# Please explain this Formalism in Linear Reponse Theory.

1. Jan 23, 2009

### jbb

I am learning linear response theory right now and I have come across a mathematical technique I have seen before but I don't understand the reason for the application. What I am talking about is the insertion of the sum of basis vectors in the commutator.
Generally speaking it looks similar to this:
$$\int \left \langle g\right|\left[H,B\right]\left|g\right\rangle dt = \int\left \langle g\right|H*B\left|g\right\rangle - \left \langle g\right|B*H\left|g\right\rangle dt$$
and since
$$\sum \left| n\rangle \langle n \left| = 1$$
then we have
$$\int \sum \left\{\left \langle g\right|H\left| n\rangle \langle n \left|B\left|g\right\rangle - \left \langle g\right|B\left| n\rangle \langle n \left|H\left|g\right\rangle\right\} dt$$
where H is a hamiltonian operator and B is some other operator.
I have seen that insertion in another context before, so I know this is a common thing to do. I do not understand how this helps, though. Could the operators not operate on one another? They are matrices of identical dimensions, aren't they?

Thank you for taking the time.