Quantum mechanics
The Hydrogen Atom
One proton, one electron, and the exact solutions of the Schrödinger equation.
the maths
Exact analytic orbitals sampled as |ψ|² point clouds: the radius comes from an inverse-CDF of r²R², the direction from rejection sampling on |Y|². Gold and violet mark the two signs of the real ψ — the nodal plane between lobes is where the wavefunction crosses zero.
Every dot above is one measurement. The electron is never at a single point — it is the wavefunction ψ, and each dot is drawn from |ψ|² for the exact orbital you pick. Watch long enough and the fuzzy cloud settles into the orbital’s true shape: spheres for s, dumbbells for p, four-leaf clovers and doughnuts for d and f. The two hues mark where the real-valued wavefunction is positive or negative — the sign flips across every node, and the gaps you see are surfaces where the electron can never be found.
Nothing here is pre-drawn. The radial part Rn,l(r) comes from Laguerre polynomials and the angular part from the real spherical harmonics Yl,m; points are sampled from their product by inverse-CDF and rejection sampling, then re-sampled a little each frame so the cloud shimmers like repeated experiments. Step through the orbitals in the toggle and notice the average radius ⟨r⟩ grow with n — the readout compares the sampled mean against the closed-form value.