= Solution
<Separation of variables> in the central <Coulomb potential> gives $\psi_{nlm}=R_{nl}(r)Y_l^m(\theta,\phi)$. The <radial wavefunction> depends on \b[$n$ and $l$], independently of $m$. For the <hydrogen radial wavefunction in the first excited shell>, put $\rho=r/a_0$. The <Bohr radius> is $a_0=4\pi\epsilon_0\hbar^2/(\mu e^2)$, where $\mu$ is the <electron>–<proton> <reduced mass>. Choose the real <phases>
$$
R_{20}=\frac{(2-\rho)e^{-\rho/2}}{2\sqrt2\,a_0^{3/2}},\qquad R_{21}=\frac{\rho e^{-\rho/2}}{\sqrt{24}\,a_0^{3/2}}.
$$
Near the origin, $R_{20}\to1/(\sqrt2a_0^{3/2})$, whereas $R_{21}\sim r/(\sqrt{24}a_0^{5/2})$. More generally, regular <hydrogen> <radial wavefunctions> behave as $r^l$. At large $r$, both displayed functions have a <polynomial> factor proportional to $r$ multiplying $e^{-r/(2a_0)}$; $R_{20}$ approaches zero from below and $R_{21}$ from above with these <phase> choices.
The $2s$ <radial wavefunction> crosses zero once, at $r=2a_0$, and has its negative minimum at $4a_0$. The $2p$ <radial wavefunction> has no <radial node> at positive $r$, and reaches its maximum at $2a_0$. The count is $n-l-1$; neither the origin nor the limit at infinity is counted as a <radial node>. These are sketches of $R$, rather than of $rR$ or the <radial probability density> $r^2|R|^2$.
\Image[/past-exam-of-the-quantum-physics-course-of-the-university-of-oxford/2016-hydrogen-radial-functions.png]
{title=Hydrogen radial wavefunctions in the first excited shell}
{description=The horizontal coordinate is measured in Bohr radii; the vertical coordinate is the radial wavefunction multiplied by the Bohr radius to the power three halves.}
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