A ball can’t roll through a hill it lacks the energy to climb. But a quantum particle is a WAVE, and a wave doesn’t stop dead at a wall — it leaks in, fading, and a sliver comes out the far side. That impossible leak is why the sun shines and how a flash drive remembers. Slide the wall and watch the ghost pass through.
For a particle of energy E meeting a barrier of height V > E and width L, the wavefunction doesn’t vanish at the wall — it decays exponentially inside as e^(−κx) with κ = √(2m(V−E))/ℏ, and emerges on the far side with a small but non-zero amplitude. The transmission probability ≈ e^(−2κL) falls fast with barrier width and height but is never zero. A fail-loud self-check throws unless the transmission is between 0 and 1 and shrinks as the barrier is made wider or taller — the leak classical physics forbids.
A rectangular-barrier approximation (the exact formula adds a prefactor); real tunnelling underlies alpha decay, fusion in stars (through the Coulomb barrier), scanning-tunnelling microscopes, and flash memory. The exponential suppression is exact for this model.