Quantum Tunnelling: How Particles Pass Through Walls

The Schrödinger equation inside a barrier

Outside a barrier (V = 0), the time-independent Schrödinger equation gives an oscillating plane wave ψ(x) = Ae^(ikx) + Be^(−ikx). Inside a barrier of height V₀ > E, the sign flips and the solutions become real exponentials rather than waves — a decaying evanescent wave that would fall to zero over an infinitely thick barrier, but for a barrier of finite width L it emerges on the far side with reduced but non-zero amplitude:

κ = √(2m(V₀−E)) / ħ           ← decay constant inside the barrier
T ≈ e^(−2κL)                  ← transmission for κL ≫ 1
T_exact = [1 + (k²+κ²)²sinh²(κL)/(4k²κ²)]⁻¹

Three things set T: barrier width L (doubling it squares T, crushing it), the energy deficit V₀−E (deeper under the barrier means smaller T), and particle mass m (heavier particles tunnel exponentially less easily at the same energy). Heisenberg's Δx·Δp ≥ ħ/2 gives an intuitive picture: confining a particle to a thin barrier region forces a momentum spread large enough that it can "borrow" the energy needed to cross for a fleeting time Δt ≤ ħ/(2ΔE) — though the rigorous story is entirely in the Schrödinger equation.

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From tunnel diodes to stellar fusion

Leo Esaki's 1957 tunnel diode exploits electron tunnelling through a thin p-n junction to switch in picoseconds. The scanning tunnelling microscope (Binnig & Rohrer, 1981) measures a tunnelling current I ∝ e^(−2κd) so sensitive that a 0.01 nm tip-surface change alters it by ~10%, resolving individual atoms. Flash memory writes and erases bits via Fowler-Nordheim tunnelling through a thin oxide layer. On the nuclear scale, George Gamow's 1928 WKB treatment of alpha decay through the Coulomb barrier explains why polonium-212 decays in microseconds while uranium-238 takes 4.5 billion years — a ~4 MeV energy difference translating into 24 orders of magnitude in half-life.

Most remarkably, the Sun's core sits at a thermal energy of only ~1.3 keV, against a ~550 keV Coulomb barrier between protons — fusion should be classically impossible. Quantum tunnelling through the barrier's tail, weighted by the Boltzmann distribution into a "Gamow peak," is what actually powers stellar fusion and, ultimately, life on Earth.

Frequently asked questions

Why can a particle cross a barrier it doesn't have enough energy to climb?

Quantum particles are described by a wavefunction, not a localised billiard ball. Inside a barrier the wavefunction decays exponentially rather than oscillating, but for a finite-width barrier it doesn't reach zero before the far side, so a reduced-amplitude wave — and non-zero probability — emerges beyond it.

What does the transmission coefficient T = e^(−2κL) tell us?

T is the fraction of incoming probability flux that tunnels through, where κ = √(2m(V₀−E))/ħ. It falls exponentially with barrier width L, with the energy deficit (V₀−E), and with particle mass m — which is why tunnelling is negligible for everyday objects but dominant for electrons.

How does tunnelling let the Sun shine?

The Sun's core temperature corresponds to a thermal energy far below the roughly 550 keV Coulomb barrier between two protons, so classical physics forbids fusion there. Quantum tunnelling lets protons cross the barrier's tail at sub-barrier energies via the Gamow factor, making stellar fusion — and life on Earth — possible.

Try it live

Everything above runs in your browser — open Quantum Tunnelling and watch a Gaussian wave packet split into reflected and transmitted parts as you tune barrier height, width and momentum. Nothing is installed, nothing is uploaded.

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