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

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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

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