A quantum wavefunction can penetrate a region where the particle's classical kinetic energy would be negative. This produces tunneling through a potential barrier.

Consider a rectangular barrier of height $V_0$ from $x=0$ to $x=a$, with particle energy $E<V_0$. Inside the barrier, the time-independent Schrödinger equation gives exponentially varying solutions $$\psi(x)=Ae^{\kappa x}+Be^{-\kappa x},\qquad \kappa=\frac{\sqrt{2m(V_0-E)}}{\hbar}.$$ Matching the wavefunction and its derivative at both boundaries leaves a nonzero transmitted wave beyond the barrier.

For a sufficiently wide or high barrier, the transmission probability has the approximate dependence $$T\propto e^{-2\kappa a}.$$ This exponential sensitivity means small changes in barrier width, particle mass or energy can change tunneling rates by orders of magnitude.

For example, increasing the width by $\Delta a$ multiplies transmission approximately by $$e^{-2\kappa\Delta a}.$$ The effect is therefore important on atomic scales but rapidly becomes negligible for macroscopic masses and distances.

Tunneling does not mean the particle temporarily has a classical negative kinetic energy, nor does it require borrowing energy in violation of conservation. The stationary quantum state extends through the classically forbidden region and matches continuously onto a transmitted state on the other side.

Tunneling underlies alpha decay, scanning tunneling microscopy, semiconductor tunnel junctions and many molecular processes.