Quantum Tunneling

Last session our electron lived behind walls of infinite height, and the rule was absolute: ψ = 0 at the wall. Real walls are never infinite. The oxide in a transistor, the vacuum gap between two metals, the electric repulsion around a nucleus: each is a potential barrier, a region where the potential energy V rises above the particle's total energy E by a finite amount over a finite distance. Classically, the kinetic energy E − V can never be negative, so a ball rolling toward a hill it cannot climb slows, halts at the turning point where E = V, and rolls back. Everything beyond is a classically forbidden region.

Quantum mechanics reads the same landscape through the Schrödinger equation of chapter 10. Where E exceeds V, the curvature of ψ points back toward the axis and the wavefunction oscillates. Where V exceeds E, the curvature points away from the axis, so ψ grows or decays exponentially. A growing solution could never be normalized in an endless barrier, and even in a thick finite one the decaying part dominates. So the wavefunction does not slam to zero at the edge. It enters, and it fades: ψ(x) ≈ ψ(0)e^(−κx), where x is the distance into the barrier and the Greek letter κ stands for the decay constant.

The decay constant is set by the energy deficit V − E, the particle's shortfall below the top. In words, κ is the square root of twice the mass times the deficit, divided by ħ; in symbols, ħκ = √(2m(V − E)). Its inverse, 1/κ, is the decay length, over which ψ shrinks by a factor of e, about 2.718. For an electron one electron-volt short of the top, κ ≈ 5.12 per nanometre and 1/κ ≈ 0.195 nm, comparable to the radius of an atom. The probability density, |ψ|², fades twice as fast, as e^(−2κx), halving every 0.068 nm. The infinite wall of chapter 11 is the limit in which κ is infinite and the fade is instantaneous.

Now give the barrier a far side: a rectangle of height V and width L, with free space on either side. An electron wave arrives from the left. Part reflects, part enters and fades, and if the barrier is thin enough the fading tail is still alive at the far edge. Beyond it ψ oscillates again with its original wavelength, and a transmitted wave travels away. Its amplitude was cut by roughly e^(−κL) during the crossing, and probability goes as amplitude squared, so the transmission probability, the chance that a particle sent at the barrier is found beyond it, obeys T ≈ e^(−2κL). Passing through a barrier you lack the energy to climb is called quantum tunnelling.

Matching ψ and its slope at both edges gives the exact result for a rectangle. Once the exponent 2κL exceeds about 4, it simplifies to T ≈ 16(E/V)(1 − E/V)e^(−2κL). The prefactor never exceeds 4, so nearly all the action lives in the exponential. Take an electron with E = 1.0 eV meeting a barrier of height V = 2.0 eV. A barrier 0.5 nm wide transmits about 2.4 percent of electrons; double the width to 1.0 nm and T falls to 0.014 percent, 166 times smaller. Doubling the width squares the exponential factor, and the prefactor barely changes. Whatever is not transmitted is reflected: R + T = 1, where R is the reflection probability.

The exponent also hides the mass, because κ grows as the square root of m. A proton is 1,836 times heavier than an electron, so its κ is larger by a factor of √1836, about 42.85, and through the barrier 0.5 nm wide its transmission is about 10⁻⁹⁵: effectively never. Tunnelling rewards thin barriers, light particles and energies close to the top. Electrons tunnel constantly in electronics and chemistry, hydrogen nuclei tunnel measurably in some chemical reactions, and heavier things almost never do. For smooth barriers, the WKB approximation, introduced in 1926 by Gregor Wentzel, Hendrik Kramers and Léon Brillouin, replaces 2κL by twice the area under the curve of κ(x) across the forbidden region.

Tunnelling's first triumph came from the nucleus. Many heavy nuclei decay by emitting an alpha particle, two protons and two neutrons bound together, identical to a helium-4 nucleus. Inside the nucleus the strong nuclear force holds the alpha in; just outside, the electric repulsion of the remaining protons shoves it away. Together they form the Coulomb barrier, roughly 30 MeV high around uranium. Yet uranium-238 emits alpha particles carrying only about 4.2 MeV, while scattering experiments analysed by Ernest Rutherford in 1927 showed that alpha particles with about twice that energy, fired at uranium, were turned back by electric repulsion without ever getting in. How can a particle escape a wall that a projectile with twice its energy cannot climb?

