- Problem Statement
- In the Scully's Quantum Optics textbook, section 9.5, an equation 9.5.11 is given to calculate the time derivative of the operator $$(a^\dagger) ^ma^nO_A$$. I try to derive this equation by myself, however, there are several problems.

- Relevant Equations
- Eq-9.1.1, Eq-9.3.5, Eq-9.5.5~10

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 $$\frac{i}{\hbar}[H_R+H_{FR},(a^\dagger) ^ma^nO_A]=\langle\frac{d}{dt}((a^\dagger) ^ma^n)\rangle_RO_A$$. Left side of this equation is something dependent on $\hat{b}$, but the right side is not. So how can I prove this equation?

(This is my first time to post a problem in Physics Forum, so if I did something wrong remind me please and I will correct.)

Thanks very much!

(This is my first time to post a problem in Physics Forum, so if I did something wrong remind me please and I will correct.)

Thanks very much!