 Nodeless Coulomb radial eigenstate

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 $a_0=4\pi {ϵ}_0{\hslash }^2/(m{e}^2)$. 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.