Quantum Tunnelling — How Particles Pass Through Walls

1. Wave-Particle Duality

The conceptual foundation of tunnelling is that quantum objects — electrons, protons, even whole atoms — are not localised billiard balls. They are described by a wavefunction ψ(x, t), a complex-valued field whose squared modulus |ψ|² gives the probability density of finding the particle at position x at time t.

This was established experimentally over decades: electrons diffract through crystal lattices (Davisson-Germer, 1927), single electrons build up interference patterns in double-slit experiments even when sent one at a time, and neutrons from a beam are reflected and transmitted by thin films just like light through glass.

The wave nature has a radical consequence. When a wave meets a boundary where its speed would change — a potential step — it is partly transmitted and partly reflected even when the particle has more than enough energy to cross. Conversely, the transmitted wave extends a little way into regions the particle classically cannot enter. In a finite-width barrier, that leaking wave can emerge on the far side: tunnelling.

2. The Schrödinger Equation

The time-independent Schrödinger equation governs how a particle of mass m behaves in a potential V(x):

where ℏ = h/(2π) ≈ 1.055 × 10⁻³⁴ J·s is the reduced Planck constant and E is the particle's total energy. The equation has different character in different regions:

Region I: Free particle (V = 0, E > 0)

The solution is a plane wave:

The first term is a wave travelling right, the second left. |k| is the wave number (radians per metre).

Region II: Inside the barrier (V = V₀ > E)

When E < V₀ the argument of the square root becomes negative. Defining:

the solutions are real exponentials, not oscillating waves:

The decaying term C·e^(−κx) is the evanescent wave — it falls exponentially inside the barrier. The growing term D·e^(+κx) would diverge for an infinitely thick barrier, so D = 0 in that limit, but for a finite barrier of width L, both terms matter and the wavefunction emerges on the far side with reduced but non-zero amplitude.

3. Transmission Coefficient T = e^(−2κL)

Matching the wavefunction and its derivative continuously at both walls of the barrier (x = 0 and x = L) gives the exact transmission coefficient — the fraction of incoming probability flux that gets through. For a rectangular barrier in the limit κL ≫ 1 (thick or high barrier), it simplifies to:

This is the most important formula in tunnelling physics. Three things control T:

• Barrier width L — T falls exponentially with L. Double the barrier width and T is squared, reducing it dramatically.
• Energy deficit (V₀ − E) — deeper under the barrier means larger κ and smaller T. Particles near the top of the barrier tunnel much more easily.
• Particle mass m — heavier particles have larger κ for the same energy deficit. Proton tunnelling is exponentially rarer than electron tunnelling at the same energy.

The full exact result (without the thick-barrier approximation) also includes pre-exponential factors that account for reflections at the barrier walls:

For κL ≫ 1, sinh(κL) ≈ e^(κL)/2 and the exact result reduces to the simple exponential.

4. Heisenberg Uncertainty and Tunnelling

Werner Heisenberg's uncertainty principle is often invoked to "explain" tunnelling, though the full Schrödinger equation needs no supplementary explanation. Nevertheless the uncertainty principle provides powerful intuition. In its most precise form:

where Δx is the standard deviation of position and Δp is the standard deviation of momentum. Because position and momentum are conjugate variables, localising a particle in space (small Δx) necessarily implies a spread in momentum (large Δp), and vice versa.

Inside a thin barrier, the particle's position is constrained to a narrow region 0 ≤ x ≤ L, so Δx ≈ L. The resulting momentum uncertainty is Δp ≥ ℏ/(2L), meaning the particle's kinetic energy has uncertainty ΔE ≈ (Δp)²/(2m) ≥ ℏ²/(8mL²). For a very thin, very high barrier this fluctuation is large enough that the particle "effectively" has sufficient energy to cross.

The energy-time uncertainty relation is also relevant:

A particle can "borrow" energy ΔE for a time Δt ≤ ℏ/(2ΔE) — insufficient to cross a macroscopic barrier, but sufficient to traverse a nanometre-scale potential hill. This is a heuristic picture; the rigorous treatment uses the Schrödinger equation throughout.

5. The WKB Approximation

The rectangular barrier is the simplest case. Real barriers — the Coulomb barrier around a nucleus, the potential well holding an electron in a molecule — have arbitrary shapes V(x). The Wentzel-Kramers-Brillouin (WKB) approximation generalises the transmission coefficient to smoothly varying potentials.

