Quantum Wavepacket

The idea

A quantum particle between measurements is described by a wavefunction ψ, and the cloud you are orbiting is |ψ|² — the probability density of where a measurement would find it. Opacity is that density; colour is the wave's phase, warm where the real part of ψ is positive and cool where it is negative, so you can see the internal oscillation streaking through the envelope. This is the thing a static orbital picture cannot show: quantum states move.

The free packet demonstrates dispersion, the most quantum thing about motion. A localized packet is built from many momenta added together (that is the uncertainty principle in construction form: tight in position means wide in momentum), and in vacuum each momentum component travels at a different speed — so the packet inevitably spreads as it goes. The tighter you start it, the faster it comes apart.

The other two scenarios are the exceptions that prove the rule. A coherent state in a harmonic trap is the famous non-spreading packet: the trap's restoring force exactly balances dispersion, and the Gaussian sloshes back and forth on the classical trajectory forever — "the most classical state quantum mechanics allows". And a particle in a box, prepared in a superposition of two energy states, does not travel at all: its density sloshes wall to wall at the beat frequency (E₂−E₁)/ħ — a clock run by an energy difference.

What to look for

• In 'free', watch the phase stripes move through the envelope faster than the envelope itself moves — phase velocity and group velocity are genuinely different things, and here you can see both at once.
• Also in 'free': as the packet spreads it dims. Total probability is conserved at exactly 1, so a wider cloud must be a thinner one.
• In 'coherent', the packet reaches the turning points, pauses, and swings back without changing width — compare that directly against the free packet's decay.
• In 'box', the density never crosses the walls, and the sloshing is periodic: after one full beat the state reconstructs itself exactly.

Getting it right

• The cloud is not the particle smeared out like butter. Any single measurement finds the whole particle at one point; run the measurement many times on identically prepared systems and the hit pattern reproduces this cloud. |ψ|² is a forecast, not a substance.
• Spreading is therefore not the particle physically inflating — it is the forecast getting less specific with time, at a rate quantum mechanics dictates exactly.
• Only differences of energy are observable as motion. The box state's beat runs at (E₂−E₁)/ħ; shift both energies by the same amount and nothing visible changes. An energy eigenstate alone would sit perfectly still — which is precisely why superposition is where the action is.

Turn the knobs

• scenario is the physics selector: 'free' for dispersion, 'coherent' for the trapped classical-like slosh, 'box' for two-state quantum beats.
• density scales the cloud's opacity — thin it to see the phase structure deep inside the envelope, thicken it to read the envelope's outline.