The wavefunction describes the state of the system. The state of the system can be stationary, in which case the wavefunction is an eigenstate of the Hamiltonian.

The stationary state wavefunctions have a special property, the temporal and spatial parts of the wavefunction are separable as a product.

The expectation value of any observable in this system corresponds to the average of a large number of measurements on identically prepared systems. The time dependent expectation value of any time independent operator (observable) in a stationary state is:

Note: the time dependent functions cancel each other out. Thus, the time dependent expectation value is always equal to the time averaged value:

This is why the energy eigenstates of the system are called stationary; The expectation values in these states are all constant.

If the state of the system is NOT stationary, then it must be in a superposition state. A superposition state is, in general,

Or, if the sum over n is the sum over all possible quantum numbers and C_n are complex coefficients:

A superposition state wavefunction is a 'mixture' of stationary states. Suppose we want a 50/50 mixture of the two lowest energy eigenstates of the system. Then, the wavefunction would look like

But, this 'guessed' wavefunction is not normalized

We could evaluate this normalization braket because of the properties of the stationary state wavefunctions called orthonormality:

To 'fix' this normalization, we must multiply our guessed wavefunction by 2^1/2, i.e.

Note that there are other, perfectly valid ways to construct a fifty-fifty mixture of the same two stationary states. For example PHI_b and PHI_c are also valid wavefunctions:

The difference between these superposition states is in the relative phase of their constituent basis functions.

What about the time dependence of expectation values in a superposition state?

Now, in general, the 'cross terms' do NOT VANISH, i.e.

(These terms can be zero, but only in special cases. Can you guess what these cases are? )

The time dependence of the expectation value now becomes clear. Expanding the stationary state wavefunction into their spatial and temporal parts

This can be rewritten as:

Or, the expectation value of this 50-50 superposition state of the stationary states with n=1 and n=2 is the average of the expectation value in each of those states PLUS a term that oscillates in time at the frequency that corresponds to the energy difference between those two stationary states divided by (h_bar)