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I have a stupid question and sorry for that. In jellium model, why can we disregard the kinetic energy of the ions? The approximation is that we take the density to be constant, \begin{align} \langle\psi^\dagger(x)\psi(x)\rangle=\mathrm{constant} \end{align} Is there any reason to forbid the change the phase to get kinetic energy?

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    $\begingroup$ You can do that, and you will get phonons. $\endgroup$
    – Hossein
    Sep 14, 2022 at 9:35
  • $\begingroup$ In the jellium model you are interested in the dynamics of electrons, the positive ions are just a neutralizing background (external potential). Then, to simplify even more, they are treated as uniform positive charge distribution), no big reasoning behind this (just a working assumption to have a simple toy model). This is my understanding but I might be wrong. The fact that you have a uniform positive charge is the reason behind the name "jellium". $\endgroup$
    – Quillo
    Sep 14, 2022 at 11:47
  • $\begingroup$ @ Hossein, nice. If I can argue that the kinetic energy of photon is small than the electron, then it is ok. It make sense, just like the Cooper pair. $\endgroup$
    – thone
    Sep 14, 2022 at 13:13
  • $\begingroup$ @thone it's more than that: not only there is no phase variation, but also no density variation! In a crystal you forget about the lattice and you have phonons because you are doing perturbations around the ground state, where electrons are organized in Block states (and this allows you to get rid of the periodic potential of ions)... but here electrons are in the "jellium" since the very start (no Block states to start with), so it is just a toy model that is not directly related to the physics of a crystal. $\endgroup$
    – Quillo
    Sep 15, 2022 at 15:57

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