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

__toc__ Let be the average of operator (observable) . This average is accessible to the experimentator (see (references)). The case where is proportional to is treated in (references) Case where is proportional to is treated here. Consider following problem: Find such that: with and evaluate: 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. 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)) }