node

A node is a point where a wavefunction crosses through zero, so the probability of finding the particle exactly there momentarily vanishes. In a one-dimensional trap the wavefunction wiggles up and down, and each place where it passes through the axis is a node. They are the quantum analogue of the still points on a plucked guitar string, where the string does not move while everything around it does.

Nodes are tightly tied to energy. The ground state, the lowest-energy bound state, has no nodes between the walls; its wavefunction keeps a single sign across the trap. Each successive excited state adds exactly one more node. More nodes mean the wavefunction wiggles faster, and faster wiggling means greater curvature, which in quantum mechanics directly corresponds to higher kinetic energy. So counting nodes is a quick way to rank the energy levels.

This node-counting rule is more than a curiosity; it is a reliable guide that holds for a wide range of potentials, not just simple boxes. Given the wavefunctions of a trap, you can order them in energy just by counting their zeros. It also reflects something deep: extra nodes are the price a wavefunction pays for carrying more energy while still fitting neatly into the same confined space.