In condensed matter physics, the independent electron approximation is a simplification used in complex systems, consisting of many electrons, that approximates the electron–electron interaction in crystals as null.
It is a requirement for both the free electron model and the nearly-free electron model, where it is used alongside Bloch's theorem.
[1] While this simplification holds for many systems, electron–electron interactions may be very important for certain properties in materials.
For example, the theory covering much of superconductivity is BCS theory, in which the attraction of pairs of electrons to each other, termed "Cooper pairs", is the mechanism behind superconductivity.
[citation needed] For an example of the Independent electron approximation's usefulness in quantum mechanics, consider an N-atom crystal with one free electron per atom (each with atomic number Z).
Neglecting spin, the Hamiltonian of the system takes the form:[1] where
is the reduced Planck constant, e is the elementary charge, me is the electron rest mass, and
The electron–electron interaction term, however, prevents this decomposition by ensuring that the Hamiltonian for each electron will include terms for the position of every other electron in the system.
is any reciprocal lattice vector (see Bloch's theorem).