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I am studying surface plasmons on a nanosphere using classical electrodynamics but I was wondering if surface plasmons were or not eigenstates of the nanoparticle Hamiltonian. I think they are not because you really have to add some boundary conditions on the external field to compute them. Is this true? Can you provide me a detailed explanation?

Thank you in advance EDIT What I meant is that the Hamiltonian: \begin{equation} H=\sum_{i}\frac{p^{2}_{i}}{m_{i}}+\frac{1}{2}\sum_{i,j}\frac{1}{\left|r_{ij}\right|}-\sum_{\gamma,i}\frac{1}{R_{\gamma i}} \end{equation} where $\gamma$ runs over the nuclei and $i,j$ over the electrons won't be enough because:

There is no reference to the surface and the surface plasmon frequency depends on the external zone;

There is no reference to the fields produced by the moving charges (retardation effects); Plus I'm not really sure how one would ensure the boundary conditions on the surface in the solution of quantum mechanical equations

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  • $\begingroup$ Since there is a surface, you need constraints on the surface. Those lead to surface plasmons, but they are perfectly valid solutions to the interaction of EM waves with the material. $\endgroup$
    – Jon Custer
    Mar 9, 2020 at 12:38
  • $\begingroup$ Yeah, I understand that, but that suggests to me that in a quantum mechanical treatment of the Surface plasmons I need to describe the fields too, and I can not expect surface plasmons to pop out just from the Coulomb interactions between the electrons as, for example, is the case with bulk plasmons. I still need to plug the external field in and then I would get an eigenstate that is a surface plasmon. Right? $\endgroup$
    – Yepman
    Mar 9, 2020 at 14:55
  • $\begingroup$ But the plasmons come about by interactions of the electrons with the EM field. This really is no different than all the various surface elastic waves - the presence of the surface allows for modes that are confined to the region near the surface. Whether you couple into a mode or not depends on the incident wave properties. $\endgroup$
    – Jon Custer
    Mar 9, 2020 at 15:01
  • $\begingroup$ Years ago (nearly 40) I did experiments on coupling to surface plasmon modes on metallic diffraction gratings. I also did calculations of the coupling - the surface plasmon modes fall out naturally in the math. $\endgroup$
    – Jon Custer
    Mar 9, 2020 at 15:02
  • $\begingroup$ Ok, so what you are saying is that the surface plasmons are proper excited states coming from the diagonalization of the Hamiltonian of an electron gas confined in a region. Then, the excited state I get into just depends on which excitation I use exactly as well as in atomic spectroscopy. $\endgroup$
    – Yepman
    Mar 9, 2020 at 15:28

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