A particle meets a wall it hasn't the energy to climb — and slips through anyway, with odds that fade exponentially in the wall's width. There is no classical hop: inside the barrier the wave does not oscillate, it decays, and the sliver that survives at the far side is a real, computable amplitude. Down the center, data flows: E, V, a go in, the engine computes T, the transmission comes out. The blue team builds and defends it; the red team tries to break it.
source G. Gamow, “Zur Quantentheorie des Atomkernes”, Z. Physik 51, 204 (1928); R. W. Gurney & E. U. Condon, Nature 122, 439 (1928) — barrier penetration / alpha decay. No stable open link (paywalled journals) — cited by author/title/year. AMBER Rendered, not quoted.
Match a plane wave onto a square barrier of height V and width a, for a particle of energy E < V. Solving Schrödinger inside gives not a wave but a decaying exponential:
ψ(x) = e−κx, with κ = √(2m(V−E))/ħ. The bigger the shortfall V−E, the faster it dies.
The far-side amplitude is what leaks past. In the thick-barrier limit the transmission is T ≈ e−2κa — and the exact square-barrier T (with its sinh factor) is what the panel computes live.
Live for the current setting:
| quantity | value (ħ=m=1) |
|---|
Gamow, 1928: the barrier the-schrodinger-equation allows is penetrated by the evanescent tail that the-wavefunction carries across it.
Same month, Gurney & Condon reached it independently. The T ≈ e−2κa fade — ferocious sensitivity to width and to V−E — is why alpha half-lives span 1024: the Geiger–Nuttall law is this exponential in disguise. Each sphere is the next one's premise.
The blue team's live check: re-derive the barrier physics from the pure functions — that T>0 for E<V, that doubling width squares T, and that log T falls linearly in a. If red flips the exponent, this badge is where it shows.
Three numbers fully specify a square-barrier tunnelling problem, in natural units ħ = m = 1:
| symbol | means | role |
|---|---|---|
| E | particle energy | must be < V to tunnel |
| V | barrier height | the wall it can't climb |
| a | barrier width | how far it must reach |
From these the engine forms the decay rate κ = √(2m(V−E)) and the transmission T. That is the whole input — feed it to the panel below.
Drag any control — κ, the exact T, and the wave picture recompute on the spot, never looked up.
Left: incoming wave (oscillatory). Middle band: inside the barrier — a decaying exponential, not a wave. Right: the transmitted sliver, amplitude √T.
What the machine produces, proven: a nonzero transmission through a classically forbidden wall, fading as T ≈ e−2κa — double the width, square the (tiny) T; raise E toward V and T climbs; widen the wall or heavy the particle and T → 0. The current T is above; the exponential law is the output.
The blue team's witness (left) re-derives these live; the red team (right) tries to flip the exponent.
WKB itself fails near the turning points (where V=E and κ→0). And an exponential half-life needs an attempt frequency too — T alone is not a decay rate. The square barrier is not the physics; it is the first exactly-solvable proof that a forbidden wall still leaks.
“The particle borrows energy from the vacuum to hop over the wall.” Cut. No energy is borrowed. The stationary state has one definite energy E<V everywhere; inside, the wave is evanescent (e−κx), never a classical hop above V.
“Tunnelling is instantaneous — faster than light.” Cut · AMBER. Traversal time is genuinely subtle; group-delay saturation (Hartman) is real but carries no superluminal signal.
“A thicker wall lets a bit more leak through eventually.” Cut. Exactly backwards — T falls exponentially with width. This is the exact lie the red team plants next door.
The red team's move: flip the sign in the exponent — make T ≈ e+2κa — so a thicker barrier wrongly transmits more. The blue team's witness (window 7) is watching.
Flip the sign and the decay-with-width law inverts: widen the wall and T grows. The witness recomputes, finds T(2a)>T(a), and turns red. Nothing is faked; the attack is real and it is caught.