The WKB tunnelling probability through an arbitrary barrier between classical turning points x₁ and x₂ (where E = V(x)) is:

where the local decay constant is now position-dependent:

The integral in the exponent is called the WKB action integral or tunnelling action. For a rectangular barrier it reduces to κL, recovering our earlier result. The WKB approximation is valid when the potential varies slowly compared to the local de Broglie wavelength — an excellent approximation for most real barriers far from the classical turning points.

6. Real Applications

Tunnel Diode

Invented by Leo Esaki in 1957 (Nobel Prize 1973), the tunnel diode exploits electron tunnelling through a thin p-n junction depletion layer. At small forward bias, electrons tunnel from the n-side conduction band to empty states in the p-side valence band, producing a current before conventional diode conduction begins. As bias increases, the energy alignment worsens and current drops — creating a region of negative differential resistance unique among electronic components. Tunnel diodes switch in picoseconds and are used in microwave oscillators and high-speed logic.

Scanning Tunnelling Microscope (STM)

Heinrich Rohrer and Gerd Binnig invented the STM at IBM Zürich in 1981 (Nobel Prize 1986). A metal tip is brought within ~0.5 nm of a conducting surface. The tunnelling current I ∝ e^(−2κd) depends so sensitively on tip-surface separation d that a 0.01 nm change (one-hundredth of an atom's diameter) changes I by roughly 10%. By scanning the tip and maintaining constant current via a feedback loop, the STM traces atomic-scale topography with sub-ångström resolution — sufficient to image, and even manipulate, individual atoms.

Flash Memory

The floating-gate transistors in NAND flash memory (SSDs, USB drives, smartphones) store bits by trapping electrons on an electrically isolated polysilicon island. Writing and erasing are accomplished by Fowler-Nordheim tunnelling — electrons tunnel through a thin (~10 nm) silicon dioxide layer under a strong applied electric field. Without tunnelling, solid-state storage as we know it would not exist.

7. Alpha Decay and the Gamow Factor

Inside a heavy nucleus, an alpha particle (two protons + two neutrons, i.e., a helium-4 nucleus) is held by the strong nuclear force in a potential well. Outside the nucleus, the dominant interaction is the repulsive Coulomb potential between the alpha and the daughter nucleus:

At the nuclear surface (r ≈ 1-10 fm), the strong force is attractive and deep; outside, the Coulomb barrier rises to a maximum and then falls as 1/r. An alpha particle with energy E below the barrier maximum is classically trapped forever. Yet experimentally, nuclei decay with half-lives ranging from microseconds to billions of years.

George Gamow (1928) applied the WKB approximation to compute the tunnelling probability through the Coulomb barrier. The result is expressed as the Gamow factor:

Or equivalently, G = exp(−2γ) where γ = √(E_G/E) and E_G = 2m_α(παZ₁Z₂e²/ℏ)² is the Gamow energy (of order MeV for typical alpha emitters). The decay rate λ = ν · P is the product of the collision frequency ν (the alpha bouncing against the barrier ~10²¹/s) and the tunnelling probability P ∝ G.

This single formula explains the enormous range of half-lives: Polonium-212 (E_α = 8.95 MeV) decays in 0.3 μs; Uranium-238 (E_α = 4.27 MeV) takes 4.5 billion years. A roughly 4 MeV difference in alpha energy translates to a factor of 10²⁴ in half-life — pure exponential sensitivity to the Gamow factor.

8. Nuclear Fusion in Stars

The Sun converts hydrogen into helium at its core, releasing the energy that sustains life on Earth. But there is a puzzle: the core temperature is about 1.5 × 10⁷ K, corresponding to a thermal energy kT ≈ 1.3 keV. The Coulomb barrier between two protons has a height of about 550 keV at contact — more than 400 times the typical thermal energy. Classical physics predicts that fusion should be essentially impossible.

Quantum tunnelling saves the day. The Gamow factor allows protons to tunnel through the tail of the Coulomb barrier even at sub-barrier energies. The fusion rate depends on the Gamow peak — a convolution of the Boltzmann energy distribution (favouring low energies) and the tunnelling probability (favouring high energies):

where S(E) is the astrophysical S-factor (a slowly varying cross-section stripped of the Gamow factor), f(E) is the Maxwell-Boltzmann distribution, and E₀ is the Gamow peak energy — the energy at which most reactions occur. For p-p fusion in the Sun, E₀ ≈ 6 keV, much lower than the barrier height.

Without quantum tunnelling, the Sun would not shine, stars would not exist, and the heavy elements forged in stellar cores — carbon, oxygen, iron — would never have been produced. Tunnelling is not an exotic laboratory curiosity; it is what makes the universe chemically rich and life possible.