Slice through any hydrogen orbital, with radial probability, Monte-Carlo electron samples and the photon of every level jump.
The electron in a hydrogen-like atom is described by ψnlm(r,θ,φ) = Rnl(r) Ylm(θ,φ) with energy En = −13.606 eV / n². The radial part is Rnl = N e−ρ/2 ρl Ln−l−12l+1(ρ) with ρ = 2r/(n a₀) (generalised Laguerre polynomial), and Ylm are real spherical harmonics, so m > 0 orbitals use cos(mφ) and m < 0 use sin(|m|φ).
P(r) dr = r² R² dr. The 1s peak is at exactly one Bohr radius a₀; the mean radius is ⟨r⟩ = a₀ (3n² − l(l+1)) / 2.n − l − 1; angular nodal surfaces = l.1/λ = RH |1/nf² − 1/ni²| (Lyman, Balmer, Paschen series).Checks. The page integrates the radial density numerically (normalisation = 1, mean radius vs the exact formula), counts nodes from sign changes, integrates the angular density over the sphere, compares the Monte-Carlo mean radius of 20000 sampled electrons with theory, and recomputes the H-alpha line from the Rydberg formula.
Simplifications. Non-relativistic Schrödinger solution for a point nucleus: no spin, fine structure, Lamb shift or hyperfine splitting; the electron is not a tiny orbiting ball - the dots are samples of a probability distribution. The pictured plane is a cut through the cloud, not a trajectory. Energies use the infinite-mass limit while wavelengths use the hydrogen Rydberg constant.