How does quantum tunneling differ from the predictions of classical physics?

In classical physics, a particle's ability to overcome a potential barrier is strictly determined by its energy ($E$). If a particle's energy is less than the height of the potential barrier ($V_0$), classical mechanics dictates that the particle cannot pass through and will remain confined by the barrier indefinitely.1

In contrast, quantum mechanics reveals that particles possess wavelike properties, described by a quantum-mechanical wave function. Because these particles behave as waves, they have a non-zero probability of penetrating or "tunneling" through an energy barrier, even when their energy is lower than the barrier's height. This phenomenon occurs because the particle's wave function does not abruptly end at the barrier but instead decays within it, allowing a portion of the wave to emerge on the other side.231

This fundamental difference is essential to the operation of a scanning tunneling microscope, which relies on the tunneling of electrons between a probe tip and a surface to map atomic structures. Unlike the deterministic predictions of classical physics, quantum tunneling is a probabilistic process that allows subatomic particles to traverse regions that would be impassable under classical laws.21