Homework Help: QM: Product expectation value

1. Jul 28, 2012

Niles

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

I have read a paper, where they want to find the average number of photons in a cavity. They have an expression for $\langle{\hat a}\rangle$, and then they use
$$\langle{\hat a}\rangle^* = \langle{\hat a^\dagger}\rangle$$
to find $\langle{\hat a^\dagger \hat a}\rangle$. I agree with the above relation, however what I don't agree with is the following equality
$$\langle{\hat a^\dagger \hat a}\rangle = \langle{\hat a}\rangle^*\langle{\hat a}\rangle$$
Am I right? I mean, one can't just factorize an expectation value like that.

Best,
Niles.

2. Jul 29, 2012

Fightfish

You are right in that the equality doesn't hold in general. It is a commonly made approximation in order to "close" the set of correlation functions.

3. Jul 29, 2012

Niles

Thanks! I read it here, on page 12/13, where they do as I wrote in my OP: http://mediatum2.ub.tum.de/doc/652711/652711.pdf

I can't see however why the approximation is valid in this case.

4. Jul 31, 2012

Fightfish

If I'm not mistaken the equality holds because the authors are considering the steady state case.

5. Jul 31, 2012

Niles

It is not obvious to me why it should be valid in steady state.