# Squeezed Packet

Quantum · ψ under V(x)

`V(x) = (1/2) · ω²x²`

[Open in the app](https://www.wavelace.com/app#p=146) · [This page](https://www.wavelace.com/presets/squeezed-packet)

### What you see

The formula is the potential, `½ω²x²`, the parabola that holds a mass on a spring, its stiffness left as a slider. It opens at `ω = 2`, so the bowl is four times as steep as the plain `x²/2`.

The packet is released at `Packet centre` 3 with `Momentum k₀` 0, at rest and off centre, with `Packet width` 0.7. It slides down through the middle, up to `x = −3`, and back, and the round trip takes `2π/ω ≈ 3.14` seconds at `Speed` 1. Watch the shape rather than the position: thin as it crosses the middle, fat at each end.

### The swing and the breathing

Because `V` is quadratic, Ehrenfest's theorem makes the mean position obey the classical equation exactly, so it swings at `2π/ω` whatever shape the packet has. Measured: 3.16 seconds at `ω = 2`, 6.29 at 1, 2.12 at 3, each within about a per cent.

The width is not carried along with it. A Gaussian released at rest here has its width swing between `σ` and `1/(2σω)`, widest at the turning points and narrowest in the middle, so twice in each round trip. With `σ = 0.7` and `ω = 2` those are 0.70 and 0.36, and the blob on the plot traces 0.35 to 0.70. They coincide only when `ω = 1/(2σ²)`, which for 0.7 is `≈ 1.0`. There the released Gaussian is the bowl's own ground state moved sideways, a *coherent state*, and it slides without changing at all. That is the preset Harmonic Well. Anywhere else the packet is squeezed, so that preset's stillness is a coincidence of its width rather than a property of the bowl.

### History

Schrödinger looked in 1926 for a solution of his new equation that would follow a classical orbit, and found this one, a Gaussian of the oscillator's own width that neither spreads nor changes shape. States whose width instead breathes, narrow in position at one moment and in momentum a quarter period later, were named squeezed states much later. They are made in the laboratory now, to push interferometer noise below what an ordinary packet allows.

### Try

- Drag `ω` to 1. The breathing stops, the width holds near 0.70 swing after swing, and the round trip doubles to 6.29 seconds. That is the preset Harmonic Well.
- Drag `ω` to 3. Over the first round trip the width swings from 0.23 in the middle to 0.70 at the ends, and that trip takes 2.12 seconds.
- Leave `ω` at 2 and set `Packet width` to 0.5, which is `1/√(2ω)`, this bowl's own ground state width. The breathing nearly stops, staying near 0.5.
- Drag `ω` to 0. With no bowl there is nothing to pull the packet back, so it simply spreads, from 0.70 to about 4.3 in ten seconds, until the walls at `±8` have it and it fills the box.

### Read more

- [Squeezed coherent state](https://en.wikipedia.org/wiki/Squeezed_coherent_state)
- [Coherent state](https://en.wikipedia.org/wiki/Coherent_state)
- [Quantum harmonic oscillator](https://en.wikipedia.org/wiki/Quantum_harmonic_oscillator)
- [Ehrenfest theorem](https://en.wikipedia.org/wiki/Ehrenfest_theorem)