# Are surface plasmons real eigenstates of the metal Hamiltonian?

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: $$$$H=\sum_{i}\frac{p^{2}_{i}}{m_{i}}+\frac{1}{2}\sum_{i,j}\frac{1}{\left|r_{ij}\right|}-\sum_{\gamma,i}\frac{1}{R_{\gamma i}}$$$$ where $$\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

• 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. Mar 9, 2020 at 12:38
• 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? Mar 9, 2020 at 14:55
• 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. Mar 9, 2020 at 15:01
• 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. Mar 9, 2020 at 15:02
• 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. Mar 9, 2020 at 15:28