Potential steps and finite wells

Classically, a particle with energy less than a barrier's height simply cannot enter the barrier region at all — it reflects with certainty. Solve the Schrödinger equation for the same situation and a different picture emerges. Inside the classically forbidden region, the wavefunction does not suddenly vanish; it satisfies an equation whose solution is a decaying exponential, not zero. The wavefunction, and the probability of finding the particle, both fall off smoothly and continuously with depth into the barrier rather than stopping dead at the boundary. This same decaying-tail behaviour is the mechanism behind quantum tunneling: if the barrier is thin enough that the exponential tail has not yet decayed to nothing by the far side, there is a nonzero probability that the particle is found beyond the barrier, on the classically inaccessible side.

A finite-depth potential well — a region of low potential energy bounded on both sides by finite (not infinite) barriers — shows a related contrast with the idealized infinite well. An infinitely deep well forces the wavefunction to be exactly zero at its walls, producing an infinite ladder of allowed energy levels, evenly related by the same boundary condition at every level. A finite well relaxes that boundary condition: the wavefunction is allowed to leak into the walls as a decaying exponential, similar to the tunneling case, which changes the allowed energies and — crucially — limits how many bound states can exist at all. Above a certain energy set by the well's depth, a particle is no longer bound: it is a scattering state, free to leave the well region entirely, and the discrete ladder of bound levels simply ends. A shallow or narrow enough well may support only a single bound state, or even none.

The general lesson is that quantum boundaries are never perfectly sharp: the allowed wavefunction always extends somewhat beyond a real physical barrier, and how far it extends, and how many discrete bound states a given well can hold, both depend on the barrier's finite height and width — not just on its existence.