# Quartic Well

Quantum · ψ under V(x)

`V(x) = 0.5x² + 0.1x⁴`

[Open in the app](https://www.wavelace.com/app#p=178) · [This page](https://www.wavelace.com/presets/quartic-well)

### What you see

The formula is the potential, `V = 0.5 · x² + 0.1 · x⁴`: the parabola of a mass on a spring with a quartic term that stiffens the walls. At `x = ±4` the quartic part is 25.6 and the parabola only 8.

The packet starts at `Packet centre` −4 with `Momentum k₀` 2 and `Packet width` 0.5. A classical particle released the same way has energy 35.6 and turns back at `x ≈ ±4.07`, so the walls at ±10 never come into it.

For a few swings the packet sloshes from side to side like a ball in a bowl. Then it comes apart. By `t ≈ 11` at `Speed` 1 its centre has settled at 0 and its spread has grown from 0.5 to about 2.8. It stays smeared across the well, rippling, with no sign of the swing it began with. Run on to `t = 500` and its spread never falls back below 2.1: the partial revivals some wells show do not come here.

### Why it comes apart

In a parabola every swing takes the same time, `2π ≈ 6.28` seconds, whatever its size. The quantum levels are evenly spaced to match, so every part of the packet returns in step, and the packet reforms forever. That is the Harmonic Well.

The quartic term breaks the rule. A swing at this energy takes about 2.61 seconds, and a swing with a little more energy is quicker still. The packet holds a spread of energies, so its faster parts pull ahead of its slower ones and it shears out round the well. In quantum terms the levels are no longer evenly spaced, and the waves that make up the packet drift out of phase. This is *dephasing*, and it is why a packet that keeps its shape is special to the parabola.

### History

The quartic oscillator is the standard test of perturbation theory. In 1969 Carl Bender and Tai Tsun Wu showed that the series for its energy levels in powers of the quartic coefficient diverges, however small the coefficient.

### Try

- Change 0.1 to 0: the plain parabola, and the packet swings back and forth every 6.28 seconds, breathing a little but never smearing.
- Change 0.1 to 0.01: the walls are nearly parabolic, and the packet keeps its swing for about 12 seconds and is smeared by `t ≈ 25`.
- Set `Packet centre` to −1 and `Momentum k₀` to 0: the packet stays low in the well, where the quartic term hardly matters, and its spread stays near 1 for 30 seconds.

### Read more

- [Anharmonicity](https://en.wikipedia.org/wiki/Anharmonicity)
- [Quantum harmonic oscillator](https://en.wikipedia.org/wiki/Quantum_harmonic_oscillator)
- [Wave packet](https://en.wikipedia.org/wiki/Wave_packet)
- [Quantum revival](https://en.wikipedia.org/wiki/Quantum_revival)