The wave equation of quantum mechanics. Give it a potential and a boundary, and it hands back the allowed energies as its eigenvalues. For a particle trapped in a box the answer is exact: standing sine waves psi_n = sin(nπx/L), energies E_n = n²π²ħ²/2mL². Down the center, data flows: the operator and the box go in, the engine solves Hψ = Eψ, the spectrum comes out. The blue team builds and defends it; the red team tries to break it.
source Schrodinger, Quantisierung als Eigenwertproblem (Erste Mitteilung), Ann. Phys. 79 (1926) 361 — doi.org/10.1002/andp.19263851302. Rendered, not quoted.
Not a fit — a solved operator equation. The Hamiltonian is
Four facts the engine holds to 1e-6, checked live:
1 the box eigenstate satisfies Hψ=Eψ at E_n (finite-difference residual vanishes). 2 its density |ψ|² is stationary — the phase e−iEt/ħ cancels in the modulus. 3 the equation is linear, so superpositions of solutions are solutions. 4 a free plane wave obeys the dispersion ħω = ħ²k²/2m.
| check | residual | ≤ 1e-6 |
|---|
Schrodinger, 1926: the matter waves of the-de-broglie-wavelength (λ = h/p) were real, but obeyed no equation. This is the equation. Feed it the box and out fall the eigenstates and quantised energies.
What Hψ=Eψ produces — the complex, normalisable ψ — is exactly what the-wavefunction solves for and Born reads as a probability. Each sphere is the next one's premise: de Broglie's wave → this equation → the wavefunction.
The blue team's live check: recompute the eigenvalue residual of the box states n = 1, 2, 3 and confirm each vanishes to the finite-difference tolerance. If red tampers with the kinetic term, this badge is where it shows.
Feed the engine two things:
1. the operator — H = −(ħ²/2m)∂²/∂x² + V(x). The first term is kinetic energy (curvature of ψ); V is the potential. 2. the box — an infinite square well on [0, L]: V = 0 inside, ∞ at the walls, forcing ψ(0) = ψ(L) = 0.
Natural units: ħ = 1, m = 1, L = π (stated). SI constants defined in code (h, ħ, k_B, c, m_e, e). Those boundaries are the whole input — everything below is derived, never looked up.
t = 0. The real part of ψ oscillates in time; the density |ψ|² does not — that is what "stationary state" means.
Every number is computed on the spot: the residual from a central-difference second derivative, the norm from a Simpson integral. Change n or drag time — nothing is stored, nothing is faked.
What the machine produces, proven: the box eigenstates ψ_n = √(2/L) sin(nπx/L) and their energies E_n = n²π²ħ²/2mL² — here n²/2: 0.5, 2, 4.5, 8, 12.5, 18… a quadratic ladder. Each state is normalised (Born integral = 1), its density stationary, and the free-particle limit obeys ħω = ħ²k²/2m.
The blue team's witness (left) recomputes these residuals live; the red team (right) tries to make the operator lie.
It is also single-particle here. For N particles ψ lives in 3N-dimensional configuration space — the cost grows exponentially, which is why quantum many-body is hard. And the smooth unitary evolution below says nothing about measurement: the collapse of |ψ|² to one outcome is bolted on, not derived. The box's infinite walls are an idealisation.
"Schrodinger derived the equation rigorously." Cut. It was an inspired ansatz — guided by Hamilton–Jacobi optics and de Broglie — justified after the fact by matching the hydrogen spectrum, not deduced from prior axioms.
"ψ is a real, physical wave in space." Cut. ψ is complex and lives in configuration space; Born (1926) fixed its meaning: |ψ|² is a probability density, not a substance.
"The energies are evenly spaced." Cut, corrected. The box ladder is quadratic (n²): 0.5, 2, 4.5, 8… It is the harmonic oscillator whose levels (n+½)ħω are evenly spaced — a different potential.
The red team's move: corrupt the kinetic prefactor — drop the 2 in −ħ²/2m, so H uses −ħ²/m. Now the box eigenstate no longer solves Hψ=Eψ at the true E_n. The blue team's witness (window 7) is watching.
Halve-the-mass and the residual |Hψ−Eψ| jumps from ~1e-7 to order 1 — the witness recomputes, disagrees, and turns red. Nothing is faked; the attack is real and it is caught.