This Demonstration shows the basics of the fermion Hubbard model applied to one-dimensional systems such as chains with sites with either open or closed boundary conditions and any phase at the ends. The Hubbard model provides a very simple but physically powerful description of electronic many-body effects in quantum mechanical systems. This can be understood by considering the form of its Hamiltonian, expressed in second quantization formalism: , where and are site indices, are all pairs of first nearest neighbors sites, gives the number of electrons on site with spin , and are electron creation and annihilation operators, respectively, and and are positive interaction constants, respectively. The term is a single-particle (tight-binding) term (thus it contains no many-body features) and describes the hopping of electrons localized on atomic-like orbitals between nearest neighbor sites and models the kinetic energy of the system. The term gives a potential energy contribution to the Hamiltonian and models the Coulomb repulsion between two electrons with opposite spin in the same orbital, hence it is the effective many-body (two-body) term. Another related operator is the velocity operator , where is the lattice parameter of the chain and is the Planck constant (since these are normalization constants, they have been set equal to 1 in the program). The term represents the current operator, which is relevant for investigating optical conductivity properties.