The whole question is: (No typo this time I think...)
In section 9.5 of Scully's quantum optics, an atom in a damped cavity is researched. The Hamiltonian is
$$H=H_F+H_A+H_{AF}+H_R+H_{FR}$$
The subscript F represents Field, and A -> Atom, R ->reservoir and AF -> interactions between atom and...
Sorry for these mistakes. I will present this question in a self-contained way tomorrow or the day after tomorrow because the book is not at hand now. Thank you for your reply.
Firstly, I don't know in which Picture this equation holds (if I hadn't missed some words in the previous text...). I think it may be the Heisenberg Picture. But if it is, the rest target is to prove $$\frac{i}{\hbar}[H_R+H_{FR},(a^\dagger) ^ma^nO_A]=\langle\frac{d}{dt}((a^\dagger)...