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Unformatted text preview: Physics 212: Statistical mechanics II, Fall 2006 Midterm solutions : test given Thursday, 11/15/06 1. (a) To get the ritical exponent , we need to compute the response to a magnetic field when T- T c . The Landau free energy in a small magnetic field is F ( m ) = ahm + bm 6 (1) where the quartic term is absent because we are describing a tricritical point. Minimizing gives m 5 h , or = 5. (b) A good set of sketches should show the following information. For T > T c , the response is linear near h = 0. For T = T c , the response shows the power-law related to for small h . For T < T c , there is a jump in the magnetization at H = 0, and this jump is of magnitude going as ( T c- T ) . 2. (a) The value of h S x i in the appropriate density matrix is zero. (b) The value of h S z i is h 2 tanh( h z /k B T ) . (c) If the perturbation commutes with the original Hamiltonian, as in this case, then it induces no transitions, and expectation values are unchanged....
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