Radial Wavefunctions

Solving the radial equation produces a family of functions $R_{n\ell}(r)$ , one for each allowed pair $(n, \ell)$ . These functions carry all the information about how the electron's probability is distributed in radius. Here we identify them and learn to read their structure.

The Bohr radius sets the scale

The natural length scale of hydrogen is the Bohr radius,

a_0 = \frac{4\pi\epsilon_0\,\hbar^2}{\mu\, e^2} \approx 0.529\,\text{\AA} = 0.529 \times 10^{-10}\,\text{m}.

Every radial wavefunction is most cleanly written in terms of the dimensionless variable $r/a_0$ .

The form of the solutions

A bound-state solution of the radial equation has three factors:

R_{n\ell}(r) = \underbrace{N_{n\ell}}_{\text{normalization}}\; \underbrace{\left(\frac{r}{a_0}\right)^{\!\ell}}_{\text{origin behaviour}}\; \underbrace{L_{n-\ell-1}^{2\ell+1}\!\left(\frac{2r}{n a_0}\right)}_{\text{polynomial}}\; \underbrace{e^{-r/(n a_0)}}_{\text{decay}}.

Reading left to right: the $(r/a_0)^\ell$ factor is the small- $r$ behaviour forced by the centrifugal barrier; the $L_{n-\ell-1}^{2\ell+1}$ are the associated Laguerre polynomials, which create the wiggles; and the exponential $e^{-r/(n a_0)}$ enforces decay at large $r$ . The larger $n$ is, the slower the decay, so higher states reach further out.

The first few radial functions

The lowest states are worth memorizing:

R_{10}(r) = \frac{2}{a_0^{3/2}}\, e^{-r/a_0},

R_{20}(r) = \frac{1}{2\sqrt{2}\,a_0^{3/2}}\left(2 - \frac{r}{a_0}\right) e^{-r/(2a_0)},

R_{21}(r) = \frac{1}{2\sqrt{6}\,a_0^{3/2}}\,\frac{r}{a_0}\, e^{-r/(2a_0)}.

The ground state $R_{10}$ is a simple decaying exponential — no nodes. The $2s$ state $R_{20}$ has a factor $(2 - r/a_0)$ that vanishes at $r = 2a_0$ : one radial node. The $2p$ state $R_{21}$ starts at zero (because of the $r$ factor) but has no node at finite nonzero $r$ .

Counting nodes

The number of radial nodes (zeros at finite $r > 0$ , not counting the origin) is

n_r = n - \ell - 1.

This is the degree of the Laguerre polynomial. So for fixed $n$ , increasing $\ell$ removes radial nodes: the $3s$ state has $2$ radial nodes, $3p$ has $1$ , and $3d$ has $0$ . The total number of nodes (radial plus angular) is always $n - 1$ , mirroring the 1D pattern where the $k$ -th state has $k$ nodes.

The takeaway

The hydrogen radial functions $R_{n\ell}(r)$ are an $(r/a_0)^\ell$ factor times an associated Laguerre polynomial times $e^{-r/(n a_0)}$ , scaled by the Bohr radius $a_0$ . The polynomial degree $n_r = n - \ell - 1$ counts the radial nodes and forces $\ell \le n-1$ .