In classical physics, a particle cannot pass through a barrier if it lacks sufficient energy. In quantum mechanics, the particle exists as a wave function that penetrates the barrier — and can emerge on the other side.

The probability of tunneling depends on barrier height, barrier width, and particle energy. Thinner barriers and higher energy = more tunneling.

T ≈ e^(-2κL)

κ = √(2m(V-E)) / ℏ
L = barrier width

— Transistors: electrons tunnel through thin oxide layers
— Nuclear fusion: protons tunnel in the Sun
— DNA mutations: protons tunnel between base pairs
— Scanning tunneling microscope: images individual atoms