Confine a quantum particle between two walls and its energy stops being a dial — it becomes a ladder. Only certain standing waves fit, so only certain energies are allowed: En = n²π²ℏ²/2mL². The lowest rung is above zero — a confined particle can never sit perfectly still. Down the center, the box goes in, the engine solves the wave, the quantized ladder comes out. The blue team builds and defends it; the red team tries to break it.
source The infinite square well — standard quantum mechanics; D. J. Griffiths, Introduction to Quantum Mechanics 2e (2005), §2.2; the first system in which the Schrödinger equation (1926) predicts quantization. No single stable primary link — AMBER. Rendered, not quoted.
Inside the well V=0; outside V=∞, so the wave must vanish at both walls: ψ(0)=ψ(L)=0. Only sine waves with a whole number of half-wavelengths fit:
ψn(x) = √(2/L)·sin(nπx/L), and the Schrödinger equation then fixes the energy at En = n²π²ℏ²/2mL². The energy is quantized because the geometry is — a rung for every extra half-wave.
Live, in natural units ℏ=m=L=1 — the ladder scales as n²:
| n | En | En/E1 | nodes |
|---|
Confinement quantizes. The box is the simplest bound solution of the-schrödinger-equation — plug an infinite well into Ĥψ=Eψ and the boundary conditions do the rest.
The zero-point floor E1>0 is not a quirk of this well; it is the-uncertainty-principle made visible — pin Δx≈L and Δp can't be zero, so neither can the energy. Each sphere is the next one's premise.
The blue team's live check: recompute the ladder ratios, integrate |ψ|² and the mode overlap, and confirm against the known truth (4, 9; norm=1; overlap=0). If red tampers, this badge is where it shows.
Feed the engine a well: a particle of mass m free to move only in 0 < x < L, walls infinitely high. That is the entire input — one length L and the demand that the wave die at each wall.
| region | potential V(x) | ψ allowed? |
|---|---|---|
| x ≤ 0 | ∞ | ψ = 0 |
| 0 < x < L | 0 | ψ = √(2/L)·sin(nπx/L) |
| x ≥ L | ∞ | ψ = 0 |
The two walls are the whole boundary value problem — everything the panel computes is forced by them.
The wall spacing sets the whole spectrum. Every energy below is integrated live — nothing is looked up.
Left: the energy ladder (rungs ∝ n²), the selected rung lit. Right: ψn(x) — the standing wave, with n−1 nodes, pinned to zero at both walls.
What the machine produces, proven: a discrete spectrum En = n²·E1 (so E₂/E₁=4, E₃/E₁=9), modes that are normalized (∫|ψ|²=1) and mutually orthogonal (∫ψmψn=0), a non-zero floor E₁>0, and the whole ladder rising as 1/L² when the box shrinks.
The blue team's witness (left) confirms these live; the red team (right) tries to make them wrong.
It is also 1D, non-relativistic, single-particle, and time-independent — it says nothing about spin, interactions, or a particle you actually watch move. "Exact" here means exact for the idealization, which is why the panel lets you shrink L and watch the failure mode of confinement appear as ever-larger energies.
"A confined particle can have zero energy." Cut. E₁ = π²ℏ²/2mL² > 0 — the zero-point floor. Confinement forbids rest; the witness verifies E₁ > 0 live.
"The levels are evenly spaced." Cut. That is the harmonic oscillator (En=(n+½)ℏω). The box grows as n²: gaps 3,5,7,… widen. Tamper (window 6) forces the false even spacing and is caught.
"Doubling the box halves the energy." Cut, corrected. E ∝ 1/L² — doubling L cuts every level to a quarter; halving L quadruples it. Verified by the witness.
The red team's move: replace the n² law with a linear one, En=n·E₁ — the "evenly-spaced" lie from the graveyard. The blue team's witness (window 7) is watching.
Linearize the ladder and the ratios collapse from 4, 9 to 2, 3 — the n²-scaling check fails, the witness recomputes and turns red. Nothing is faked; the attack is real and it is caught.