= Coulomb radial origin condition
{c}
{title2=$u=rR,\quad u(0)=0,\quad R(r)\sim r^\ell$}
= Regular origin condition for a Coulomb radial wavefunction
{synonym}
For the usual three-dimensional <Coulomb potential> <Hamiltonian operator>, a separated <radial wavefunction> $R(r)$ uses the reduced function $u=rR$, whose radial <norm> has measure $dr$. The regular physical <operator domain> imposes $u(0)=0$. Near the origin the leading radial equation gives the <indicial equation> $c(c-1)-\ell(\ell+1)=0$ for $u\sim r^c$, with roots $c=\ell+1$ and $c=-\ell$.
The regular root gives $R\sim r^\ell$. The other root gives $R\sim r^{-\ell-1}$, and its radial <probability> integral $\int_0^\varepsilon r^2|R|^2dr\sim\int_0^\varepsilon r^{-2\ell}dr$ diverges for $\ell\geq1$. At $\ell=0$, $R\sim1/r$ is locally <square-integrable> in three dimensions, but violates $u(0)=0$ and has divergent <kinetic energy> integral $\int_0^\varepsilon r^2|R'|^2dr\sim\int_0^\varepsilon r^{-2}dr$. Thus local <wavefunction normalization> alone is insufficient to select the physical branch in this exceptional case; the standard Coulomb energy domain supplies the missing condition.
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