I have a spatial wave function here $$\psi(x)=\frac{1}{\sqrt{a\sqrt{\pi}}}e^{-(x^2/2 a^2)+ikx} \quad .$$
I calculated its position expectation value, and it's zero, as expected since it's a stationary state as $|\psi|^2$ depends only on $x$.
Why is the Ehrenfest theorem not applicable here? That is, why $\langle p\rangle$ is non-zero if $\langle x\rangle$ is constant?
Your wave function $$\psi(x)=\frac{1}{\sqrt{a\sqrt{\pi}}}e^{-(x^2/2 a^2)+ikx}$$ does not depend on time $t$ and as such is not a solution of the time-dependent Schrödinger equation $$i\hbar \frac{\partial\psi(x,t)}{\partial t}=H\psi(x,t).$$
But Schödinger's equation is a requirement of Ehrenfest's theorem. Therefore you cannot apply Ehrenfest's theorem with your time-independent wave function.
