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# Jellium

Jellium is the model of interacting electrons in which a uniform background of positive charge exists. In this model at zero temperature the system properties are dependent only on the charge density of electrons. This allows for the simplistic calculation of the electron-electron coupling energy being a ratio between the free-electron kinetic energy and the Coulomb potential energy.

The model works with the atoms as if they were actually blobs of jelly, hence the name. The Hamiltonian of the system consists of three parts:
H = Hel + Hback + Helback
where Hel signifies the Hamiltonian of the electrons, Hback the Hamiltonian of the positive background (jellium) and Helback the Hamiltonian of the interaction of the electrons and the background. Hback and Helback are constant and $H_{el}=\sum_{i=1}^N\frac{p_i^2}{2m}+\frac{1}{2}\sum_{i\neq j=1}^N\frac{e^2}{|r_i-r_j|}$, with:
N : The number of the electrons in the jellium.
pi : The momentum of the i-th electron.
m : The mass of the electron.
e : The charge of the electron.

With this Hamiltonian and the principles of quantum-mechanics, the energy of the system can be calculated, for example in the approximation of Hartree-Fock or in the theory of Bohm and Pines.

The jellium model is also used in nuclear physics, and has been used to try to explain the properties of superatoms.