# Please explain this Formalism in Linear Reponse Theory.

• jbb
In summary, the insertion of the sum of basis vectors in the commutator is a common technique in linear response theory that allows for easier manipulation of operator components.
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.

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.

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.

## 1. What is Formalism in Linear Response Theory?

Formalism in Linear Response Theory is a mathematical framework used to describe the response of a system to an external perturbation. It allows us to understand how a system will change in response to a small change in its environment.

## 2. How does Formalism in Linear Response Theory work?

In Formalism in Linear Response Theory, we start with a Hamiltonian, which is a mathematical representation of a system's energy. We then use the Hamiltonian to calculate the system's response to an external perturbation by solving equations of motion.

## 3. What are the key assumptions of Formalism in Linear Response Theory?

The key assumptions of Formalism in Linear Response Theory are that the system is in thermal equilibrium, the perturbation is small, and the response is linear. Additionally, the system is assumed to be in a stable state, and the perturbation does not change the system's equilibrium state.

## 4. What are the applications of Formalism in Linear Response Theory?

Formalism in Linear Response Theory has many applications in physics, chemistry, and materials science. It is used to study the behavior of various systems, such as solids, liquids, and gases, under different conditions. It is also used to determine the properties of materials, such as electrical and magnetic properties.

## 5. What are the limitations of Formalism in Linear Response Theory?

Formalism in Linear Response Theory has limitations in its application to complex systems and strong perturbations. It also assumes that the system is in thermal equilibrium, which is not always the case in real-world situations. In some cases, more advanced mathematical methods may be needed to accurately describe the system's response.

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