Homework Set 1

Problem 4: Phonons in one-dimensional chain with two different atoms per unit cell

Consider an infinite one-dimensional chain described by classical Hamiltonian:

$H=\sum _j[\frac{p_j^2}{2m_j}+\frac{k}{2}(x_j-x_{j-1}{)}^2]$

whose even sites host atoms of mass $m_{2j}=m$ and odd sites host atoms of mass $m_{2j+1}=M$.

(a) Find classical normal mode frequencies and sketch their dispersion curves <math> \omega(k) <\math>. (b) What is the gap in the excitation spectrum? (c) Write the diagonalized Hamiltonian in second quantized form and discuss how you might arrive at your fi�nal answer. You will now need two types of creation operators.