From Waves to Quanta
A ripple tank showing two overlapping wave sources and an electron passing through two slits produce the exact same striped pattern — because underneath quantum mechanics is, in a precise mathematical sense, wave physics applied to probability instead of water or sound.
1. One mathematical language, two physics
Classical waves — sound, light, water ripples, a plucked string — all obey the same core mathematical idea: superposition. Two overlapping waves add together at every point in space, and where their crests align you get reinforcement (constructive interference); where a crest meets a trough, they cancel (destructive interference).
Quantum mechanics uses the exact same superposition mathematics
— but instead of describing a physical displacement of water or
air, the quantum wavefunction ψ describes a
probability amplitude. Squaring its magnitude,
|ψ|², gives the probability of finding a particle
at a given point. The same addition-and-cancellation rules that
create ripple patterns in a pond create the probability patterns
that govern where electrons, photons, and atoms are likely to be
found.
2. The classical wave equation
The classical wave equation describes how a disturbance — a plucked guitar string, a sound wave, a ripple in water — propagates through space over time:
Here u(x, t) is the physical displacement (of a
string, of air pressure, of water height) and c is
the wave's propagation speed. Solutions are sinusoidal waves
that can be added together — superposed — to build up any more
complex waveform, which is exactly what happens in the
wave interference
simulation: two circular wave sources overlapping produce
the classic hyperbolic fringe pattern of alternating
reinforcement and cancellation.
3. From the wave equation to Schrödinger's equation
The time-dependent Schrödinger equation — the central equation of non-relativistic quantum mechanics — has the same structural DNA as the classical wave equation: a second-order spatial derivative (via the Laplacian) related to a time derivative of the same field.
| Classical wave equation | Schrödinger equation |
|---|---|
| Displacement u(x,t) — real-valued | Wavefunction ψ(x,t) — complex-valued |
| Second time derivative (∂²u/∂t²) | First time derivative, but with an imaginary coefficient (i∂ψ/∂t) |
| Fixed wave speed c | Speed emerges from mass m and potential V(x) |
| |u| = physical amplitude | |ψ|² = probability density |
The crucial difference — a complex-valued field governed by a first-order (in time) equation with an imaginary unit — is exactly what produces uniquely quantum behaviour like tunneling and quantized energy levels. But the wave-like machinery underneath (superposition, interference, dispersion) is directly inherited from classical wave theory.
4. The double-slit experiment: interference made visible
Fire electrons, photons, or even large molecules one at a time through two narrow slits, and — despite each particle arriving individually — an interference pattern of alternating bright and dark bands builds up on the detector screen, identical in structure to what light waves (or water waves) produce through the same two slits.
The double-slit simulation shows exactly this: each particle's wavefunction passes through both slits simultaneously and interferes with itself, just as a classical wavefront does in the wave interference simulation. The pattern only vanishes if you measure which slit the particle actually went through — collapsing the superposition destroys the interference, a purely quantum phenomenon with no classical wave analogue.
The fringe spacing in both the classical ripple-tank pattern and the quantum double-slit pattern is governed by the same geometric formula relating wavelength, slit separation, and screen distance — proof that the interference machinery is identical, even though one case is a physical medium vibrating and the other is a probability amplitude.
5. Quantum tunneling: waves don't stop at walls either
Classical waves already do something that looks almost "impossible": an evanescent wave can leak a short distance into a barrier region where it would classically be forbidden (this happens with total internal reflection of light, for example) — decaying exponentially rather than stopping abruptly at the boundary.
The quantum tunneling simulation shows the particle-physics version of exactly this behaviour: a particle's wavefunction hitting a potential barrier taller than its energy doesn't drop instantly to zero inside the barrier — it decays exponentially, and if the barrier is thin enough, a non-zero amplitude survives on the far side. Squaring that surviving amplitude gives a real, measurable probability that the particle appears beyond a barrier it classically could never cross.
The throughline: whether it's a light wave evanescently leaking past a boundary or an electron's wavefunction tunneling through a potential barrier, the same exponential-decay mathematics governs both — because both are, structurally, waves meeting a region where propagation is classically forbidden.
🌊 Explore the connection yourself
Watch classical interference build fringes in the wave tank, then compare it directly to the probability fringes of the double-slit experiment and the exponential leak of quantum tunneling.
Open Wave Interference →