# Free Packet

Quantum · ψ under V(x)

`V(x) = 0`

[Open in the app](https://www.wavelace.com/app#p=68) · [This page](https://www.wavelace.com/presets/free-packet)

### What you see

The formula is the potential, and here it is `0`. Nothing pushes the particle about, so there is no silhouette standing on the plane and no energy line across it, only the packet.

It starts as a Gaussian at `Packet centre` `−4`, of width `σ = 0.5`, carrying `Momentum k₀` 3: `ψ₀ = exp(−(x + 4)²/4σ²) · exp(3ix)`. Momentum is also speed here, so at `Speed` 1 the hump slides right at 3 units a second. It passes the origin at `t ≈ 1.33`, and meets the wall at `x = 8` near `t = 4`, where it bounces.

### Why it spreads

A hump that narrow has to be built from a range of momenta, `Δk = 1/2σ = 1` either side of 3. Each of them travels at its own speed, so the packet smears as it goes: `σ(t) = σ · √(1 + (t/2σ²)²) = 0.5 · √(1 + 4t²)`. It is twice as wide by `t ≈ 0.87`, and four times by `t = 2`. Nothing is lost while it flattens. The area under the curve is fixed, and the norm readout holds at 1.0000, because the solver is exactly unitary.

### Where T settles

Nothing turns this packet back, so `T` climbs through 0.19 at `t = 1` and 0.63 at `t = 1.5`. It then levels near 0.96 rather than 1, because by that time the packet is wide enough to leave a tail behind the origin. Once the wall has returned it, the two readouts settle near 0.5 apiece.

### Try

- Raise `Packet width` to 2: less momentum spread, so it holds together, only half as wide again by `t = 4`.
- Set `Momentum k₀` to 0: the packet stays where it is and spreads both ways at once.
- Put something in its path: `3 · (abs(x) < 0.4)` is a barrier, and most of it comes back.
- Press `Top` in the deck: the trailing past of the curve reads as a wake, widening as it goes.

### Read more

- [Wave packet](https://en.wikipedia.org/wiki/Wave_packet)
- [Free particle](https://en.wikipedia.org/wiki/Free_particle)
- [Uncertainty principle](https://en.wikipedia.org/wiki/Uncertainty_principle)
- [Schrödinger equation](https://en.wikipedia.org/wiki/Schr%C3%B6dinger_equation)
- [Crank–Nicolson method](https://en.wikipedia.org/wiki/Crank%E2%80%93Nicolson_method)