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

Let $<A>(t)$ be the average of operator (observable) $A$. This average is accessible to the experimentator (see (references)). The case where $H(t)$ is proportional to $sin(\omega t)$ is treated in (references) Case where $H(t)$ is proportional to $\delta (t)$ is treated here. Consider following problem: Template:IMP/prob Template:IMP/rem Using the interaction representation\footnote{ This change of representation is equivalent to a WKB method. Indeed, $\widetilde{\psi (t)}$ becomes a slowly varying function of $t$ since temporal dependence is absorbed by operator $e^{\frac{i{H}_{0}t}{\hslash }}$} Template:IMP/eq and Template:IMP/eq Quantity $<qZ>$ to be evaluated is: Template:IMP/eq Template:IMP/eq At zeroth order: Template:IMP/eq Thus: Template:IMP/eq Now, $\tilde{\psi }$ has been prepared in the state ${\psi }_0$, so: Template:IMP/eq At first order: Template:IMP/eq thus, using properties of $\delta$ Dirac distribution: Template:IMP/eq Let us now calculate the average: Up to first order,

$\begin{array}{ccc}<qZ>& =& <{\tilde{\psi }}^0+\widetilde{{\psi }^1}|e^{\frac{i{H}_{0}t}{\hslash }}qZ{e}^{\frac{i{H}_{0}t}{\hslash }}|{\tilde{\psi }}^0+\widetilde{{\psi }^1}>\\ & =& <{\tilde{\psi }}^0|e^{\frac{i{H}_{0}t}{\hslash }}qZ{e}^{\frac{i{H}_{0}t}{\hslash }}|{\tilde{\psi }}^1>+<{\tilde{\psi }}^1|e^{\frac{i{H}_{0}t}{\hslash }}qZ{e}^{\frac{i{H}_{0}t}{\hslash }}|{\tilde{\psi }}^0>\end{array}$

Indeed, $<{\tilde{\psi }}^0|qZ|{\tilde{\psi }}^0>$ is zero because $Z$ is an odd operator.

$\begin{array}{c}<{\tilde{\psi }}^0|e^{\frac{i{H}_{0}t}{\hslash }}qZ{e}^{\frac{i{H}_{0}t}{\hslash }}|{\tilde{\psi }}^1>=\\ & =& <{\tilde{\psi }}^0|e^{\frac{i{H}_{0}t}{\hslash }}qZ{e}^{\frac{-i{H}_{0}t}{\hslash }}|{\psi }_k><{\psi }_k|{\tilde{\psi }}^1>\end{array}$

where, closure relation has been used. Using perturbation results given by equation ---pert1--- and equation ---pert2---: Template:IMP/eq We have thus:

$\begin{array}{ccc}<qZ>(t)& =& 0\text{ if }t<0\\ <qZ>(t)& =& e^{i{\omega }_{0k}t}<{\psi }^0|qZ|{\psi }^k>\frac{1}{i\hslash }<{\psi }^k|W_i^c|{\psi }^0>+CC\text{ if not }\end{array}$

Using Fourier transform\footnote{ Fourier transform of: Template:IMP/eq and Fourier transform of: Template:IMP/eq are different: Fourier transform of $f(t)$ does not exist! (see (references)) }