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,  R_n and P_n 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.