Please explain this Formalism in Linear Reponse Theory.

[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:
$$\int \left \langle g\right|\left[H,B\right]\left|g\right\rangle dt =<br /> \int\left \langle g\right|H*B\left|g\right\rangle - \left \langle g\right|B*H\left|g\right\rangle dt$$
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.

 

[azntoon]

The insertion of the sum of basis vectors in the commutator is a useful technique when you want to express operators in terms of their components. This allows you to manipulate the components of the operator and make calculations more straightforward. For example, if you have a Hamiltonian operator H and some other operator B, you can use this technique to rewrite the commutator [H,B] as a sum of component-wise products, which is often easier to work with. This is especially useful when dealing with operators that have multiple components, such as spin operators or angular momentum operators.