A free particle in two dimensions is described by a cloud of probability that spreads out on its own. This sim lets you watch the cloud spread and then measure the particle's position, which is where quantum probability meets a single result.

What's on screen

• Blue surface (3D view): its height at (x, y) is |ψ|² at that point.
• Heatmap (the other view): the same |ψ|² as colour, with a colour bar on the right.
• Green flash: the surface turns green for a moment when you press Measure.
• Statistics: time, peak |ψ|², probability readouts, and the starting position.
• Recent Collapses: the Final X and Final Y of your last 10 measurements.

Press ⌖ on a row to see where that control sits in the sim.

Fixed: ħ = m = 1, no potential, and the display covers x, y ∈ [−10, 10]. The packet has zero mean momentum, so its centre does not drift.

The surface height, and the colour of the heatmap, are one quantity: the probability density |ψ(x, y, t)|². It is probability per unit area, so a single value is not a probability. Probabilities come from adding up density over a region, and the volume under the whole surface is exactly 1 at every moment.

• Total Probability reads 1, the volume under the whole surface. Visible Probability is the part inside the display square. It falls only when the tails of a very wide packet reach the edge.
• The sim evaluates only |ψ|². It never shows the phase of ψ.
• The heatmap colour scale is fixed to the opening peak 1/(2πσ²). A packet that spreads fades toward dark blue instead of being rescaled.

Nothing pushes the packet, so it only widens. It stays a Gaussian, and its width after time t (in units with ħ = m = 1) is:

• The volume under the surface stays 1, so a wider packet must be lower: the peak falls as 1/σ(t)².
• The peak halves at τ = 2σ², when the width has grown by √2. The width doubles at t₂ = 2√3 σ².
• A packet that starts narrower spreads sooner: τ is four times shorter when σ is halved.

A measurement returns one point (x_m, y_m), not a patch. Where it falls is random, but the odds are set by the density: the probability of landing in a region A is |ψ|² added up over A (the Born rule). The sim draws x and y from the exact Gaussian, restricted to the display square, so its odds are |ψ|² renormalised on that square.

• Bright regions are likely and dark regions are rare, but dark does not mean forbidden.
• The most likely single point is the centre. The most likely distance from the centre is r = σ(t), because rings farther out have more area.
• P(r < kσ(t)) = 1 − e^−k²/2. That is 39% inside σ(t) and 86% inside 2σ(t). The familiar 68% is for one coordinate alone.
• Each Measure replaces the packet (next section), so a second press in the same run does not sample this hump. Start a fresh packet for each outcome.

After the outcome (x_m, y_m), the sim replaces the packet with a new Gaussian of width σ, the Measurement Width, centred on the outcome. The old density is gone.

• This models a measurement with finite resolution σ. A perfectly sharp position state cannot be normalised, and real detectors have finite resolution.
• A repeat measurement straight away is not guaranteed to agree with the first. It lands within about σ of it.
• One slider sets both widths. Measuring just after Play barely narrows the packet, because it has not spread yet. Measure after it has spread and the width drops back to σ.
• The sim applies the textbook collapse rule. Interpretations of quantum mechanics differ on what this step is physically, and the sim takes no side.

The new packet is free, like the first one, and it spreads the same way from the moment of the measurement t_m:

• The same σ sets the new packet's τ = 2σ², so its peak halves again after the same time as the first packet's did.
• Nothing pulls the packet back toward the original centre. It is centred on the outcome, and it widens from there.
• Measure again long after the first outcome and the new outcomes scatter over a patch of size about σ(t − t_m). That patch grows with the waiting time.

Restoring units, the doubling time is t₂ = 2√3 mσ²/ħ. It grows as the mass m and as the square of the width.

• Electrons. A free electron prepared with a width of 1 nm doubles it in about 30 fs. Tight localisation is short-lived unless something holds the electron in place.
• Heavy objects. A 1 µg grain localised to 1 µm needs about a million years to double. On any human timescale it does not spread, which is why everyday objects look classical.
• Matter-wave interferometry. Interferometers for atoms and molecules rely on each packet being wider than the spacing between the paths it must pass through.

Key points

• The packet's width grows as σ(t) = √(σ² + (t/2σ)²), so its peak falls while the volume stays 1.
• One outcome per measurement, drawn from |ψ|²: 39% inside r = σ(t), 86% inside 2σ(t).
• After a measurement the packet has width σ, sits on the outcome, and spreads again from t − t_m = 0.

Common mix-ups

• "A measurement just reads off where the particle already was." In the rule the sim uses, the outcome is drawn from |ψ|² and the packet is then replaced. Whether the particle had a position beforehand is an interpretation question the sim does not settle.
• "The dark surround is forbidden." |ψ|² is small there but not zero. Rare is not impossible.
• "A measurement pins the particle down for good." The new packet starts spreading at once, exactly as the first one did.

Things to try on your own

• Pick a width σ, work out τ = 2σ², press Play, and check Time Elapsed when Peak |ψ|² reaches half its opening value.
• Let the packet spread past t₂, press Measure, and compare Peak |ψ|² before and after.
• Repeat Reset, Play, Measure twenty times. Count how many Final X, Final Y pairs have √(X² + Y²) below σ = 0.75 and compare with 39%.