Barrier or Well
V(x) = V₀ · (abs(x) < abs(w))
Open in the app The dials and keys named below are the app's.
What you see
The formula is the potential. abs(x) < abs(w) is 1 where it holds and 0 where it does not, so V is a rectangle of half width abs(w) standing on the origin, and V₀ is its height. A negative w draws the same rectangle as a positive one. It opens at w = 1 and V₀ = −6, so the rectangle hangs below the ground instead of above it: a hole 2 wide and 6 deep.
The packet starts at Packet centre −8 with Momentum k₀ 2, and the dashed line across the plate is its energy, k₀²/2 + 1/4σ² = 2 + 0.10 = 2.10.
The two sides of zero
Drag V₀ up and the rectangle becomes a wall, and the number that matters is the energy 2.10. T reads 0.88 at V₀ = 1, 0.33 at 2 and 0.05 at 3: a factor of nineteen over two units of the slider, as the wall rises past the packet's energy.
Drag it down instead and the rectangle is a hole, which a classical particle would cross without noticing. This one does not: T falls to 0.66 at V₀ = −6, so a third of the packet turns round at a drop in the ground.
Why a hole reflects, and then stops
Any abrupt change in the potential reflects part of a wave, and the well has two edges. Inside it the wavenumber is √(2(E − V₀)), and when a whole number of half wavelengths fits across the full width 2w, the reflections from the two edges cancel and everything goes through. For this energy and width that falls at V₀ ≈ −2.8 and ≈ −9.0, and T agrees: 0.97 near −3 and 0.94 near −9, with the dip at −6 between them. Deepening the well lets less through, then more.
History
Carl Ramsauer and John Sealy Townsend found the same thing in 1921, independently, firing slow electrons through noble gases: at one particular electron energy the gas was almost transparent. Nothing in classical collision theory allows that. Treating the atom as a well of finite depth does, and the minimum is this same cancellation, found by moving the energy rather than the depth.
Try
- Drag V₀ to 3. The rectangle stands above the packet's energy now, and only about a twentieth of it tunnels through.
- Set V₀ to −3, then −6, then −9, and watch T: 0.97, then 0.66, then 0.94 again.
- Set Packet width to 0.7. A narrower packet carries a wider spread of momenta, and the cancellation smears out: T is 0.86 at V₀ = −3 against 0.64 at −6.
- With V₀ at 3, raise Momentum k₀ to 3. The energy is 4.6 and the wall is well below it, yet T is only 0.81: a wave is partly turned back by any sudden step, even one it clears.