Introduction to Mathematical Physics/Quantum mechanics/Linear response in quantum mechanics

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Let be the average of operator (observable) . This average is accessible to the experimentator (see ([#References|references])). The case where is proportional to is treated in ([#References|references]) Case where is proportional to is treated here. Consider following problem:

Problem:

Find such that:

with

and evaluate:

Remark: Linear response can be described in the classical frame where Schr\"odinger equation is replaced by a classical mechanics evolution equation. Such models exist to describe for instance electric or magnetic susceptibility.

Using the interaction representation\footnote{ This change of representation is equivalent to a WKB method. Indeed, becomes a slowly varying function of since temporal dependence is absorbed by operator }

and

Quantity to be evaluated is:

At zeroth order:

Thus:

Now, has been prepared in the state , so:

At first order:

thus, using properties of Dirac distribution:

Let us now calculate the average: Up to first order,

Indeed, is zero because is an odd operator.

Failed to parse (unknown function "\lefteqn"): {\displaystyle \begin{matrix} \lefteqn{ \mathrel{<} \tilde{\psi}^0|e^{\frac{iH_0t}{\hbar}}qZ e^{\frac{iH_0t}{\hbar}}|\tilde{\psi}^1\mathrel{>} =}\\ &=& \mathrel{<} {\tilde{\psi}}^0| e^{\frac{iH_0t}{\hbar}}qZe^{\frac{-iH_0t}{\hbar}}|{\psi}_k\mathrel{>} \mathrel{<} {\psi}_k|{\tilde{\psi}}^1\mathrel{>} \end{matrix}}

where, closure relation has been used. Using perturbation results given by equation ---pert1--- and equation ---pert2---:

We have thus:

Using Fourier transform\footnote{ Fourier transform of:

and Fourier transform of:

are different: Fourier transform of does not exist! (see ([#References|references])) }