A problem in time-dependent perturbation theory

I'm trying to solve this problem from (Baym, Problem 4, Chapter 12):

1- Consider a system in an energy eigenstate $|0\rangle$ at time $t = 0$, acted on by a peturbation $V'(t)$. Let θ be some observable of the system with $\langle 0|\cal{O}|0\rangle = \cal{O}_0$. Show that the expectation value of an operator $\cal{O}$ at time $t$ is given to the first order in $V'$ by

$$\frac{-i}{\hbar}\int_0^t \langle 0 |[\cal{O}(t),V'(t')] |0\rangle dt'$$

plus zeroth order term, where the operators and states are written in the interaction picture.

2- Calculate the time-dependent dipole mement induced in a charged particle moving forced to move along the $x$ axis in a 1d simple harmonic oscillator potential, by an electric field pointing in the $x$ direction with magnitude $E\cos\omega t$.

When I start from the interaction picture, I got energy eigenstate $t$ instead of $0$. Can anyone help me tackle this problem?

Thank you

• Since $|0\rangle$ is an energy eigenstate for the unperturbed Hamiltonian, say $H_0$, what's the action of $\exp(-i\;H_0 t/\hbar)$ on it? – udrv Feb 21 '17 at 6:55