In 1928 George Gamow, and independently Ronald Gurney and Edward Condon, answered: it does not go over, it goes through. The alpha rattles inside the nucleus, striking the barrier about 10²¹ times per second, each strike with a minuscule, fixed chance of tunnelling out. A nucleus therefore has no memory and no clock: decay times are random, and a large sample decays exponentially with a definite half-life, the time in which half the nuclei decay. Because electric repulsion fades as 1/r with distance r from the centre, a uranium-238 alpha must tunnel from the nuclear surface, about 9 fm out, to about 61 fm, where the hill sinks below its energy (a femtometre is 10⁻¹⁵ metres). The WKB recipe gives odds of a few times 10⁻³⁹ per strike: billions of years.

The model's glory is the Geiger–Nuttall law, found by Hans Geiger and John Nuttall in 1911: the more energetic an emitter's alpha particles, the shorter its half-life. Polonium-212 emits alphas of 8.78 MeV and has a half-life of 0.29 μs; uranium-238 emits alphas of 4.20 MeV and lasts 4.47 billion years. Barely a factor of two in energy, a factor of about 5×10²³ in lifetime! The more energetic alpha meets the 1/r wall where it is thinner, needing to tunnel only out to about 26 fm, and because width sits in an exponent, a modest change in energy becomes an astronomical change in lifetime. Bismuth-209, long thought stable, was caught alpha-decaying in 2003; its half-life, 2.0×10¹⁹ years, is over a billion times the age of the universe.

The Sun fuses hydrogen into helium in its core, and the first step is the hardest: two protons must come within a femtometre or so before the strong force can grab them, and their electric repulsion at that range is a barrier of roughly 1.0 MeV. At a core temperature of about 15.7 million kelvin, the matching thermal energy scale kT, where k is Boltzmann's constant, is just 1.35 keV, and a typical proton carries about 2 keV, some five hundred times too little. The fraction of protons with enough energy to climb the barrier is of order e^(−739), about 10⁻³²¹. Even with about 10⁵⁷ protons in the Sun, classically not one pair would fuse in the age of the universe.

A proton with just a few thousand electron-volts has a tiny but real chance of slipping through, a chance that climbs steeply with energy while the number of protons at each energy falls steeply. Their product peaks in a narrow window, the Gamow peak, near 6 keV in the solar core: most fusing pairs tunnel through a barrier more than a hundred times higher than their energy. Arthur Eddington, who had argued in 1920 that stars burn hydrogen, told critics who said the stars were not hot enough to go and find a hotter place. In 1929 Robert Atkinson and Fritz Houtermans vindicated him with Gamow's tunnelling, and in 1938 and 1939 Hans Bethe, partly with Charles Critchfield, worked out the reaction chains that power the stars, work that won Bethe the 1967 Nobel Prize in Physics.

Even with tunnelling the first step is extraordinarily slow, because it also needs the weak nuclear force to turn a proton into a neutron. That bottleneck is a blessing: the Sun shines steadily for about ten billion years instead of burning out before life could evolve.

Exponential sensitivity, a curse for anyone predicting lifetimes, is a gift for anyone measuring distances. In March 1981, at IBM's research laboratory near Zurich, Gerd Binnig and Heinrich Rohrer first measured a controlled tunnelling current across a vacuum gap, the decisive step toward their scanning tunnelling microscope, or STM. A sharp metal tip is held within about a nanometre of a conducting surface, never touching it, and a small voltage is applied. The vacuum gap is a barrier whose height is roughly the work function, the minimum energy needed to pull an electron out of a metal, typically 4 to 5 eV. Electrons tunnel across, giving a current of around a nanoampere that falls as e^(−2κd) with the gap d.

For a barrier of 4.5 eV, κ ≈ 10.9 per nanometre, so retracting the tip by just 0.1 nm cuts the current about ninefold. A feedback loop moves the tip up and down to hold the current constant as it scans, and the record of those movements is a contour map of the surface with height resolution at the picometre level. The same exponential delivers sideways resolution: if the tip ends in a single atom, that atom sits a fraction of a nanometre closer than its neighbours and carries almost all the current. Binnig and Rohrer shared the 1986 Nobel Prize in Physics with Ernst Ruska, the pioneer of the electron microscope.

