Bound States of the Finite Square Well

Trap a quantum particle in a box with walls of finite height and only a handful of energies are allowed. Inside the well the wavefunction oscillates, outside it leaks into the classically forbidden region as a decaying exponential, and a state can only exist if these pieces join smoothly, matching in value and slope at both walls. That smoothness is a strict condition, satisfied at just a discrete set of energies: the bound states. Writing $z = kL/2$ and $z_0 = \frac{L}{2}\sqrt{2mV_0}/\hbar$, the matching collapses to two transcendental equations, $z\tan z = \sqrt{z_0^2 - z^2}$ for the even states and $z\cot z = -\sqrt{z_0^2 - z^2}$ for the odd ones, and the panel below solves them the way you would by hand: the even and odd branches climb while a circle of radius $z_0$ falls, and every crossing is an allowed level. Deepen or widen the well and the circle grows, sweeping past more branches and admitting more states, the count being $\lfloor z_0/(\pi/2)\rfloor + 1$. No matter how shallow, there is always at least one. The top panel draws the levels inside the well with their wavefunctions, the ground state nodeless and even, each higher one adding a node and flipping parity. Click a level to follow it in both panels.