finite square well quantum
In an infinite box the walls are endlessly high, so standing waves can be stacked without limit. A real well has a finite depth V0. Then the states that can be trapped, those whose energy E lies between −V0 and 0, come in a finite number. The wave function leaks past the wall as e−κ|x|. κ is the same decay constant from the tunneling lesson, and a bound state has E below 0. The first picture changes the depth and shows ladder rungs appear and vanish.
One dimensionless number set by the depth V0 and the half-width a is the strength z0 = a·√(2 m V0)/ℏ. The number of bound states is about 1 + floor(z0 / (π/2)), and there is at least one however shallow the well. Raise V0 and a new rung rises inside the well each time z0 crosses a multiple of π/2. The picture draws the well walls together with the level lines inside.
Look closely at the shape of one bound state. Inside the well it oscillates with wavenumber k = √(2 m (E + V0))/ℏ, and outside the wall it fades smoothly as e−κ|x|. Here κ = √(−2 m E)/ℏ, well defined because E is below 0. Pull E toward 0, which is the same as making the well shallow, and the tails grow fatter and reach farther. The particle leaks into a region it cannot enter classically.
How do the allowed energies come out. Even states satisfy k·tan(ka) = κ, and odd states satisfy −k·cot(ka) = κ, a transcendental equation. Solve it as a picture. Overlay a quarter-circle of radius z0 with the branches of tan and −cot on one plane, and the meeting points are the allowed energies. Raise z0 and the quarter-circle grows, so the number of crossings grows.
Move E slowly so that it passes 0. While E is below 0 the tails fade on both sides, a bound state, a particle that stays in the well. Once E passes 0 and becomes positive it is no longer trapped and becomes a scattering state stretching without end to both sides. Then the energy is continuous rather than separated. E < 0 is a discrete spectrum, and E > 0 is a continuous spectrum.
Last, make the well deeper and deeper. Raise V0 and κ grows, the tails vanish quickly, and the waveform inside approaches the sin shape of the infinite box that is 0 at both ends. The energy ladder too approaches the infinite-box ratio Eₙ ∝ n². The infinite-box result is drawn over the finite-well waveform, so the limit V0 → ∞ returning to the familiar box is plain at a glance.