radial wavefunction

The radial wavefunction is the part of an atomic orbital that depends only on the distance from the nucleus, written R(r). Because the Coulomb potential is the same in every direction, the full wavefunction splits neatly into two factors: one that handles direction and shape, and this radial one that handles how the electron's probability rises and falls as you move outward. It answers the question, how does the cloud thin out with distance.

Its form carries fingerprints of the quantum numbers. The principal number n sets how far out the cloud reaches and how many times the radial wave crosses zero on the way; each such crossing is a spherical shell where the electron is never found, called a radial node. The number of these nodes climbs with n and falls as the orbital's angular momentum grows, which is why a 3s orbital ripples in and out more than a 3p or 3d.

To find where the electron is most likely to be, you do not look at R(r) alone but at the radial probability — R(r) squared multiplied by the surface area of a sphere at that distance. That extra factor accounts for there being more room far out, and it shifts the most-likely distance outward from where R alone peaks. In the hydrogen ground state this radial probability peaks neatly at the Bohr radius, tying the rigorous theory back to the old model's lucky guess.