S. Scandolo, ICTP

Lesson number 10( Electronic states, Bloch theorem)

This lesson continues with the solution of the electronic problem of a chain of atoms, recalling the results obtained in the previous lesson. In a chain of atoms ordered periodically with the same distance *a *the position of atom J is R_{J }= *a*J, thus the solution for the coefficient can be written as *a _{J }= *exp

*(ikR*where the

_{J})**wave vector**–π/

*a*<

*k*= q/

*a*< π/

*a*is introcuced. In order to face the problem of placing in the band an infinity number of electron the system is change from an infinity system to a finite system of length L =

*aN*where

*N*is the number of atoms and then periodic boundary condition are imposed, to recover the initial system the thermodynamic limit (L goes to infinity) is approached, this mathematical trick is called

**Born–von Karman boundary condition**. Periodic boundary condition implies that the wave vector can takes only discrete values k = 2 π

*m*/L where

*m*is an integer number. As – π/

*a*<

*k*< π/

*a*it will be a total number of

*N*states within this range. If the atoms are hydrogen atoms then the electronic band will be half filled because we can put two electrons for each state with spin up and down. Then it was shown that the wave function at any point of the chain can be written as a periodic function times a phase. Finally the

**Bloch theorem**for the wave function of a particle placed in a periodic potential was introduced and some symmetries properties of the Hamiltonian and its wave functions were discussed.