
=\boxed{13e^{-4}\simeq0.238}.
$$
A substantial <probability> lies beyond the classical turning radius; the bound-state <wavefunction> decays there rather than vanishing.
\Image[/past-exam-of-the-physics-a-course-of-the-university-of-cambridge/2012/paper-1-hydrogen-tail.png]
{title=Original hydrogen ground-state radial probability density with its classically forbidden tail shaded}
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