What is quantum tunnelling?

It is a particle passing through an energy barrier it does not have enough energy to climb over — something impossible in classical physics. Because a particle is described by a wavefunction, and that wave does not stop dead at the barrier but decays exponentially inside it, a small amplitude survives to the far side, so there is a non-zero probability the particle emerges. For a 1 eV electron meeting a 2 eV, 0.5 nm-wide barrier the exact transmission probability is about 2.4% — small, but far from zero.

Why does tunnelling depend so sharply on distance?

Because the transmission probability falls off exponentially with barrier width: T ≈ e^(−2κa), where κ = √(2m(V0−E))/ℏ. Widening the 2 eV barrier from 0.5 nm to 1 nm drops the odds for our electron from 2.4% to 0.014% — a factor of ~170 — and to under one in a million by 1.5 nm. For a realistic 4 eV barrier the current drops about eightfold for every extra 0.1 nm. That ferocious sensitivity is exactly what a scanning tunnelling microscope exploits: the tunnelling current between its tip and the surface changes so much with tiny height differences that it can map individual atoms.

What happens when the particle has more energy than the barrier?

Classically it would just sail over, and mostly it does — but quantum mechanically the transmission is not a flat 100%. It oscillates with energy, dipping below 1 and returning to exactly 1 at resonant energies where the barrier width is a whole number of half-wavelengths (k₂a = nπ). For a 2 eV, 0.5 nm barrier the first perfect-transmission resonance is near 3.5 eV. This above-barrier reflection and resonance is another purely wave effect with no classical analogue.

Quantum

Quantum Tunnelling

By Lucian — builder & engineer, LK Forge

Roll a ball at a hill it hasn’t the speed to climb and it rolls back, every time. Fire an electron at an energy barrier it hasn’t the energy to cross and… sometimes it’s simply on the other side. The wavefunction leaks through, and the odds obey a startlingly steep law. We took the tunnelling lab’s exact barrier formula and mapped it.

September 10, 2026 · 6 min read · exact rectangular-barrier transmission

e^(−2κa)

tunnelling odds fall exponentially with barrier width

×7.8 / 0.1 nm

the gap sensitivity that lets an STM see atoms

T > 0

the particle crosses a wall it classically cannot

T = 1

above the barrier, transmission resonates to 100%

The exponential wall

On a logarithmic axis the transmission-versus-width curve is very nearly a straight line — the hallmark of exponential decay, T ≈ e^(−2κa). The steeper line is the taller barrier: a 3 eV wall kills the wave faster than a 2 eV one, because the decay rate κ grows with how far below the barrier top the particle sits. Every tenth of a nanometre matters, which is precisely why tunnelling is a nanoscale affair and invisible at everyday sizes.

Transmission vs width on a log scale: straight-ish lines = exponential decay. The 3 eV barrier decays faster than the 2 eV one; a nanometre of barrier spans six orders of magnitude in probability.

Below the barrier, then above it

Sweep the particle’s energy toward a fixed 2 eV barrier and two quantum surprises appear. Below 2 eV, where classical physics allows nothing through, transmission is small but rises steadily — that whole region is pure tunnelling. Cross 2 eV and the particle now has enough energy to pass, yet transmission does not lock at 100%: it overshoots, dips, and returns to a perfect 1.0 only at resonant energies where the barrier is an exact number of half-wavelengths wide.

Transmission vs energy for a 2 eV, 0.5 nm barrier. The shaded region left of 2 eV is classically forbidden yet has T > 0; above it, transmission resonates back to 100% near 3.5 eV instead of simply saturating.

Where the leak matters

A 2.4% leak sounds negligible until you notice what rides on it. The scanning tunnelling microscope turns the exponential width-sensitivity into a ruler fine enough to image single atoms, because a fraction of a nanometre changes the current by an order of magnitude. Alpha decay is tunnelling: an alpha particle rattling inside a nucleus escapes only by tunnelling through the Coulomb barrier, and because the odds are exponential in the barrier, half-lives span from microseconds to billions of years across a small range of energies. The Sun shines by it too — protons fuse only because they tunnel through their mutual repulsion, since none has enough thermal energy to climb it. And every flash drive stores bits by pushing electrons through a thin oxide barrier. The wall that shouldn’t be crossable is, just often enough.

Reproduce it yourself

The exact transmission is one expression (electron: ℏ²/2m = 0.0381 eV·nm²).

const HB2M = 0.0381;   // hbar^2 / 2m for an electron, eV·nm^2
function T(E, V0, a) {  // energy eV, barrier eV, width nm
  const k = Math.sqrt((V0 - E) / HB2M);           // decay constant, 1/nm
  const s = Math.sinh(k * a);
  return 1 / (1 + V0*V0 * s*s / (4*E*(V0 - E)));   // E < V0
}
T(1, 2, 0.5);   // 0.0235   (2.4%)
T(1, 2, 1.0);   // 1.42e-4  — double the width, ~170x rarer

The transmission formula (both E < V₀ and E > V₀ regimes) is lifted straight from the tunnelling solver; the width, height and energy sweeps are the small script behind this post.

Send a wave at a wall

Open the lab, set the energy below the barrier, and watch the wavefunction decay through it and emerge shrunken on the far side — then widen the barrier and watch the transmission collapse.

Open the tunnelling lab →

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