Here, a particle in a 1-D infinite box (PIB) is taken as the working example. You can compare a single energy eigenstate (whose probability density is frozen in time) against a superposition of two or three eigenstates (whose probability density keeps changing). Units are chosen so that ħ = m = L = 1, so energies are Eₙ = n²π²/2.
A single eigenstate ψₙ(x)e^(−iEₙt/ħ). The phase factor spins, but |ψₙe^(−iEₙt/ħ)|² = |ψₙ|² never changes.
An energy eigenstate ψₙ evolves only by an overall phase, Ψ(x,t) = ψₙ(x)e^(−iEₙt/ħ). Because probability density is |Ψ|², and phase cancels exactly. No observable moves, no matter how long you wait! Here "stationary" means: stationary in probability (or property), not in phase.
A superposition Ψ = Σ cₙψₙ e^(−iEₙt/ħ) mixes the underlying phases that rotate at different rates. Their cross terms survive in |Ψ|², oscillate at the beat frequencies ω_mn = (E_m − Eₙ)/ħ. The quantities like <x>(t) change. This is quite similar to "classical-looking" motion.
Ehrenfest's theorem says expectation values obey classical-looking equations of motion:
d<x>/dt = <p>/m d<p>/dt = −<∂V/∂x>
Inside the box V(x)=0, you may think <p> to be conserved. But, it is not, once you mix states. It oscillates too! A single stationary state trivially satisfies both equations with everything constant at zero net motion; but a superposition is the first place you actually see the classical-mechanics analogy.
Try it: mix n=1 with n=2 and then add a third, a more distant state. Watch the motion turn less sinusoidal and more "wavepacket-like." It starts building a superposition of many eigenstates.