# Double Slit

Quantum 2D · ψ under V(x, y)

`V(x, y) = 8(abs(x) < 0.2) · (abs(abs(y) − 1) > 0.4)`

[Open in the app](https://www.wavelace.com/app#p=73) · [This page](https://www.wavelace.com/presets/double-slit)

### What you see

The formula is the potential over the plane, and it is a product of three things. `abs(x) < 0.2` stands a wall 0.4 thick along the `y` axis. `abs(abs(y) − 1) > 0.4` punches two openings in it, each 0.8 wide, centred on `y = ±1`, so the centres are 2 apart. The 8 is how tall the wall is. Against that the packet's energy is `k₀²/2 + 1/2σ² = 4.5 + 0.5 = 5`, so the floor under the wall is tinted as ground it has no classical right to enter.

The packet starts at `(−4, 0)` with `Packet width` 1 and `Momentum k₀ (along x)` 3. Its wavelength is `2π/k₀ ≈ 2.09`, and it reaches the wall about 1.3 seconds in at `Speed` 1.

### The pattern

Two openings, one wave. Past the wall the pair act as two sources in step, and in directions where the two paths differ by half a wavelength they cancel. With centres `d = 2` apart that is `sin θ = λ/2d ≈ 0.52`, about 32° off the axis, and on an arc of radius 6 the density does fall to near nothing there. It is the only dark line the geometry allows, since the next would need `sin θ = λ/d ≈ 1.05`. Long waves through close slits give a broad central lobe with one dark line each side, not a train of fringes.

### Where T settles

`T` settles near 0.82, far more than two narrow slits can pass. The wall is thin and not very tall: at an energy of 5 against a height of 8 the wave dies inside it over `1/√(2(8 − 5)) ≈ 0.41`, about the wall's own thickness. Much of the packet therefore tunnels straight through it. Seal the openings and `T` is still 0.55. That leak also fills in the dark lines.

### Try

- Change the 8 to 30: the wall goes opaque, `T` drops to about 0.53, and the dark lines deepen into real nulls.
- Close one slit, `8 · (abs(x) < 0.2) · (abs(y − 1) > 0.4)`, and the pattern collapses to a single fan.
- Set `Momentum k₀ (along x)` to 2: the wavelength grows to `π ≈ 3.14` and the dark line swings out past 50°.
- Press `Top` for the plan view, where the two dark lines are easiest to place.

### Read more

- [Double-slit experiment](https://en.wikipedia.org/wiki/Double-slit_experiment)
- [Matter wave](https://en.wikipedia.org/wiki/Matter_wave)
- [Wave interference](https://en.wikipedia.org/wiki/Wave_interference)
- [Quantum tunnelling](https://en.wikipedia.org/wiki/Quantum_tunnelling)