Homework Set 1: Difference between revisions
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Created page with "== Problem 1: Three-dimensional isotropic harmonic oscillator == == Problem 2: Dynamics of the Bloch vector == == Problem 3: Coordinate representation of coherent state..." |
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whose even sites host atoms of mass <math> m_{2j} = m </math> and odd sites host atoms of mass <math> m_{2j+1} = M </math>. | whose even sites host atoms of mass <math> m_{2j} = m </math> and odd sites host atoms of mass <math> m_{2j+1} = M </math>. | ||
(a) Find classical normal mode frequencies and sketch their dispersion curves <math> \omega(k) < | (a) Find classical normal mode frequencies and sketch their dispersion curves <math> \omega(k) </math>. | ||
(b) What is the gap in the excitation spectrum? | (b) What is the gap in the excitation spectrum? | ||
(c) Write the diagonalized Hamiltonian in second quantized form and discuss how you might arrive | (c) Write the diagonalized Hamiltonian in second quantized form and discuss how you might arrive | ||
at your | at your final answer. You will now need two types of creation operators. | ||
Revision as of 21:49, 9 September 2019
Problem 1: Three-dimensional isotropic harmonic oscillator
Problem 2: Dynamics of the Bloch vector
Problem 3: Coordinate representation of coherent state
Problem 4: Phonons in one-dimensional chain with two different atoms per unit cell
Consider an infinite one-dimensional chain described by classical Hamiltonian:
whose even sites host atoms of mass and odd sites host atoms of mass .
(a) Find classical normal mode frequencies and sketch their dispersion curves . (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.
