quantum tunnelling
A ball rolled at a hill it lacks the energy to climb rolls back down -- always. A quantum particle fired at a barrier taller than its energy sometimes appears on the far side, as if it passed through the hill rather than over it. This is tunnelling: because the wavefunction does not stop dead at a wall but decays smoothly into the classically forbidden region, a thin enough barrier leaves a nonzero amplitude on the other side, and with it a real probability of transmission.
For a particle of energy E hitting a rectangular barrier of height V_0 > E and width a, the wavefunction inside the barrier is a decaying exponential exp(-kappa x) with decay constant kappa = sqrt(2 m (V_0 - E))/hbar. The transmission probability in the thick-barrier limit is T is approximately exp(-2 kappa a) = exp(-2 a sqrt(2 m (V_0 - E))/hbar). Read this formula carefully: tunnelling is exponentially sensitive to the barrier width a, to the square root of the mass m, and to the energy shortfall V_0 - E. Double the width and you square (shrink) the tiny transmission; use a heavier particle and it plummets. Nothing is 'borrowed and repaid' -- the particle never has negative kinetic energy; the amplitude simply penetrates and re-emerges.
Tunnelling is not a rare curiosity but a workhorse of nature and technology. Alpha decay is an alpha particle tunnelling out through the Coulomb barrier of a nucleus, and its exponential sensitivity explains the staggering 20-order-of-magnitude range of nuclear half-lives (the Gamow theory). Hydrogen fusion in the Sun proceeds only because protons tunnel through their mutual repulsion. The scanning tunnelling microscope reads a surface atom-by-atom from the exponential tunnelling current across a vacuum gap, and flash memory and tunnel diodes are engineered around it.
In a scanning tunnelling microscope, the tunnelling current across the tip-sample gap falls by roughly a factor of 10 for every extra 0.1 nm of gap. That exponential dependence is why the instrument can resolve individual atoms: a single-atom step change in height changes the current by an order of magnitude, an easily measured signal.
The exponential sensitivity of tunnelling to distance is what lets the STM image single atoms.
Tunnelling does not mean the particle briefly has negative kinetic energy or 'borrows energy' -- energy is conserved throughout, and the barrier interior is a region where the wavefunction is evanescent, not where a classical particle sits. The exponential exp(-2 kappa a) is a large-barrier approximation; for thin or low barriers you must use the full transmission formula.