Quantum tunnelling

A wave packet with energy E = 5 meets a barrier of height V₀ = 6. Classically it should come back entirely; on screen part of it appears on the other side.

Inside the barrier the wave function is not oscillatory but exponentially decaying, ψ ∝ e^(−κx) with κ = √(2(V₀−E)). As long as the barrier is not infinitely wide, a tail survives and keeps propagating.

The steady-state transmission T ≈ [1 + V₀² sinh²(κa)/(4E(V₀−E))]⁻¹ is exponentially sensitive to width and to V₀−E. That sensitivity is what makes scanning tunnelling microscopy resolve fractions of a nanometre.