Quantum Tunnelling
Quantum tunnelling is the phenomenon in which a quantum particle has a nonzero probability of passing through a potential energy barrier higher than the particle's own energy — a transition forbidden entirely in classical mechanics, but permitted because the particle's wavefunction extends into and through the barrier.
Curious
In classical mechanics, if a ball doesn't have enough energy to get over a hill, it simply cannot get to the other side — full stop. Quantum particles break this rule. A particle's wavefunction doesn't sharply stop at a barrier's edge; it decays exponentially inside the barrier but doesn't reach exactly zero, meaning there's a small but genuine, calculable probability of finding the particle on the far side, having passed straight through a barrier it classically shouldn't have been able to cross.
This isn't the particle briefly "borrowing" extra energy to get over the top — it never has enough energy to be inside the barrier in the classical sense at all. It's a purely wave-mechanical effect, with no classical analogue beyond loose (and imperfect) comparisons to how sound or light can partially penetrate a barrier that should fully block them.
Exploring
For a rectangular potential barrier of height V₀ and width L, with a particle of energy E < V₀ incident on it, the approximate transmission probability (the "tunnelling probability") for a wide or tall barrier is:
The exponential dependence on barrier width L and on √(V₀−E) means tunnelling probability drops off extremely fast as barriers get taller or wider — which is why tunnelling is dramatically important at atomic and subatomic scales but utterly negligible for everyday macroscopic objects (a person cannot tunnel through a closed door; the exponential suppression for an object of ordinary mass makes the probability astronomically, immeasurably small).
Worked example: An electron with 1 eV less energy than a barrier height, facing a barrier 1 nm wide, has κ = √(2 × 9.109×10⁻³¹ × 1.602×10⁻¹⁹)/1.0546×10⁻³⁴ ≈ 5.12×10⁹ m⁻¹. Then T ≈ e⁻²⁽⁵·¹²ˣ¹⁰⁹⁾⁽¹⁰⁻⁹⁾ = e⁻¹⁰·²⁵ ≈ 3.6 × 10⁻⁵ — a small but very much non-negligible tunnelling probability at this scale, which is exactly the regime exploited in real electronic tunnelling devices.
Deep Dive
George Gamow's 1928 paper was the first application of quantum tunnelling to a real physical problem: explaining alpha decay, in which an alpha particle escapes a nucleus despite classically lacking the energy to overcome the strong nuclear force's confining potential. Gamow's tunnelling calculation correctly predicted the extraordinarily wide range of observed alpha-decay half-lives (from microseconds to billions of years) as a consequence of the exponential sensitivity of tunnelling probability to barrier height and width — a striking early success of applying quantum mechanics to nuclear physics, developed independently and almost simultaneously by Ronald Gurney and Edward Condon.
Technological applications: The scanning tunnelling microscope (STM), invented by Gerd Binnig and Heinrich Rohrer in 1981 (earning them the 1986 Nobel Prize in Physics), exploits the exponential distance-dependence of tunnelling current between a sharp conducting tip and a surface to image individual atoms with sub-nanometre resolution. Tunnel diodes, flash memory (which stores data by tunnelling electrons through an insulating barrier onto a floating gate), and the Josephson junction (used in superconducting qubits) are further major technologies built directly on controlled quantum tunnelling.
Exact solution context: The approximation T ≈ e⁻²κL given above is the WKB (Wentzel-Kramers-Brillouin) approximation, valid for barriers that vary slowly or are sufficiently wide/tall; the exact rectangular-barrier transmission coefficient (solved directly from matching wavefunctions and their derivatives across the barrier boundaries) is somewhat more complex but reduces to the same exponential form in the relevant limit.
Concept Relationship Map
Requires: particle in a box, wave function
Enables: nuclear alpha decay, scanning tunnelling microscopy, tunnel diodes and flash memory
Commonly confused with: the false idea that tunnelling requires the particle to temporarily violate energy conservation
Sources
- Gamow, G. (1928). Zeitschrift für Physik, 51(3–4), 204–212.
- Binnig, G., Rohrer, H. (1982). "Scanning Tunneling Microscopy." Helvetica Physica Acta, 55, 726–735.
- Griffiths, D.J., Schroeter, D.F. (2018). Introduction to Quantum Mechanics, 3rd ed. Cambridge University Press, §2.5.