I've spent quite a while looking into classical models for electric susceptibility, including some semi-classical extensions to the idea, based around concepts such as effective electron mass. However, I've recently moved onto looking into the susceptibility of atomic vapours, potentially near the transition levels, where it seems necessary to take a much more quantum-based line of approach, as well as the effects of broadening.

If you look up the derivation of relatively classical electric susceptibility, it's quite easy to find derivations of the theory - which makes it quite easy to see where simplifications (such as the Drude model) are made. However, when it gets onto trying to take into account quantum effects, I can only find the basic information on the prediction of the position of lines, as derived from the Schrodinger equation, and nothing to do with broadening, or the actual shape of the spectrum as a function of frequency. If I try from the other end, to look up broadening about various spectral lines, and try to work backwards from various papers, it can be quite hard to follow the trail of research, and even if I can, inevitably the paper for which I'm looking is behind a pay-wall. I'm aware that there almost certainly isn't a closed-form solution to this problem, otherwise I would have seen reference made to it already, but if I can see a derivation, that would put me in a position to at least understand which terms are left out, and/or simulate, numerically, solutions.

So, my question: Are there any sources which you would recommend, from which I'd be able to find the derivation - or at least a starting point. I am willing to pay to get past a pay-wall, but only want to do so if I really have to.

  • $\begingroup$ There is the first-order derivation of the Stark effect. (It does not converge when taken to infinite order, because of electron tunneling.) This is probably in most quantum physics textbooks. $\endgroup$
    – user137289
    Mar 11, 2019 at 21:25

1 Answer 1


The problem is that there are a number of calculations one would do, depending on the physics one wishes to include in the problem. First is a calculation of the eigenstate energy levels. Next is a calculation of the transition probabilities via Fermi’s Golden Rule. Then you need to think about broadening, which could involve multiple mechanisms. For example, inhomogeneous broadening includes Doppler broadening (a Gaussian lineshape, dependent on temperature), and homogeneous broadening includes lifetime broadening (Lorentzian lineshape, dependent on couplings to other states).

Finally, perform these calculations for all of the relevant states, and put them together to find the energies, amplitudes, and shapes of the lines in the spectrum.

  • $\begingroup$ Would this be in addition to, or as a quantum replacement to, the more classical treatment of the calculation of susceptibility. And also, most of this stuff to do with broadening etc tends to refer to the emission/absorption spectra, rather than directly to the real part of the susceptibility, it seems, to me at least, to find the imaginary part well, but to fall when looking at the real part? To my limited understanding, at least. $\endgroup$ Mar 12, 2019 at 16:52
  • $\begingroup$ @DoublyNegative This would be to compute the parameters underlying the classical treatment. Treating the atom as a classical oscillator gives you the Lorentzian lineshape of a transition (if it’s only homogeneously broadened), and this gives you the real part too from Kramers-Kronig. But you still need to supply the photon energy, amplitude, and linewidth, which must be calculated with quantum mechanics as I described. $\endgroup$
    – Gilbert
    Mar 12, 2019 at 19:34
  • $\begingroup$ Sorry, I probably didn't describe what I was looking for too well in my initial question. I think I found what I was looking for in this paper (phys.ksu.edu/personal/wysin/notes/dielectricsA.pdf), but thank you very much for your time - and your post above was helpful! $\endgroup$ Mar 12, 2019 at 20:35

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