Tunnelling Barrier
V(x) = 3(abs(x) < 0.4)
Open in the app The dials and keys named below are the app's.
What you see
The formula is the potential. abs(x) < 0.4 is 1 where it holds and 0 where it does not, so V is a rectangular barrier 0.8 wide and 3 tall sitting on the origin. The dashed line drawn across it is the packet's energy, k₀²/2 + 1/4σ² = 2 + 0.51 = 2.51, and it passes below the top.
The packet starts at Packet centre −4 with Momentum k₀ 2, so it takes about two seconds at Speed 1 to reach the barrier. A ball with that energy would bounce off. Watch what the packet does instead: it splits, and part of it appears on the far side.
The physics
Inside a barrier it cannot climb, the wave stops oscillating and decays, at the rate κ = √(2(V − E)) = 0.99. A plane wave at the packet's central energy therefore comes out the far side with about e^(−2κ · 0.8) ≈ 0.21 of its probability. Thinner or lower barriers pass far more, and the dependence on width is exponential. The packet is not one plane wave, though. Its width of 0.7 gives it a momentum spread of 1/2σ ≈ 0.71, and a quarter of it carries more than the 3 the barrier asks for and simply goes over the top. Weighting the exact barrier formula across those momenta gives 0.36.
Where T settles
The collision is over by t ≈ 4, and there T reads 0.375 against R 0.625, close to the 0.36 the momentum sum predicts. That is the number to read early. The reflected half returns from the wall at x = −8 around t = 7, and after that the two mix again.
History
In 1928 George Gamow, and independently Ronald Gurney and Edward Condon, used this to explain alpha decay. An alpha particle inside a nucleus is held by a barrier it has no energy to climb, so it leaks out instead. Because the leak is exponential in the barrier, a narrow range of decay energies produces half-lives spread over twenty orders of magnitude.
Try
- Raise the barrier, 6 · (abs(x) < 0.4): T falls to about 0.08.
- Double its width, 3 · (abs(x) < 0.8): about 0.20 gets through, so width bites harder than height.
- Set Momentum k₀ to 3. The energy is now 5, well above the top, and still only about 0.71 goes on: a wave is partly reflected by any sudden change, even a favourable one.
- Use Step to walk the collision frame by frame.