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Quantum Tunnelling: Through the Wall

Watch a particle cross a barrier it has too little energy to surmount, derive the transmission coefficient, and see why this quantum quirk powers stars, microscopes and memory chips.

The classically impossible

Roll a ball at a hill. If its energy is less than the height of the hill, it rolls back down — always. A quantum particle does not. Facing a potential barrier taller than its energy, its wavefunction penetrates the barrier and there is a finite probability it emerges on the far side, still with its original energy. This is quantum tunnelling, and it has no classical analogue whatsoever.

A wave through a barrier

Model the simplest case: a rectangular barrier of height V_0 and width a, with an incoming particle of energy E<V_0. Split space into three regions — incoming plus reflected wave on the left, exponential decay inside, transmitted wave on the right — and match \psi and \psi' at both edges. Inside, the wave is governed by the same decay constant as the finite well.

\kappa = \frac{\sqrt{2m(V_0-E)}}{\hbar}, \qquad \psi_{\text{inside}}(x) \sim e^{-\kappa x}

The wavefunction decays inside the barrier with penetration depth 1/κ.

Carrying the matching through gives the exact transmission probability T — the fraction of incident particles that get through. It depends on the energy, the barrier height and, crucially, the width.

T = \left[\,1 + \frac{V_0^{2}\sinh^{2}(\kappa a)}{4E(V_0-E)}\,\right]^{-1}

Exact transmission through a rectangular barrier for E < V₀.

Drag the barrier taller or wider and watch the transmission collapse. The amplitude decays inside the barrier and re-emerges smaller — the height matters, but the width matters exponentially more.

Thick barriers and WKB

For any barrier that is thick or tall, \kappa a \gg 1, the \sinh becomes a growing exponential and the transmission is dominated by a single exponential factor. This is the form worth memorizing: transmission falls off exponentially with the barrier width and with the square root of the particle's mass.

T \approx 16\,\frac{E}{V_0}\!\left(1-\frac{E}{V_0}\right)e^{-2\kappa a} \;\sim\; e^{-2\kappa a}

Thick-barrier limit: the exponential factor e^(−2κa) dominates.

Real barriers are rarely rectangular. For a smoothly varying potential, the WKB approximation generalizes the result: the exponent becomes an integral of the local decay rate across the whole forbidden region. This one formula estimates tunnelling in problems from nuclear decay to chemical reactions.

T \approx \exp\!\left(-\frac{2}{\hbar}\int_{x_1}^{x_2}\sqrt{2m\,[V(x)-E]}\;dx\right)

The WKB tunnelling estimate for a smooth barrier between turning points x₁ and x₂.

Where tunnelling runs the world

Tunnelling is not a curiosity — it is a workhorse. It explains alpha decay (an alpha particle tunnels out of the nucleus, and Gamow's exponential explains the enormous spread in half-lives). It lets fusion proceed in the Sun's core, where nuclei tunnel through their mutual Coulomb repulsion at temperatures far too low to surmount it classically. The scanning tunnelling microscope turns the exponential width-sensitivity into atomic-resolution imaging. And tunnel diodes, Josephson junctions and the erase step of flash memory all rely on it.