The left panel is a classical stand-in: a mass on a spring with the same total energy as level n, oscillating between the turning points ±An. It is not the quantum state itself — a true energy eigenstate |n⟩ is stationary (|ψn(x)|² is constant in time); the moving dot only shows where a classical particle of that same energy would be, for comparison. The right panel is the energy-level diagram: the parabola is V(x)=½x², the dashed rails are the quantized energies En=(n+½)ħω, and the highlighted curve is ψn(x) = Hn(x)·e−x²/2 (normalized), riding on the En rail it belongs to. Step levels exactly with the ladder operators â / â†, or drag the weight and let go — that models an energy measurement on the coherent state the pull creates, with the panel below showing the actual (Born-rule) probability of landing on each n.
Click and drag the weight, then release. This is not a real relaxation process — it simulates measuring the energy of the coherent state your pull creates, with the outcome n drawn at random from the probabilities shown under the diagram.
Hₙ(x) is the bare polynomial — unbounded, so it's deliberately left to run off the chart outside the turning points (dotted lines). Multiplying it by the Gaussian e−x²/2 is what tames it into ψₙ(x), the actual normalizable wavefunction. |ψₙ(x)|² is the position probability density.
Releasing the mass at rest at displacement x₀ makes a coherent state |α⟩ with α = x₀/√2 — a superposition of many n, not a single eigenstate. Its number distribution is Poissonian, P(n) = e−|α|²|α|2n/n!. Free evolution would keep oscillating classically forever without collapsing; only an actual energy measurement forces a single random outcome n, with exactly these probabilities. That's why a small pull can still occasionally measure n=2 or n=3 — it doesn't deterministically round to the "nearest" level.
N atoms in a ring, each linked to its two neighbors by identical springs (stiffness K), with atom N linked back to atom 1 — periodic boundary conditions. In Fourier (normal-mode) coordinates the Hamiltonian decouples into N independent oscillators, one per allowed wavevector k: Ĥ = Σk [ (1/2m) p̂kp̂−k + ½ m ωk² x̂kx̂−k ], ωk² = (4K/m) sin²(ka/2). Each of the N allowed k's is exactly the single-oscillator problem from the first tab, just with its own ωk — and its own independent occupation number nk. The full state is a product over every mode, |n0, n1, …, nN−1⟩ — âk / âk† raise or lower only the one mode k you're currently inspecting, leaving every other mode's occupation number untouched.
Displacement is drawn transverse purely for visibility (the real motion is longitudinal, along the chain). The dashed arc is the periodic link — atom N connects back to atom 1. The dotted orange curve is the selected mode's own contribution in isolation; the atoms follow the sum of every mode's contribution, since each keeps its own occupation number.
One independent phonon number per mode — click a mode above, then use âₖ/âₖ† to change just that one.
Click any dot to jump to that mode — that's the other way to change k, besides the slider.
Levels are drawn in units of ħωk — every mode's ladder looks identical on this normalized axis, but the real energy spacing still scales with ωk (read it off the dispersion plot above).
k = 0 is the zero-frequency translation mode: the whole ring drifting rigidly. There is no restoring force for it, so it is not a bound oscillator and has no discrete levels.