Introduction to Mathematical Physics/Quantum mechanics/Linear response in quantum mechanics
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:
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.
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)) }