Help Understanding Response Function $$H(\omega)$$

[Diracobama2181]

Homework Statement

Let us study a system coupled tho an external field h(t) with the Hamiltonian $$H = H_0 −h(t)A $$
Assume that the perturbation is monochromatic, $$h(t) = h_0 cos \omega t$$, and take the unperturbed density operator to be thermal. With the aid of a suitable average over the period of the perturbation, show that the rate of change of the expectation value of the perturbed Hamiltonian $$H_0$$ equals $$\frac{1}{2}h_{0}^2\omegaχ_{AA}(\omega)$$.

Relevant Equations

$$χ_{AA}=\frac{1}{2\hbar} Tr{\overline{\rho}[\overline{A(t)},\overline{A(0)}]}$$
$$<B(\omega)>=\sum_{j} χ_{BAj}h_j(\omega)$$ where $$B(\omega)$$ is an operator.

$$<H(\omega)>=\sum_{j} χ_{HAj}h_j(\omega)$$
Where $$χ_{HA}=\frac{1}{2\hbar} Tr{{\rho}[{H(t)},{A(0)}]}$$.
But
$$[H(t),A(0)]=[H_o,A(0)]-[A(t)h,A(0)]=-h_0 cos(\omega t)[A(t),A(0)]$$.
So $$χ_{HA}=-\frac{1}{2\hbar}Tr(\rho h_0 cos(\omega t)[A(t),A(0)])=-h_0cos(\omega t)χ_{AA}$$.
Then $$<H(\omega)>=\sum_{j}-h_0^{2}cos^2(\omega t)χ_{AAj}$$.

Is my reasoning correct thus far? And where would I go from here? Thanks.

 

[Diracobama2181]

Nevermind, figured it out. Thank you though.