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 $\omega (k)$.

(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 final answer. You will now need two types of creation operators.