Hydrogen Radial Wavefunctions

Solving the hydrogen atom is one of the triumphs of quantum mechanics: the electron in the Coulomb pull of the proton has its wavefunction split into an angular part, the familiar orbital shapes, and a radial part $R_{nl}(r)$ that says how the probability is spread out from the nucleus. That radial piece is an associated Laguerre polynomial times a decaying exponential, and its structure is pinned by two integers, the principal number $n$ and the angular number $l \lt n$. The quantity that actually tells you where the electron is likely to be found is the radial probability density $P(r) = r^2|R_{nl}|^2$, the chance of finding it in a thin shell at radius $r$; the $r^2$ from the growing shell area competes with the wavefunction to set the peaks. It has exactly $n - l - 1$ nodes, spheres where the electron is never found, and its most probable radius marches outward roughly as $n^2$ Bohr radii, the atom swelling as it climbs to higher states. The scene shows the orbital two ways: a disk of the radial wavefunction oscillating in time, where the nodes appear as dark rings between shells of alternating sign, and beside it the probability curve $P(r)$ with its nodes, its peak, and its mean radius marked. The bottom panel is the energy ladder $E_n = -13.6\,\text{eV}/n^2$, the rungs crowding toward zero as the electron is less and less bound, all the states of a given $n$ sharing the same energy whatever their $l$, the special degeneracy of the pure Coulomb force. Step $n$ and $l$ and watch a node appear, the cloud swell, and the level shift.