A central topic in condensed matter physics is the study of electrons in a periodic potential created by ions in a lattice. The Hamiltonian for a system of ions and electrons can generally be written as –
Here, Rn and Pn are the positions and momenta of the ions, and are the positions and momenta of the electrons, is the ion-ion potential, is the electron-ion potential. Note that the electronic and lattice motions are coupled in this Hamiltonian. In general, this is a complex many-body problem even with the simplification afforded by the spatial periodicity of the system.