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.
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.
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.