# Equation of motion and operators in the interaction picture.

1. May 20, 2010

### Denver Dang

1. The problem statement, all variables and given/known data

I have a question that says:
What is the equation of motion for a general operator in the interaction picture. I.e. how does the time derivative of the operators behaves ? Show this.

And then I have to find the time development for the annihilation and creation operator ($\hat{a}$ and $${{\hat{a}}^{\dagger }}$$) in the interaction picture.

2. Relevant equations

3. The attempt at a solution

The first question I THINK this is how it is supposed to be done, but I'm not sure.
I have that:

$$$\frac{d{{A}_{I}}}{dt}=\frac{1}{i\hbar }\left[ {{A}_{I}},{{H}_{0}} \right]$$$

where:

$${{A}_{I}}={{e}^{i{{H}_{0}}t/\hbar }}{{A}_{s}}{{e}^{-i{{H}_{0}}t/\hbar }}$$

So my thought is, that I just take the derivative of ${{A}_{I}}$, and then I think, if my math is correct, I end up with something where I'm able to write the commutator as in the first equation. And then I get what is says. And if I'm now mistaken that is the equation of motion I need to find ?

The second question I'm not entirely sure about how to do.
For the Harmonic Oscillator, which is what I'm working with here, the commutator of the two operators is $\left[ {{a}_{-}},{{a}_{+}} \right]=1$.
But then I don't know what my next step is.

So if someone could help I would be grateful :)

Regards