Hydrogen Radial Wavefunctions: Formula Reference

This reference gives normalized point-Coulomb bound-state radial functions, probability measures, and useful moments. The Radial Wavefunctions teaching page develops their derivation, low-lying examples, plots, and exercises.

Normalized hydrogenic radial functions

For $\psi_{n\ell m}=R_{n\ell}(r)Y_\ell^m(\theta,\phi)$ , define

a_Z=\frac{4\pi\epsilon_0\hbar^2}{\mu Ze^2}, \qquad \rho=\frac{2r}{na_Z}.

The normalized bound-state radial function is

R_{n\ell}(r) =\sqrt{\left(\frac{2}{na_Z}\right)^3 \frac{(n-\ell-1)!}{2n(n+\ell)!}} e^{-\rho/2}\rho^\ell L_{n-\ell-1}^{(2\ell+1)}(\rho).

Also,

u_{n\ell}=rR_{n\ell}, \qquad P_{n\ell}(r)=r^2|R_{n\ell}(r)|^2=|u_{n\ell}(r)|^2,

and the number of interior radial nodes is $n-\ell-1$ .

Common Moments

\langle r\rangle_{n\ell} =\frac{a_Z}{2}[3n^2-\ell(\ell+1)],

\left\langle\frac1r\right\rangle_n=\frac{1}{a_Zn^2}, \qquad \left\langle\frac1{r^2}\right\rangle_{n\ell} =\frac{1}{a_Z^2n^3(\ell+1/2)}.

Assumptions and Conventions

$\int|Y_\ell^m|^2d\Omega=1$ and $\int_0^\infty r^2|R_{n\ell}|^2dr=1$ .
$n=1,2,\ldots$ and $0\le\ell\le n-1$ .
$\mu$ is the electron–nucleus reduced mass; $a_Z$ is not generally $a_0$ .

Symbols

Symbol	Meaning
$R_{n\ell}$	radial factor, dimensions $\mathsf L^{-3/2}$
$u_{n\ell}$	reduced radial function, dimensions $\mathsf L^{-1/2}$
$P_{n\ell}$	radial shell density, dimensions $\mathsf L^{-1}$
$L_q^{(\alpha)}$	generalized Laguerre polynomial
$a_Z$	reduced-mass, charge- $Z$ Coulomb length

Validity and Warnings

Do not normalize $R$ with $dr$ ; its measure is $r^2dr$ .
The endpoint $r=0$ is not counted as an interior radial node.
A radial function alone is not a complete orbital; attach $Y_\ell^m$ .
These are point-Coulomb bound states. Screening, relativistic structure, finite nuclear size, and continuum normalization require other treatments.

Canonical Treatment

Derivation, low-lying examples, radial matrix elements, plots, exercises, and references are at Radial Wavefunctions.

References

H. A. Bethe and E. E. Salpeter, Quantum Mechanics of One- and Two-Electron Atoms, Springer, 1957.
D. J. Griffiths and D. F. Schroeter, Introduction to Quantum Mechanics, 3rd ed., Cambridge University Press, 2018, Ch. 4.