= Probability concentration of a circular Coulomb eigenstate
{title2=$\psi_\ell\propto r^\ell\sin^\ell\theta\,e^{i\ell\phi-r/[(\ell+1)a]}$}
For a nonnegative integer $\ell$, this <Coulomb potential> <energy eigenstate> has principal quantum number $\ell+1$ and maximal magnetic quantum number. Put $b=(\ell+1)a$. Including the spherical <volume element>, its radial <probability density> is a <gamma distribution> proportional to $r^{2\ell+2}e^{-2r/b}$; its polar density is proportional to $\sin^{2\ell+1}\theta$, while its azimuth is uniform. Radial and polar variables are independent. Their moments satisfy
$$
\mu=\mathbb E[r]=\frac{2\ell+3}{2}b,\qquad\frac{\operatorname{Var}(r)}{\mu^2}=\frac1{2\ell+3},\qquad\mathbb E[\cos^2\theta]=\frac1{2\ell+3}.
$$
The radial identities follow by successive <integration by parts>; the angular identity follows from $\int_0^\pi\sin^{j+2}\theta\,d\theta=(j+1)\int_0^\pi\sin^j\theta\,d\theta/(j+2)$.
Set $h=(2\ell+3)^{-1/3}$. <Chebyshev's inequality> and <Markov's inequality> give $\mathbb P(|r/\mu-1|>h)\leq h$ and $\mathbb P(|\cos\theta|>h)\leq h$. By the <union bound>, outside an event of probability at most $2h$, the point lies at distance $O(\mu h)$ from the equatorial circle of radius $\mu$. The <quantum state> therefore concentrates near a classical circular orbit in relative units as $\ell$ grows. The absolute orbit radius and absolute transverse width both grow; this statement does not assert localization in a fixed narrow tube.
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