Plot the radial probability distribution function ψ²(r) for any hydrogen atom orbital from 1s through 4f.
The radial probability distribution function answers a deceptively subtle question: at what distance from the nucleus is an electron most likely to be found? This IChO-level quantum chemistry topic goes beyond the simple orbital-shape pictures taught in introductory courses and requires distinguishing the wave function ψ(r) from the probability density ψ²(r)4πr².
The wave function ψ(r) itself is largest at the nucleus for an s orbital — but that does not mean the electron is most likely to be found there. The radial probability distribution multiplies |ψ(r)|² by the volume of a thin spherical shell at radius r, which is proportional to 4πr². Near the nucleus this shell volume is nearly zero, so even though ψ² is large there, the probability of finding the electron in that vanishingly thin shell is also near zero. The most probable radius for a 1s electron in hydrogen turns out to be exactly one Bohr radius (52.9 pm), not r = 0.
Radial nodes — spherical surfaces where the probability of finding the electron is exactly zero — occur where the radial wave function changes sign. The number of radial nodes for any orbital equals n - l - 1. A 1s orbital (n=1, l=0) has zero radial nodes; a 2s orbital (n=2, l=0) has one; a 3p orbital (n=3, l=1) has one as well. Counting nodes correctly is a frequent source of error on IChO quantum problems, especially distinguishing radial nodes from angular nodes (which depend only on l).
Comparing radial distributions across orbitals reveals orbital penetration: a 3s electron, despite having higher average energy than a 3p or 3d electron, has significant probability density very close to the nucleus due to an inner lobe — this penetration effect is what causes the 4s orbital to fill before 3d in multi-electron atoms, explaining several periodic table exceptions.
This radial probability simulator plots ψ(r) and ψ²(r)4πr² side by side for any hydrogen orbital from 1s through 4f, letting you directly see node positions, most probable radius, and orbital penetration — visualizations rarely available outside advanced physical chemistry coursework.