A Gaussian wave packet approaching a rectangular barrier. Part reflects; part tunnels through — the transmitted packet is smaller but intact.
The Core Idea
In classical mechanics, a particle hitting a wall it can't climb just bounces back. In quantum mechanics, the particle's wavefunction leaks through the barrier. If the wall is thin enough, there's a measurable probability of finding the particle on the other side — even though it never had enough energy to go over. This is quantum tunnelling, and it powers everything from nuclear fusion in the Sun to flash memory in your phone.
The Mathematics
For a particle of energy E approaching a rectangular barrier of height V₀ > E:
Region I (x < 0): ψ = e^(ikx) + R·e^(−ikx) (incident + reflected)
Region II (0 < x < a): ψ = A·e^(−κx) + B·e^(κx) (evanescent decay)
Region III (x > a): ψ = T·e^(ikx) (transmitted)
where k = √(2mE)/ℏ and κ = √(2m(V₀−E))/ℏ. Inside the barrier, the wavefunction decays exponentially — it doesn't oscillate. But it doesn't reach zero either. The transmission coefficient is:
T ≈ e^(−2κa) = exp(−2a·√(2m(V₀−E))/ℏ)
The thicker the barrier (larger a) or the higher the barrier (larger V₀ − E), the exponentially smaller the tunnelling probability. But it's never exactly zero.
Wider barrier — the exponential decay has more room to suppress the wavefunction. Very little makes it through.
Try widening the barrier and watch the transmitted packet shrink exponentially. The barrier width slider gives you direct control over the e^(−2κa) factor — every small increase makes a dramatic difference.
Evanescent waves
The key physics is the evanescent wave inside the barrier. Classically, kinetic energy would be negative there — an impossibility. Quantum mechanically, the wavefunction simply switches from oscillatory (e^(ikx)) to exponentially decaying (e^(−κx)). The probability density |ψ|² is nonzero everywhere inside the barrier — the particle has a presence there, it just fades rapidly.
The energy balance
The transmitted particle has the same energy as the incident one. Tunnelling doesn't slow the particle down or steal energy. The barrier only attenuates the probability amplitude — the wave that emerges on the other side oscillates at the same frequency and wavenumber, just with a smaller amplitude.
Thin barrier, energy close to V₀ — significant tunnelling. A large fraction of the wave packet passes through.
Real-world tunnelling
- Sun: Protons in the solar core tunnel through their Coulomb barrier to fuse — the Sun burns because of tunnelling.
- Tunnelling diodes: Exploit negative differential resistance from resonant tunnelling through quantum wells.
- Flash memory: Electrons tunnel through an insulating oxide layer to charge/discharge a floating gate.
- STM (scanning tunnelling microscope): Maps surfaces by measuring tunnelling current between a tip and the sample — sensitive to sub-angstrom height differences.
Further reading
- Griffiths, Introduction to Quantum Mechanics — Chapter 2 derives the rectangular barrier in detail.
- Quantum tunnelling (Wikipedia)
- Gamow, Zur Quantentheorie des Atomkernes (1928) — the original tunnelling paper explaining alpha decay.