Particle in a box.
Drop a free electron into a one-dimensional cavity with infinitely-tall walls. There's no force inside — the electron is free — but the walls force its wavefunction to vanish at the boundaries, exactly like the ends of a guitar string. Only certain wavelengths fit, and energies climb as n²: gaps that widen with every step, the opposite of a harmonic series. Eigentone tunes a scale to that n² ladder.
A guitar string for an electron.
Inside the box the electron's wavefunction is a sine wave. The boundary conditions are unforgiving: ψ must be zero at the walls. Only n = 1, 2, 3 … half-wavelengths fit. Pick a state to see the wavefunction, the probability density, and the audio frequency that level produces.
Why confinement quantizes.
The "infinite square well" is the first problem in every quantum-mechanics textbook because it's the cleanest possible example of confinement producing quantization. There's no potential to think about inside — the particle is genuinely free. All the physics is in the walls.
Boundary conditions force ψ(0) = ψ(L) = 0. The only solutions are sines whose half-wavelength evenly divides L. n = 1 fits one half-cycle; n = 2 fits two; and so on. Plug those allowed wavelengths into p = h/λ and then E = p²/2m, and the n² spacing falls out.
Real systems behave like boxes when carriers are trapped — quantum dots, conjugated dye molecules, the lowest band of a semiconductor superlattice. The n² staircase is observable.
A ladder that stretches.
Unlike the harmonic oscillator (linear spacing) and the Morse potential (crowding), the box spreads. The gap between n = 1 and n = 2 is one unit; between n = 2 and n = 3 it's three. Between 6 and 7 it's thirteen. The intervals widen forever.
Sonically, this gives a tuning system unlike any other in Eigentone: low keys close together, high keys flying apart. An accelerating staircase. The opposite of a piano's logarithmic ratios.
Sines that fit.
Inside the box the Schrödinger equation reduces to ψ″ = −k²ψ, with solutions sin(kx). The boundary conditions pick out a discrete set of wavenumbers:
ψn(x) = √(2/L) · sin( n π x / L )
Plug that k = nπ/L into E = ℏ²k²/2m and the energy spectrum is:
For an electron in a 1 nm box the prefactor is ≈ 0.376 eV, so E1 ≈ 0.376 eV, E2 ≈ 1.50 eV, E3 ≈ 3.39 eV … purely n².
Make the box smaller and the energies climb (try a 0.1 nm box and you're in the keV range). Make it larger — say a one-meter macroscopic box — and the levels are so close together that the system looks classical.
Each energy level corresponds to a frequency via E = hf. E1 = 0.376 eV is about 9.1 × 1013 Hz — far infrared. Forty halvings drop it to 82.7 Hz, just above a low E on a bass guitar.
n=2: E = 1.50 eV → 331 Hz (× 4)
n=7: E = 18.4 eV → 4054 Hz (× 49)
Constants: ℏ = 1.0546 × 10−34 J·s, me = 9.1094 × 10−31 kg. The ratio En/E1 is exactly n², regardless of box size or particle mass.
What each white key plays.
Seven keys, seven values of n²: 1, 4, 9, 16, 25, 36, 49.
Notice the gap between adjacent keys widens dramatically. C→D is a jump of 3 E1; A→B is a jump of 13 E1. The high keys span more than a typical piano octave each.
Hear quantum confinement.
Particle in a Box ships with the Pro tier alongside QHO and Morse — a complete quantum-mechanical kit, $39 once.