Softening the walls
The infinite box was a useful lie: its walls were infinitely high, so the particle was sealed in forever. Real traps are never like that. The pull that holds an electron to an atom, or a marble to a real bowl, is finite — push hard enough and you can escape. So let us replace the infinite walls with a more honest trap: a finite square well, a dip of fixed depth with walls of limited height. Think of a valley with rims you could, with enough energy, climb over and walk out of.
The height of those rims is set by what physicists call potential energy — the cost, in energy, of being at a given place. Inside the well the cost is low (a comfortable valley floor); outside, beyond the rims, the cost is high (you would have to pay energy to be up there). A particle whose total energy is less than the rim height is trapped: classically, it simply does not have the energy to be outside, full stop. Hold that classical verdict in mind, because quantum mechanics is about to quietly overrule it.
The wavefunction leaks past the wall
With the infinite box, the wavefunction was slammed to exactly zero at the wall — it had no choice. With a finite wall, it cannot do that. Quantum waves are not allowed to have sudden jumps or sharp kinks; they must be smooth, joining up continuously across any boundary. This requirement is wavefunction continuity. And a smooth wave that is sizeable just inside the wall cannot instantly drop to nothing on the far side. Instead it pokes a little way into the wall before fading out.
Inside the forbidden wall, the wave does not keep wiggling — it cannot, because there is not enough energy for it to oscillate there. Instead it decays: it droops smoothly toward zero, dying away the deeper it reaches. This drooping, non-oscillating tail living inside a region the particle classically cannot enter is the evanescent wave. The wall is thick enough that the tail usually fades to nothing before reaching the outside — but it is not exactly zero just past the rim, and that small fact is the seed of something dramatic.
Fewer rungs, and softer ones
Softening the walls changes the staircase in two sensible ways. First, because the wave now spills a little beyond the rims, it effectively has a touch more room than the bare width of the well — like a string whose ends are not quite nailed down but tied with a bit of slack. A slightly longer effective wave means a slightly longer wavelength, which means slightly lower energies than the infinite box gave. The rungs sag a little.
Second, and more strikingly, a finite well holds only a finite number of bound rungs. The infinite box had an endless staircase climbing forever; a shallow real well might hold just two or three trapped states, or even only one. Any state that would need more energy than the rim height is no longer trapped at all — give the particle that much energy and it sails over the rim and leaves. There is a ceiling to how high the staircase goes, set by the depth of the well.
What about a wall with only one side — a single upward step, like a sudden cliff the particle runs into? Here the particle is not trapped at all; it is travelling, and it slams into the step. Classically, if it has less energy than the step it simply bounces straight back, every time. Quantum mechanically it mostly bounces back too — but its evanescent tail still droops a little way into the forbidden high ground before turning around. The split between what reflects and what gets through is the question of reflection and transmission, and it is the natural bridge to the next idea. Because if you make that forbidden region a wall rather than an endless cliff — thin enough that the drooping tail has not quite died before it reaches the far side — something the classical world flatly forbids becomes possible.