Consider this relation
$$\sigma(\omega_{I})=\int_{-\infty}^{+\infty} <\psi(0)|\psi(t)>e^{i(E_i/\hbar+\omega_I)t}dt$$
I know, by inverse Fourier transform, I can compute for
$$\int_{-\infty}^{+\infty} <\psi(0)|\psi(t)>e^{i\omega _It}dt$$
But I really have no idea how to take the $tE_i/\hbar$ power into account in order to determine $\sigma(\omega_{I})$; I just couldn't see the relation between the two integrals. Can someone help me?