A Finite Well Holds Only a Few Bound States
In the infinite box the walls were infinitely high, so you could stack standing waves without limit. But a real well has a finite depth V0. Then the number of states that can be trapped inside (states whose energy E lies between −V0 and 0) is a fixed, finite count. On top of that the wavefunction leaks past the walls like e−κ|x|. Here κ is the same decay constant you met in the tunneling lesson, and bound states have E below zero. In the first figure we watch ladder rungs appear and vanish as we change the depth.
A single dimensionless number set jointly by the depth V0 and 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 always at least one, no matter how shallow. Raise V0 with the slider and a new rung pops into the well each time z0 crosses a multiple of π/2. The figure draws the well walls together with the energy level lines sitting inside.
Look closely at the shape of one bound state. Inside the well it oscillates with wavenumber k = √(2 m (E + V0))/ℏ, but outside the walls it dies away smoothly as e−κ|x|. Here κ = √(−2 m E)/ℏ, well defined because E is below zero. Pull E up toward zero (make the well shallow) and the tails fatten and reach further. The particle leaks into a region it could never enter classically.
How do the allowed energies come out? The even states satisfy k·tan(ka) = κ, the odd states −k·cot(ka) = κ, transcendental equations you must solve. We solve them graphically. Overlay a quarter-circle of radius z0 with the tan and −cot branches on one plane, and the points where they meet are exactly the allowed energies. Raise z0 with the slider and the quarter-circle grows, so the number of intersections increases.
Move E slowly through zero. While E is below zero you get a localized bound state with tails dying on both sides, a particle that stays in the well. But once E rises past zero and turns positive, it is no longer trapped and becomes an oscillating scattering state stretching off to infinity on both sides. Now the energy is no longer discrete but continuous. So E < 0 is a discrete spectrum, and E > 0 is a continuous one.
Finally, make the well deeper and deeper. Raising V0 grows κ, so the tails die off fast and the waveform inside approaches the sin shapes of the infinite box that vanish at both ends. The energy ladder too approaches the infinite-box ratio Eₙ ∝ n². The figure overlays the infinite-box result on the finite-well waveform, so you see at a glance how the V0 → ∞ limit recovers the familiar box.