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Unformatted text preview: H = h ( a a + 1 / 2). c)  n i denotes the normalized eigenstate with energy E n = h ( n + 1 / 2). Show that a  n i = n + 1  n + 1 i and a  n i = n  n1 i . Problem 3: Lowtemperature specic heat in d dimensions and for nonlinear dispersion laws (AshcroftMermin problem 23.2, 15 points) Consider small lattice vibrations in a ddimensional crystal in harmonic approximation. a) For the Debye model, i.e. a linear dispersion = c  k  of all phonon modes, calculate the phonon density of states and show that it varies as d1 . What is the Debye frequency? b) Determine the phonon contribution to lowtemperature specic heat. c) Investigate what would happen for a nonlinear phonon dispersion  k  (anomalous sound). Show that the lowtemperature specic heat would vanish as T d/ in d dimensions....
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 Spring '11
 ThomasVojta
 Physics, Mass, Work

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