Quantum
Field

One wave. Written into the world.

The Schrödinger equation is often called the law of the quantum world — but its mathematics is older than that framing. A vibrating string, a drumhead, the surface of a pond, an electron in an atom: all are described by the same family of wave equations. The universe, at every scale from the subatomic to the cosmic, is written in the language of waves.

This piece superposes two vibrational states of a two-dimensional quantum oscillator — the same mathematical object that describes a particle held near equilibrium, whether that particle is an electron, a molecule, or, in the classical limit, the Earth in its orbit. The two states interfere. Where they align, the combined wave has a definite direction — a phase. Where they cancel exactly, phase has no meaning at all: a single point of pure potential, a seed from which the entire surrounding spiral unwinds.

And because the whole construction is deterministic — fixed mode numbers, a fixed mixing ratio, a fixed phase, a fixed seed — the exact same wave can be summoned again from nothing but those numbers, at any time, anywhere. Nature's equation, written once, repeatable forever.

Two states, one phase, traced everywhere

Each vibrational state ψ(n,m) of a 2D quantum harmonic oscillator is built from Hermite polynomials — the same functions that describe the resonant modes of a vibrating membrane — multiplied by a Gaussian envelope that confines the wave near the centre. Two such states, A and B, are combined with a mixing ratio and a relative phase shift into a single complex wavefunction Ψ.

Ψ has a real part and an imaginary part everywhere in the plane. Its phase — the angle between them — is what's drawn. For any chosen angle θ, the curve Re(Ψ)·sin θ − Im(Ψ)·cos θ = 0 traces every point where the wavefunction points in exactly that direction. Drawing several such angles produces a family of interlocking curves: a direct portrait of how the wave's direction rotates across space, converging toward — but never quite reaching — the single point where Ψ itself vanishes.

A second mode: harmonic resonance

The sandbox below has a switch — Field: Quantum / Harmonic. Quantum mode is the wavefunction phase described above. Harmonic mode is a different, related construction: a single field built directly from three interfering angular frequencies, the same mathematics that describes standing waves on a drum, ripples on water, or the petal-counts of a flower.

A, B, and C are the three angular frequencies — how many times each term repeats around a full turn. Try integers from Fibonacci's sequence (3, 5, 8, 13, 21) for forms that echo the spirals of shells and seed-heads. The Mix slider sets the second amplitude (ampB); the third (ampC) follows automatically at two-thirds of it, preserving the balance of the original formula.