ex07 - Solid State Theory Exercise 7 FS 11 Prof M Sigrist...

Info icon This preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon
Solid State Theory Exercise 7 FS 11 Prof. M. Sigrist Exercise 7.1 Phonons in One Dimension In this exercise you will show that a chain of atoms that are harmonically coupled to each other (and may thus oscillate around their equilibrium positions) is equivalent to a collection of harmonic oscillators. When quantized canonically, these are turned into non- interacting bosons. More specifically, consider a chain of atoms with alternating masses, such that atoms at site i with i even have mass m and those at odd sites have mass M . The potential energy is given by V = v N/ 2 X i =1 ( u 2 i - u 2 i +1 ) 2 + ( u 2 i - u 2 i - 1 ) 2 (1) a) Diagonalize the equations of motion to find the eigenmodes of the classical system. To achieve this, introduce ˜ u i = ( u 2 i , u 2 i +1 ) T , (2) where i now labels unit cells instead of atoms (˜ u i, 1 u i, 2 ) corresponds to an atom belonging to the even (odd) sublattice). Next write ˜ u j,a ( t ) = r 2 N X k X μ C k ( q ( t ) e ikj + q * ( t ) e - ikj ) , (3) where a, μ ∈ { 1 , 2 } . The q ( t
Image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Image of page 2
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern