Where the electron lives.
An electron in a hydrogen atom has no orbit. It has a wave function. The square of that wave function gives the probability density of finding it at any point. The clouds you see here are real samples of that density. The colour separates the two signs of the wave function, the feature that turns into chemical bonding once you have more than one atom.
Hydrogenic ψ_nlm · real spherical harmonics · rejection sampling on |ψ|² · Z=1 in Bohr radii
Fourteen orbitals up to n=3
Click an orbital below. The cloud rebuilds, you can spin it. The number n sets the size, the number ℓ sets the shape, the number m picks the orientation. The two-colour clouds show the sign of the wave function changing across a node. That sign is what makes molecules bond constructively or repel.
Why this matters for chemistry
Every real atom heavier than hydrogen has many electrons, but the orbital labels stay: 1s, 2s, 2p, and so on. The shapes you see here are the building blocks of every chemistry textbook diagram, of every bond geometry, of why water is bent and methane is tetrahedral. They are the visual language of the periodic table.
Three quantum numbers
The number n is the principal quantum number, n = 1, 2, 3 ... It sets the size and the energy. The number ℓ is the angular momentum, 0 to n−1. ℓ = 0 is called s, ℓ = 1 is p, ℓ = 2 is d. The number m is the magnetic quantum number, −ℓ to +ℓ. It picks the orientation. Three integers, fourteen distinct shapes up to n = 3.
Nodes are where the cloud thins out
Every orbital except 1s has n−1 radial nodes — concentric shells where the wave function passes through zero. The colour flips across each node. The 2s has one such shell, the 3s has two. The p and d orbitals add angular nodes, the planes through the nucleus where the cloud disappears. The bilobed shape of 2p is just that: one angular node between the lobes.
Real versus complex
Solving the Schrödinger equation in spherical coordinates gives complex-valued spherical harmonics. For chemistry and visualisation, we usually take real linear combinations of those: 2p_x and 2p_y instead of 2p_+1 and 2p_−1. The shapes you see here are the real ones — these are what the textbook orbital pictures actually plot.
Beyond hydrogen
These exact wave functions only solve the one-electron problem. Helium already needs approximation methods. Modern computational chemistry uses linear combinations of hydrogenic orbitals (LCAO) as basis functions. The clouds you see are not just pedagogical pictures: they are the building blocks of every density functional theory calculation you have ever read about.
Sources
- Schrödinger E. (1926). Quantisierung als Eigenwertproblem. Annalen der Physik 384, 361–376. Erste Loesung fuer das Wasserstoff-Atom.
- Griffiths D.J., Schroeter D.F. (2018). Introduction to Quantum Mechanics. 3rd ed., Cambridge University Press. Kapitel 4 zum Wasserstoff-Atom.
- Cohen-Tannoudji C., Diu B., Laloë F. (2019). Quantum Mechanics, Volume II. Wiley-VCH. Kapitel VII Komplement B zu sphaerischen Harmonischen.
- Pauling L. (1939). The Nature of the Chemical Bond. Cornell University Press. Erste systematische Anwendung der reellen Orbital-Formen auf Bindungen.