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Chemistry LibreTexts

Relation to spectra

Suppose that \( F_e (t) \) is a monochromatic field

\[ F_e(t) = F_{\omega}e^{i\omega t}e^{\epsilon t} \]


where the parameter \(\epsilon \) insures that field goes to 0 at \(\underline { t = - \infty } \). We will take \(\underline {\epsilon\rightarrow 0^+ } \) at the end of the calculation. The expectation value of  \(B\) then becomes

 

\(\langle B(t)\rangle \) $\textstyle =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$

\(\langle B\rangle _0 + \int_{-\infty}^t\;ds\;\Phi_{BB}(t-s)F_{\omega}e^{i\omega s}e^{\epsilon s}\)

 
  $\textstyle =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$

\(\langle B\rangle _0 + F_{\omega}e^{(i\omega + \epsilon)t} \int_0^{\infty}d\tau\Phi_{BB}(\tau)e^{-i(\omega-i\epsilon)\tau}\)

 


where the change of integration variables \(\underline {\tau=t-s } \) has been made.

Define a frequency-dependent susceptibility by

\[ \chi_{BB}(\omega-i\epsilon) = \int_0^{\infty}d\tau \Phi_{BB}(\tau) e^{-i(\omega-i\epsilon)\tau} \]


then

\[ \langle B(t)\rangle = \langle B\rangle _0 + F_{\omega}e^{i\omega t}e^{\epsilon t}\chi_{BB}(\omega-i\epsilon) \]


If we let \(z=\omega-i\epsilon \), then we see immediately that

\[ \chi_{BB}(z) = \int_0^{\infty}d\tau\;\Phi_{BB}(\tau) e^{-iz\tau} \]


i.e., the susceptibility is just the Laplace transform of the after effect function or the time correlation function.

Recall that

\[ \Phi_{AB}(t) = {i\over\hbar}\langle [A(t),B(0)]\rangle _0 = {i \over \hbar}\langle[e^{iH_0t/\hbar} Ae^{-iH_0t\hbar},B]\rangle _0 \]


Under time reversal, we have

 

\(\Phi_{AB}(-t)\) $\textstyle =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$

\(\underline { {i \over \hbar} \langle \left[e^{-iH_0t/\hbar}Ae^{iH_0t/\hbar},B\right]\rangle _0} \)

 
  $\textstyle =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$

\( \underline { {i \over \hbar} \langle \left(e^{-iH_0t/\hbar}Ae^{iH_0t/\hbar}B -Be^{-iH_0t/\hbar}Ae^{iH_0t/\hbar}\right)\rangle _0 } \)

 
  $\textstyle =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$

\(\underline { {i \over \hbar} \langle \left(Ae^{iH_0t/\hbar}Be^{-iH_0t/\hbar} -e^{iH_0t/\hbar}Be^{-iH_0t/\hbar}A\right)\rangle _0 } \)

 
  $\textstyle =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$

\(\underline { {i \over \hbar} \langle \left(AB(t)-B(t)A\right)\rangle _0 } \)

 
  $\textstyle =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$

\(\underline {-{i \over \hbar} \langle \left[B(t),A\right]\rangle = -\Phi_{BA}(t) } \)

 


Thus,

\[ \Phi_{AB}(-t) = -\Phi_{BA}(t) \]


and if  \(A = B \), then

\[ \Phi_{BB}(-t) = -\Phi_{BB}(t) \]


Therefore

 

\(\chi_{BB}(\omega)\) $\textstyle =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$

\(\lim_{\epsilon\rightarrow 0^+}\int_0^{\infty} dt\;e^{-i(\omega-i\epsilon t)}\Phi_{BB}(t) \)

 
  $\textstyle =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$

\(\lim_{\epsilon\rightarrow 0^+}\int_0^{\infty}dt\;e^{-\epsilon t}\left[\Phi_{BB}(t)\cos\omega t - i\Phi_{BB}(t)\sin\omega t\right]\)

 
  $\textstyle =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$

\({\rm Re}(\chi_{BB}(\omega)) - i{\rm Im}(\chi_{BB}(\omega))\)

 


From the properties of \( \Phi_{BB}(t) \) it follows that

 

 \({\rm Re}(\chi_{BB}(\omega)\) $\textstyle =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ \({\rm Re}(\chi_{BB}(-\omega)\)  
\( {\rm Im}(\chi_{BB}(\omega) \) $\textstyle =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ \( -{\rm Im}(\chi_{BB}(-\omega) \)  


so that \({\rm Im}(\chi_{BB}(\omega)) \) is positive for \( \underline { \omega > 0 } \) and negative for \( \underline { \omega < 0 } \). It is a straightforward matter, now, to show that the energy difference \( Q (\omega ) \) derived in the lecture from the Fermi golden rule is related to the susceptibility by

\[ Q(\omega) = 2\omega\vert F_{\omega}\vert^2{\rm Im}(\chi_{BB}(\omega)) \]