A stationary pure state of a time-independent Hamiltonian is an energy eigenstate. Its time evolution is multiplication by , a global phase, so probabilities for observables with no explicit time dependence remain constant.
For example, a quantum harmonic oscillator number state has a fixed spatial probability density even though its vector acquires a phase. A superposition of states with different energy eigenvalues generally develops changing relative phases and need not be stationary. Conversely, a superposition inside one degenerate energy eigenspace stays stationary. For mixed states, stationarity means , allowing mixtures of different energies as well as pure eigenstates. Stationarity does not imply that every measured observable has zero uncertainty.
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