As the state evolves in time, each individual eigenstate is multiplied by a complex exponential. But they are *different* complex exponentials, each one based on its own eigenvalue *E _{1}* or

The position probability is based on the squared modulus of the wavefunction.

How do you find the squared modulus of a complex number? You multiply the number by its complex conjuate, which you generally write by replacing every occurence of *i* with *-i*.

Multiply all that out.

Remembering that *e ^{ix} = *cos

Now expand out the other cross-term. Then rewrite the result based on the fact that the cosine is an even function, and the sine is an odd function.

When we add those two cross-terms, the sines cancel out and the cosines add. That gives us our original goal, the modulus-squared of the time-evolved wavefunction.

After all that, the real point is that the time dependence didn't cancel out, so the position probabilities are changing.