Time Dependent Position Operator

Question

How does one find the time dependent position expectation value for a wave function? I thought we could simple take the time dependent wave and apply the position operator like normal, but this gave me the wrong answer, as the time dependencies multiplied to one and left me with my solution for $t = 0$.

Added: The given solution was to use Ehrenfest's Theorem to find the expectation of momentum at a given time, then to use Ehrenfest's again to find the expected position at a time. I still don't understand why I couldn't just apply the momentum operator to the time dependent wave function to get the time dependent momentum expectation value, and do the same for position.

Answer 1

I supposed you are in a context of bound states, with normalized eigenfunctions $\psi_n(x,t) = \phi_n(x) e ^{iE_nt}$. Of course, if you calculate $\langle x(t)\rangle_{\psi_n} = \int dx \bar \psi_n x \psi_n$, you will find a position expectation value which does not depend on time.

Now, this is not the general case, if you take a linear combination of the $\psi_n$ : $\psi = \sum a_n \psi_n$, and calculate $\langle x(t)\rangle_{\psi} = \int dx \bar \psi x \psi$, you will find a positition expectation value which depends on time.