# Orbiting Packet

Quantum 2D · ψ under V(x, y)

`V(x, y) = (x² + y²)/2`

[Open in the app](https://www.wavelace.com/app#p=76) · [This page](https://www.wavelace.com/presets/orbiting-packet)

### What you see

The formula is the potential over the plane, `V = (x² + y²)/2`: a bowl, the two-dimensional harmonic well, which as `½ω²r²` has `ω = 1`. The packet's energy is `k₀²/2 + 1/2σ² = 2 + 1.02 = 3.02`, and `V` passes that at `r = √(2E) ≈ 2.46`. The floor is tinted outside that circle, the turning circle a classical particle of this energy could not cross.

The packet starts at `(0, −2)`, with `Packet width` 0.7 and `Momentum k₀ (along x)` 2, and it goes round and round.

### Why the orbit holds

A circular orbit of radius `r` in this bowl needs a speed of `ωr`. Here that is `1 · 2 = 2`, exactly the momentum given, and momentum is speed. So the packet runs a circle of radius 2, anticlockwise seen from above, once every `2π/ω ≈ 6.28` seconds at `Speed` 1.

What is remarkable is that it stays a packet. The bowl's own ground state has width `1/√(2ω) ≈ 0.7071`, and 0.7 is that width. This is a *coherent state*: over four turns its width holds at 0.70 with no spreading at all, while a free packet of the same width would have doubled. Ehrenfest's theorem gives the mean position the classical equation whenever `V` is quadratic. The coherent state is the case where the shape rides along unchanged, about as close as a wave gets to a particle on an orbit.

### T as a clock

On a circular orbit the readouts count out the turn. `T` is 0.5 at the start, then 1 a quarter turn in at `t ≈ 1.57`. It is back to 0.5 at 3.14 and down to 0 at 4.71, once round per period.

### Try

- Set `Momentum k₀ (along x)` to 1, half the circular speed: the circle becomes an ellipse, 1 across and 2 deep, still with a period of 6.28.
- Widen the orbit with `Packet centre y` `−3` and `Momentum k₀ (along x)` 3: it circles at radius 3 in the same 6.28 seconds, because in a harmonic bowl every orbit takes the same time.
- `Packet width` 0.4 is narrower than the bowl's own state, so the packet breathes as it goes round.
- Press `Top` to watch the circle from above, where its radius can be read against the tinted turning circle.

### Read more

- [Quantum harmonic oscillator](https://en.wikipedia.org/wiki/Quantum_harmonic_oscillator)
- [Coherent state](https://en.wikipedia.org/wiki/Coherent_state)
- [Ehrenfest theorem](https://en.wikipedia.org/wiki/Ehrenfest_theorem)
- [Correspondence principle](https://en.wikipedia.org/wiki/Correspondence_principle)