The solutions to Schrodinger's Time-Independent Equation,
are stationary states. Here we have plotted probability densities for the harmonic oscillator at 8 different energies. Stationary states are defined by a single energy. For example, the quantum harmonic oscillator single state energy is depicted as the darker horizontal red line at 2. Since this is the time-independent solution, the red line never moves. The lighter lines above and below it are upper and lower bounds of uncertainty. These two never move either. The darker virtical red line is the expectation value for the position of the "wave" of energy 2. The lighter lines to the left and right are boundaries for the uncertainty of the wave's position. The uncertainty never changes in time. Also, note that as you increase in energy level the position expectation value stays centered right at zero, while the uncertainties broaden.
Below we have an anharmonic oscillator. Say we are in a single state of energy 2, again with the solution being time-independent, the horizontal red line never moves. This assymetric potential shape causes different energy levels to have different position expectation values seen in the green and blue lines. Note that as the energy goes up the position expectation values get further away from the energy minimum. The uncertainties are not show here.
The point of this stationary state discussion was that none of the above wave functions change in time.