S. Scandolo, ICTP

Lesson number 9(Electronic states)

This lesson begins with the study of an extended system were instead of having two atoms we have an infinite chain of atoms ordered periodically with the same distance *a. *The goal is to determine the solution of the wave function, the energy for the electronic state of this system. In order to do it is used the same approximation that was used in the last lesson for the covalent bond fallowing the same mathematical reasoning and nearest neighbors interaction. This time instead of a 2 by 2 quantum mechanics problem it is obtained an infinite by infinite quantum mechanics problem. The infinite matrix is composed by a diagonal term and an off-diagonal term just one position out the diagonal we have a hopping term above and below all the others terms are zero.

The solution to this problem is proposed using a guess exponential function for the coefficient *a _{J }= exp(iqJ) *where

*q*is a real number,

*i*is the imaginary number and

*J*is a label for the coefficient

*a.*This give a set of infinite many solutions of the form E = E

_{0 }+ΔE

_{0 }+ 2tCos[q] one for each value of

*q*, where E

_{0 }is the atomic energy of an isolated atom, ΔE

_{0 }is the “on-site” energy and t is the so called “hopping” term as defined in the previous lesson. The periodicity of this solution on q space was discussed on detail during the lesson and it was stated that all the solutions can be found between one period of the solution that is for between

*q*-π and π.