# Scattering Disc

Quantum 2D · ψ under V(x, y)

`V(x, y) = 8(r < 1)`

[Open in the app](https://www.wavelace.com/app#p=75) · [This page](https://www.wavelace.com/presets/scattering-disc)

### What you see

The formula is the potential over the plane. `r` is fed to it as `hypot(x, y)`, the distance from the origin, so `r < 1` is a disc of radius 1 there and the 8 is how high it stands. The packet's energy is `k₀²/2 + 1/2σ² = 4.5 + 0.78 = 5.28`, well under that, so the floor is tinted right across the disc.

The packet comes in from `(−4, 0.6)`, aimed to pass above the centre of the disc but well inside its edge. It carries `Packet width` 0.8 and `Momentum k₀ (along x)` 3, and strikes about 1.3 seconds in at `Speed` 1.

### The scattering

The wavelength is `2π/k₀ ≈ 2.09` and the disc is 2 across, so the obstacle is about one wavelength wide. Nothing here behaves like a ball off a post: the wave wraps the disc. What gets past is thrown up and away from the side it struck. On an arc of radius 6 the density is heaviest between 35° and 50° above the axis, and nearly nothing below it. Straight behind the disc there is a shadow, which diffraction fills back in as the wave travels on. That angular spread, measured far from the target, is what a scattering experiment reports as a cross section.

### Where T settles

Once the collision is over, near `t = 3.5`, the readouts hold at about 0.65 and 0.35. A third of the packet has been thrown back the way it came. The norm readout stays at 1.0000, so whatever leaves one side is accounted for on the other.

### Try

- Aim it dead centre, `Packet centre y` 0. The pattern turns symmetric, and just behind the disc the axis is brighter than the shadow either side of it: the bright point Poisson derived to ridicule the wave theory of light, and Arago then found.
- `Packet centre y` 2 misses the disc altogether: `T` goes to 0.90 and the shadow is a dent in one edge of the sheet.
- Make it hard, `30 · (r < 1)`: less leaks through the disc and `T` falls to about 0.57.
- Give it something bigger to get round, `8 · (r < 2)`, and the shadow behind it widens.

### Read more

- [Scattering](https://en.wikipedia.org/wiki/Scattering)
- [Cross section (physics)](https://en.wikipedia.org/wiki/Cross_section_(physics))
- [Arago spot](https://en.wikipedia.org/wiki/Arago_spot)
- [Diffraction](https://en.wikipedia.org/wiki/Diffraction)