Quantum Tunneling
Penetration of classically forbidden barriers, and the exponential dependence on width.
A particle with energy $E$ below a potential barrier $V_0 > E$ has a nonzero probability of appearing on the other side — a purely quantum phenomenon. For a 1D rectangular barrier of height $V_0$ and width $a$, the transmission coefficient is
$$T = \frac{1}{1 + \frac{V_0^2 \sinh^2(\kappa a)}{4E(V_0 - E)}}, \qquad \kappa = \sqrt{\frac{2m(V_0 - E)}{\hbar^2}}.$$For $\kappa a \gg 1$, $T \approx 16 \frac{E(V_0 - E)}{V_0^2} e^{-2\kappa a}$. The dominant feature is exponential suppression in barrier width.
Tunneling explains $\alpha$-decay (Gamow), STM imaging, Josephson junctions, the field emission of electrons, and modern flash memory. WKB approximation generalizes: for a smooth barrier,
$$T \sim \exp\!\left(-\frac{2}{\hbar}\int_a^b \sqrt{2m(V(x) - E)}\, dx\right).$$Inside the barrier the wavefunction is real exponential, $\psi \propto e^{-\kappa x}$, with imaginary momentum.