homework8

# homework8 - H = ¯ hω a † a 1 2 c | n i denotes the...

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Physics 481: Condensed Matter Physics - Homework 8 due date: March 18, 2011 Problem 1: Damped oscillations (10 points) In a linear chain of lattice spacing a, particles of mass m are connected by nearest-neighbor springs of spring constant K . In addition to the elastic forces, each particle is subjected to a damping force F D = - Γ ˙ u n , where u n is the displacement of the nth particle from the equilibrium position. How does the damping change the frequencies ω ( k ), and what is the relaxation time of the modes? Assume (Γ /m ) 2 ± K/m and discuss the limiting cases k 0 and k π/a . Problem 2: Creation and destruction operators (15 points) Consider a quantum-mechanical harmonic oscillator with Hamiltonian H = p 2 2 M + 1 2 2 x 2 where M is the mass, ω is the frequency and p and x and the momentum and position operators fulﬁlling the commutator [ p,x ] = ¯ h/i . The destruction and creation operators are deﬁned by a = s h x + i 2 ¯ h p a = s h x - i 2 ¯ h p a) Calculate the commutator [ a,a ]. b) Show that the Hamiltonian can be written as
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Unformatted text preview: H = ¯ hω ( a † a + 1 / 2). c) | n i denotes the normalized eigenstate with energy E n = ¯ hω ( n + 1 / 2). Show that a † | n i = √ n + 1 | n + 1 i and a | n i = √ n | n-1 i . Problem 3: Low-temperature speciﬁc heat in d dimensions and for nonlinear dispersion laws (Ashcroft-Mermin problem 23.2, 15 points) Consider small lattice vibrations in a d-dimensional crystal in harmonic approximation. a) For the Debye model, i.e. a linear dispersion ω = c | k | of all phonon modes, calculate the phonon density of states and show that it varies as ω d-1 . What is the Debye frequency? b) Determine the phonon contribution to low-temperature speciﬁc heat. c) Investigate what would happen for a nonlinear phonon dispersion ω ∼ | k | ν (anomalous sound). Show that the low-temperature speciﬁc heat would vanish as T d/ν in d dimensions....
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