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:
[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.

 

[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.

 

[Entropia]

Formalism in Linear Response Theory is a mathematical framework used to study the response of a physical system to external perturbations. It is based on the assumption that the system can be described by a Hamiltonian operator, which represents the total energy of the system, and other operators that represent different physical quantities or observables.

The insertion of the sum of basis vectors in the commutator is a key step in the formalism of linear response theory. It allows us to calculate the response of the system to small perturbations by breaking it down into a sum of basis vectors, which are eigenstates of the Hamiltonian operator. This allows us to apply perturbation theory, which is a powerful mathematical tool for studying small changes in a system.

In the context of linear response theory, the commutator represents the difference between the system's response to the perturbation and its unperturbed state. By inserting the sum of basis vectors, we are essentially breaking down this difference into smaller, more manageable components. This allows us to calculate the response of the system in a more systematic and efficient manner.

Furthermore, the operators in the commutator may not necessarily commute, meaning that their order of operation can affect the result. The insertion of the sum of basis vectors helps to properly account for this non-commutativity and ensure accurate calculations.

In summary, the insertion of the sum of basis vectors in the commutator is a crucial step in the formalism of linear response theory, allowing us to accurately and efficiently calculate the response of a physical system to external perturbations.