finite square well

The finite square well is the more realistic cousin of the infinite well: a region of low potential energy bordered by walls of finite, not infinite, height. A particle with energy below the wall height is still bound and trapped, but the trapping is no longer absolute. The walls are penetrable in the quantum sense, and the wavefunction does not slam to zero at the edge.

Instead, just outside the well the wavefunction decays smoothly into the forbidden region, where classically the particle would never have enough energy to be. This decaying tail means there is a small but real chance of finding the particle slightly outside the trap. Matching the wave inside to its decaying tail outside is more delicate than the infinite case, and it is solved by requiring the wavefunction and its slope to join up continuously at each wall.

Two consequences follow. First, a finite well holds only a limited number of bound states; a shallow or narrow well may support just one, and a sufficiently weak well in three dimensions might bind nothing at all. Second, the energy levels sit slightly lower than the infinite-well values, because the leaking tail effectively gives the particle a little more room. The finite well is the natural bridge between clean textbook models and the messier traps of real atoms and materials.