Quantum harmonic oscillator.
A mass on a spring is the simplest oscillator in classical physics. Replace the mass with a quantum particle and the energy stops being continuous — it can only sit on a ladder of perfectly evenly-spaced rungs, separated by ℏω. Every chemical bond, every phonon in a crystal, every mode of the electromagnetic field starts here. Eigentone tunes a chromatic scale to that ladder.
Even rungs, real wavefunctions.
The parabola is the harmonic potential — a mass on an idealized spring. Each horizontal line is an allowed energy level. Pick a level to draw the actual wavefunction ψn(x) at that energy: the squiggle is real, computed from Hermite polynomials.
A spring with rules.
A classical mass on a spring can vibrate with any energy you give it — push it harder, it moves further. A quantum mass can't. Solving the Schrödinger equation for a parabolic potential gives a clean, surprising result: only certain energies are allowed, and they're spaced exactly evenly.
The lowest level isn't zero. Even at absolute zero a quantum oscillator still has ½ ℏω of energy — its zero-point energy. You cannot turn it off. This residual jiggle is why helium stays liquid down to absolute zero, why crystals have a finite heat capacity at low temperature, and why every quantum field in the vacuum has measurable energy.
The model that's everywhere.
Real systems are messier than a perfect parabola, but near any equilibrium point a potential looks harmonic — Taylor-expand it and the leading term is quadratic. So the QHO is the first approximation for almost everything: molecular vibrations, phonons in solids, modes of the electromagnetic field, fluctuations of the Higgs.
Every level is a multiple of ℏω above the last. Eigentone uses that as a tuning system: the audio frequency for each MIDI key climbs in equal steps, not equal ratios. Linear, not logarithmic. The opposite of a piano.
Schrödinger's parabola.
Take the time-independent Schrödinger equation with a quadratic potential V(x) = ½ m ω² x² and the energy eigenvalues drop out:
En = ℏω ( n + ½ )
The wavefunctions are Hermite polynomials multiplied by a Gaussian:
with ξ = x √(mω/ℏ)
Two consequences fall out: the spacing between adjacent levels is always ℏω, regardless of n. And the ground state has more energy than the bottom of the well — quantum mechanics never lets a particle sit still.
Eigentone uses HCl's actual harmonic frequency ωe = 2989 cm⁻¹ — the spring constant of the real hydrogen-chloride bond — as the energy quantum. Multiply by the speed of light to get the natural infrared frequency, then divide by 2³⁹ to drop into audio.
f₀ ≈ 81.50 Hz, Δf ≈ 163.00 Hz per level
Same N as the Morse Potential set by design — the QHO is the harmonic limit of Morse, so they share the same energy unit and the v=0 / n=0 ground state lands at the same ≈ 81 Hz in both. Comparing the two sets head-to-head reveals the anharmonic correction directly.
What each white key plays.
Seven white keys, seven energy levels. Linear spacing — climb a key, climb a quantum.
Because the levels are equally spaced, intervals between non-adjacent keys aren't conventional — C4→E4 (2 steps) sounds wider than C4→D4, but neither is a "third" or a "second". This set is for sound design, not melody-making in the classical sense.
Hear quantum mechanics.
The Quantum Harmonic Oscillator set ships with the Pro tier alongside Morse and Particle in a Box — all three quantum models, $39 once.