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.(adsbygoogle = window.adsbygoogle || []).push({});

Generally speaking it looks similar to this:

[tex]\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 [/tex]

and since

[tex]\sum \left| n\rangle \langle n \left| = 1[/tex]

then we have

[tex]

\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 [/tex]

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.

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# Please explain this Formalism in Linear Reponse Theory.

Can you offer guidance or do you also need help?

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