Orbiting Packet
V(x, y) = (x² + y²)/2
Open in the app The dials and keys named below are the app's.
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.