The three numbers

The quantum harmonic oscillator is a particle in a bowl-shaped potential — the quantum version of a mass on a spring. Its eigenstates are labeled n_x, n_y, n_z: how many times the wavefunction crosses zero along each axis.

The energy depends only on the total N = n_x+n_y+n_z, so states like 2,0,0 and 1,1,0 share an energy level — same energy, different shape.

The animation: superposition

Classic: the animation this page ran from 2011, Ψ = cos t·ψ_A + sin t·ψ_B with real weights. The cloud morphs A → B → A — but this is no Hamiltonian's evolution: a real wavefunction carries identically zero probability current, so the density teleports rather than flows.

Free: the true unitary superposition Ψ = (ψ_A + e^−iωtψ_B)/√2, with ω the energy difference. The weights are fixed; only the relative phase winds, and the density sloshes through the interference term ψ_Aψ_B cos ωt. Pick A and B on the same energy level and the cloud stands still.

Driven: a resonant field couples the two levels and the state Rabi-oscillates, Ψ = cos(Ωt/2)·ψ_A − i·sin(Ωt/2)·ψ_B. The weights themselves cycle, so the cloud morphs from A to B and back — the relative phase pinned at −π/2, which is why the morph is clean.

Reading the picture

Brightness follows the probability of finding the particle there. Blue marks regions where Re Ψ is positive, red where it is negative. Dark planes are nodes — the zero crossings counted by n_x, n_y, n_z.

The Slice and Cutaway displays expose structure hidden inside the cloud, and lowering the brightness thins it.

Weak measurement & current

In the Cloud display, the probe slider softens the position measurement: a Gaussian meter of width σ applied to |ψ|². Each measure click drops a marker at the outcome and recentres the probe; run repeats this with a narrowing probe, collapsing the cloud onto a wandering point while the superposition sloshes underneath.

The current switch sets a particle ensemble flowing along the Bohmian velocity j/ρ — what a weak momentum measurement returns — integrated on the GPU, with the ensemble size adapted to your hardware. In free mode the ensemble stays Born-distributed as it moves (the continuity equation), so the flowing particles are the density: the cloud sloshes because the current carries it.

In classic mode the switch runs the same honest computation on the 2011 state, and the answer is the autopsy: that state is real, so j/ρ is identically zero. The tracers stand still while the cloud morphs through them — the continuity violation that fifteen years of morphing was hiding, made visible.

In driven mode the switch stays off, honestly: for the two-level Rabi state, no spatial current transports the density — population transfer runs through the drive coupling, and j/ρ for the truncated state fails the continuity equation. The exact driven state only conserves probability once the counter-rotating corrections are kept.