The STM soon did more than look. In 1989 Donald Eigler and Erhard Schweizer at IBM's Almaden laboratory in California used the tip to slide 35 xenon atoms across a nickel crystal until they spelled the letters IBM, a feat published in 1990. Building with single atoms had begun.

The photos on your phone are stored by tunnelling. Flash memory, invented by Fujio Masuoka's team at Toshiba in the 1980s, keeps each bit as charge trapped on an isolated layer walled in by oxide only several nanometres thick. To write or erase the NAND flash used for mass storage, the chip applies roughly 15 to 20 volts, which tilts the barrier into a thin triangle that electrons tunnel through: Fowler–Nordheim tunnelling. Remove the voltage, the barrier widens again, and the charge stays put for years without power. Tunnelling can also be a nuisance: by the mid-2000s the silicon dioxide layer insulating each transistor's gate was only about 1.2 nm thick, and leakage through it pushed Intel, in 2007, to a physically thicker hafnium-based insulator.

In 1957 Leo Esaki, at the company soon renamed Sony, doped a germanium junction so heavily that the junction was only some ten to twenty nanometres wide. Electrons tunnelled straight across, and over a range of voltages the current fell as the voltage rose, a negative resistance that let tunnel diodes switch extremely fast. In 1962 Brian Josephson, a 22-year-old Cambridge graduate student, predicted that Cooper pairs, the bound electron pairs that carry a superconductor's resistance-free current, would tunnel through a thin insulator between two superconductors, a device now called a Josephson junction. A current then flows with no voltage at all, and a steady voltage makes it oscillate at 483.6 GHz per millivolt, the basis of modern voltage standards. Esaki, Josephson and Ivar Giaever, who used tunnelling to probe superconductors, shared the 1973 Nobel Prize in Physics.

Can something big tunnel? In 1984 and 1985 at the University of California, Berkeley, John Clarke, Michel Devoret and John Martinis tested exactly that with a Josephson junction on a chip about a centimetre across. A current flowed with zero voltage, a state trapped behind an energy barrier. What tunnelled out was not one electron but a single collective variable describing billions of Cooper pairs, and a voltage suddenly appeared. Below a crossover temperature the escape rate stopped depending on temperature, the fingerprint of tunnelling rather than thermal hopping, and microwaves revealed quantized energy levels. The work earned the 2025 Nobel Prize in Physics; its descendants, superconducting qubits built around Josephson junctions, are as of 2026 the heart of quantum processors at Google, IBM and elsewhere.

A tempting thought: perhaps a tunnelling particle crosses the barrier instantly. In 1962 Thomas Hartman calculated that the delay of a transmitted wave packet stops growing once a barrier is thick, implying ever-faster apparent speeds. In 1993 Aephraim Steinberg, Paul Kwiat and Raymond Chiao at Berkeley timed single photons tunnelling through a multilayer mirror 1.1 μm thick: the transmitted peaks arrived 1.47 fs earlier than those of twin photons crossing the same distance in air, an apparent speed of about 1.7 times the speed of light.

No signal outran light. The barrier reshapes the pulse, transmitting a faint copy built mostly from its leading edge, which was already ahead; the peak shifts forward, but the front, the earliest moment anything can be detected, never outruns light. Nor is tunnelling instantaneous: in 2020 Steinberg's group in Toronto sent rubidium atoms cooled to about two nanokelvin through a barrier of laser light 1.3 μm thick and found that transmitted atoms spent about 0.6 ms inside. What happens in between is interpretation-dependent: standard quantum mechanics assigns no path, while the pilot-wave theory of chapter 22 gives every particle a definite trajectory.

Finally, strolling through your bedroom wall. Treat a person of 70 kg as a single particle, grant a generous energy deficit of one joule and a wall 10 cm thick: 2κL ≈ 2.2×10³⁴, so T is about 1 divided by a number with 10³⁴ digits. Trying once a second for the age of the universe, about 4×10¹⁷ seconds, changes nothing. Worse, you are not one isolated quantum object but about 7×10²⁷ atoms, jostled by air and light, which destroys the coherence any collective tunnelling would need. Mwahaha, measurably: the wall wins.