In Marc Bee's book, he has described the principle of spectroscopy with reservoir (the material) and the probe as interacting systems with their own hamiltonians $H_R$ and $H_p$ respectively and $H_c$ being the interaction hamiltonian. I have difficulty in evaluating one particular expression which obtains from Fermi Golden rule

$$ W_{nm} = \sum_{n'}\sum{m'}\frac{1}{Z_R}\text{e}^{-\beta E_{m'}}|\left<n'|\bar H_c|m'\right>|^2\delta(\omega_{n'm'}-\omega) $$

where $\hbar\omega_{n'm'} = E_{m'} - E_{n'}$ and $\hbar\omega = E_m - E_n $.

After using integral representation of the delta function, he goes on to write this can be evaluated to

$$ W_{nm} =\frac{1}{\hbar^2}\int_{-\infty}^{\infty}\text{Tr}\{\hat \rho_R \bar H_c^{\dagger}(0)\bar H_c(t)\}\text{e}^{-i\omega t} \text dt. $$

  • $\begingroup$ $\uparrow$ Which page? $\endgroup$ – Qmechanic Aug 5 '15 at 13:03
  • $\begingroup$ page 29, chapter 2 of Quasi-elastic neutron scattering ! $\endgroup$ – user35952 Aug 6 '15 at 4:47

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