3D harmonic oscillator eigenfunctions
Ray-marched 3D plots of the quantum harmonic oscillator — the same WebGL engine as the atomic orbitals, with a twist: instead of a static eigenstate, the plot animates a superposition of two states A and B. Three evolutions to play with: the classic 2011 animation this page ran for fifteen years (real weights cos t and sin t — preserved as the bug it was, its zero probability current exposed by the tracer ensemble), free evolution, where the weights are fixed and only the relative phase winds (the cloud sloshes through its interference term), and a resonant drive, where the state Rabi-oscillates and the cloud morphs from A to B and back. In the Cloud display you can watch the probability current carry the sloshing — or refuse to carry the classic morph — and run weak position measurements that collapse the cloud onto a wandering point. The story of what was wrong with the original animation is in A superposition is not a morph, and the drive physics in the Rabi oscillation article.
The one-dimensional twin
Everything the 3D states do, the 1D eigenstates do without the ray-marching in the way. This widget plots the position probability of the first several Hermite–Gaussian states — and lets a classical and a relativistic oscillator answer the same question, for contrast.
The superposition in one dimension
The A+B animation on this page is the 3D version of a much simpler picture. Here it is in 1D: pick the two states, flip between the 2011 morph and honest free evolution, and watch the probability current underneath — the arrows that carry the slosh in one mode and vanish identically in the other.
ω = 1: fixed weights, the current carries the slosh, ∂tρ + ∂xj ≈ 0.
The phase-space view
One more way to see the same oscillator: a joint distribution over position and momentum, carried by the classical harmonic flow. The distribution shears and rotates rigidly — Liouville’s theorem made visible — and the two marginals along the edges are the position and momentum probabilities of the panels above.