Hydrogen

Exact solutions of the Schrödinger equation, sampled point by point.

Orbital

—

n principal 4 ℓ angular 2 m magnetic +1

Measured

Energy—

⟨r⟩—

Nodes—

Beat—

Measure

A measurement never returns a cloud. It returns one location. Each click re-prepares the atom in the same state and finds the electron once — the cloud is what the outcomes add up to.

Outcomes0

Last—

Hide the cloud — rebuild it from measurements alone

Superposition

Mix two states and evolve in time

A single eigenstate is stationary — |ψ|² never changes. Mix two and the cross term beats at (E₂−E₁)/ℏ: the cloud sloshes, and that oscillating dipole is what radiates a photon.

n₂1 ℓ₂0 m₂0 Speed1.00×

Render

Quality ray steps 180 Opacity 38.0 Detail volume grid 128³ Points 500,000 Size Exposure Style Colour Real orbitals (p_x, d_xy…) Flow probability current 0.35 rad/s Slice cross-section solid Cutaway — remove a quadrant to expose the shells Auto-rotate

Reading it

Every dot is one draw from |ψ|² — where the electron would be found.
The grains drift along the real probability current, v_φ = ℏm/(μr sinθ). A stationary state is stationary, not static: the density never changes while the electron keeps circulating.
Hue (in glow style) is the phase of ψ, winding once per unit of m.
Dark shells are nodes, where ψ passes through zero: n−ℓ−1 radial, ℓ angular.
Scale grows as n² — 8g is ~40× wider than 1s.

Hydrogen only. One electron means the Schrödinger equation is exactly solvable, so nothing here is fitted or approximated. Helium onward has no closed form — two electrons repelling each other needs Hartree–Fock or DFT, and “orbital” stops being an exact object.