Hydrogen Radial Wavefunctions

Formula

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.

Related Lookup

• Hydrogen Spectrum
• Hydrogenic Ions
• Laguerre Polynomials