# Double Well

Quantum · ψ under V(x)

`V(x) = (x² − 4)²/8`

[Open in the app](https://www.wavelace.com/app#p=72) · [This page](https://www.wavelace.com/presets/double-well)

### What you see

The formula is the potential, `V = (x² − 4)²/8`. It is zero at `x = ±2` and rises to `V(0) = 2` between them: two wells with a hill in the middle, the classic shape for a thing that can be in one of two places. With the walls at `±6`, where `V` reaches 128, the wells are drawn almost flat. Pull `Span of x` in and they show.

The packet starts in the left well, at `Packet centre` `−2`, at rest, with `Packet width` 0.5.

### Why it leaks

Near the bottom of a well the potential is a parabola. At `x = −2` it is `V ≈ 2(x + 2)²`, which is `½ω²u²` with `ω = 2`. That well's own ground state has width `1/√(2ω) = 0.5`, exactly the width the packet is given. So it settles in almost at rest and only breathes a little, out to about 1.0 and back, because the true well is not quite parabolic.

Its energy is the readout's `1/4σ² = 1` of motion plus about 0.52 of potential where it sits, so roughly 1.52. That is below the 2 at the top of the hill, and it should stay where it is forever. Instead `T` creeps up: 0.07 by `t = 3`, 0.10 by 12, 0.21 by 20 at `Speed` 1. The particle is tunnelling through the middle, and given long enough the two wells share it.

### Where it turns up

Ammonia is built like this. The nitrogen atom of `NH₃` sits above or below the plane of its three hydrogens, two wells with a barrier between, and it tunnels through at 23.8 GHz. That line is what the first maser amplified, built by Gordon, Zeiger and Townes in 1954.

### Try

- Lower the hill, `(x² − 4)²/16`: the top drops to 1, no longer above the packet's energy, so it spills over instead of tunnelling and `T` reaches 0.83 by `t = 20`.
- Push the wells apart, `(x² − 9)²/8`, with `Packet centre` `−3`: the barrier is now 10 tall and six wide, and almost nothing crosses.
- Start on top of the hill: `Packet centre` 0. It splits evenly and stays even, half in each well.
- Pull `Span of x` in to 4 to see the shape of the two wells.

### Read more

- [Double-well potential](https://en.wikipedia.org/wiki/Double-well_potential)
- [Quantum tunnelling](https://en.wikipedia.org/wiki/Quantum_tunnelling)
- [Ammonia maser](https://en.wikipedia.org/wiki/Ammonia_maser)
- [Quantum harmonic oscillator](https://en.wikipedia.org/wiki/Quantum_harmonic_oscillator)