Problem: Tight-Binding Model

 

Electrons propagating in a lattice are often described in terms of electron creation and annihilation operators. Consider the one-dimensional Hamiltonian

where site i is at the position (a is the lattice spacing), and the operator creates a filled electronic state of spin s in site i.

(a) Solve the Schrödinger equation with a one-electron wavefunction,

(Hint: A single, delocalized electron ``wave" is described by the wavefunction , where N is the number of sites.)

(b) Show that for two electrons the electron energies add up. The trial wavefunction is

(c) Discuss what happens if k=k' and s=s'.

  (d) Sometimes it is convenient to define ``two-electron excitations" by considering two electrons with the total fixed. Plot the possible energies of the two-electron excitations as a function of .