Squeezed Packet
V(x) = (1/2) · ω²x²
Open in the app The dials and keys named below are the app's.
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.