= Nodeless Coulomb radial eigenstate
{title2=$R=A r^\ell e^{-r/[a_0(\ell+1)]}$}
For the nonrelativistic point-Coulomb hydrogen <Hamiltonian>, the lowest radial state at fixed <orbital angular momentum> $\ell$ has <principal quantum number> $n=\ell+1$ and no positive-radius <wavefunction nodes>. Substituting the displayed radial shape cancels the centrifugal inverse-square term and fixes $a_0=4\pi\epsilon_0\hbar^2/(me^2)$. With $b=a_0(\ell+1)$ and unit-normalized <spherical harmonic>, $A=(2/b)^{\ell+3/2}/\sqrt{(2\ell+2)!}$. Excited radial states require additional polynomial factors.
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