infinite square well
The infinite square well is the precise mathematical setup behind the particle-in-a-box picture: a region where the potential energy is zero inside and instantly jumps to infinity at the walls. The infinite walls mean the particle can never be found outside, so its wavefunction is forced to be exactly zero there and at the boundaries. This is the first quantum system most students solve completely from start to finish.
Because the walls are infinitely high, the rule is simple and rigid: the wavefunction must vanish at each end. The allowed solutions are clean sine waves that complete a whole number of half-cycles across the well. Counting these solutions gives the quantum numbers, and squaring that count gives the energy of each level. There is no leakage, no decay outside, and no ambiguity — just a tidy spectrum of standing waves.
Its value lies in being exactly solvable and utterly transparent. Every deeper idea — discrete energy levels, the ground-state energy that cannot be zero, the growing number of nodes as energy rises — appears here in its purest form, uncluttered by approximation. Physicists keep returning to it as a sanity check and a teaching tool, even though no real trap has walls of literally infinite height.
Zero potential inside, infinite walls outside; the solutions are sine waves that vanish at both ends.
The infinitely high walls are a useful fiction. They make the wavefunction vanish abruptly, but a real potential is always finite, so the sharp edge is an approximation, not a feature of nature.