Soft Corral
V(x, y) = 6(1 + erf(3(r − 3)))
Open in the app The dials and keys named below are the app's.
What you see
The formula is the potential, and it depends only on r, the distance from the centre, so everything about it is circular. erf runs from −1 to +1, which makes 6(1 + erf(3(r − 3))) climb from 0 to 12.
The 3 subtracted from r puts the middle of that climb at r = 3, where the potential is 6. The 3 multiplying it sets how fast the climb happens: the potential is 0.20 at r = 2.5 and 11.80 at r = 3.5, so the wall goes up across roughly one unit of radius. Inside r = 2.5 the floor is flat and empty.
A box, not a spring
That flat floor is what makes this different from a harmonic trap. Momentum k₀ (along x) 3 and Packet width 1 give an energy of k₀²/2 + 1/(2σ²) = 4.5 + 0.5 = 5, and the potential reaches 5 at r = 2.95. That radius is the edge of the untinted floor, and about nine tenths of the packet stays inside it.
A packet in a harmonic well keeps its gaussian shape for ever, sloshing without spreading, because every one of its parts oscillates at the same frequency. This well has no such property. The packet spreads across the flat floor, reflects from the round wall and runs back into itself. Its peak density climbs from 0.32 to 0.62 by t ≈ 4 at Speed 1 as it briefly refocuses, then falls away again.
History
Crommie, Lutz and Eigler built the physical version in 1993, standing 48 iron atoms in a ring on a copper surface and imaging the electron waves trapped inside. Their corral was 14 nanometres across.
Try
- Harden the wall with 6(1 + erf(10(r − 3))). The climb now takes a third of the distance, and the reflections come back sharper.
- Shrink the ring with 6(1 + erf(3(r − 2))). A smaller box means a shorter round trip, so the packet refocuses sooner and more often.
- Press Top. Looking straight down, the wall is a clean circle and the pattern inside it is plainly circular too.
- Lower Momentum k₀ (along x) to 1. The energy drops to 1, the turning radius moves in to about 2.7, and the packet drifts rather than bouncing.