# 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.