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

Let $$ \mathrel{<} A\mathrel{>} (t)$$ 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:

Using the interaction representation\footnote{ This change of representation is equivalent to a WKB method. Indeed, $$\tilde{\psi(t)}$$ becomes a slowly varying function of $$t$$ since temporal dependence is absorbed by operator $$e^{\frac{iH_0t}{\hbar}}$$}

and

Quantity $$ \mathrel{<} qZ\mathrel{>} $$ to be evaluated is:

At zeroth order:

Thus:

Now, $$\tilde{\psi}$$ has been prepared in the state $$\psi_0$$, so:

At first order:

thus, using properties of $$\delta$$ Dirac distribution:

Let us now calculate the average: Up to first order, $$\begin{matrix} \mathrel{<} qZ\mathrel{>} &=& \mathrel{<} \tilde{\psi}^0+\tilde{\psi^1}|e^{\frac{iH_0t}{\hbar}} qZe^{\frac{iH_0t}{\hbar}}|\tilde{\psi}^0+\tilde{\psi^1}\mathrel{>} \\ &=& \mathrel{<} \tilde{\psi}^0| e^{\frac{iH_0t}{\hbar}}qZe^{\frac{iH_0t}{\hbar}}|\tilde{\psi}^1\mathrel{>} + \mathrel{<} \tilde{\psi}^1|e^{\frac{iH_0t}{\hbar}} qZe^{\frac{iH_0t}{\hbar}}|\tilde{\psi}^0\mathrel{>} \end{matrix}$$ Indeed, $$ \mathrel{<} \tilde{\psi}^0|qZ|\tilde{\psi}^0\mathrel{>} $$ is zero because $$Z$$ is an odd operator. $$\begin{matrix} { \mathrel{<} \tilde{\psi}^0|e^{\frac{iH_0t}{\hbar}}qZ e^{\frac{iH_0t}{\hbar}}|\tilde{\psi}^1\mathrel{>} =}\\ &=& \mathrel{<} {\tilde{\psi}}^0| e^{\frac{iH_0t}{\hbar}}qZe^{\frac{-iH_0t}{\hbar}}|{\psi}_k\mathrel{>} \mathrel{<} {\psi}_k|{\tilde{\psi}}^1\mathrel{>} \end{matrix}$$ where, closure relation has been used. Using perturbation results given by equation ---pert1--- and equation ---pert2---:

We have thus: $$\begin{matrix} \mathrel{<} qZ\mathrel{>} (t)&=& 0 \mbox{ if } t < 0\\ \mathrel{<} qZ\mathrel{>} (t)&=& e^{i\omega_{0k}t} \mathrel{<} {\psi}^0|qZ|{\psi}^k\mathrel{>} \frac{1}{i\hbar} \mathrel{<} {\psi}^k|W^c_i|{\psi}^0\mathrel{>} +CC \mbox{ if not } \end{matrix}$$ Using Fourier transform\footnote{ Fourier transform of:

and Fourier transform of:

are different: Fourier transform of $$f(t)$$ does not exist! (see (references)) }