5.A: Stationary states
In general, wave functions evolve in time like the ones you saw for the particle bouncing between two walls. However, each physical system has a set of special wave functions for which the probability is a constant in time. These special states are called stationary states or eigenstates. The term eigenstates comes from the German eigen which means ``it's own''. In other words, each physical system - a particle bouncing between two walls, the hydrogen atom, the helium atom, and so on - has ``it's own'' set of stationary states or eigenstates.
The stationary states for a particle moving between two walls are shown here. Also indicated are the relative energies of a particle prepared in these eigenstates. The eigenstate with the lowest energy is called the ground state. Each state has an arbitrary label n called a quantum number. Quantum numbers are man-made labels that often arise by historical accidents. They are just names by which physical scientists refer to the eigenstates.