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p212midterm06 - Physics 212 Statistical mechanics II Fall...

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Physics 212: Statistical mechanics II, Fall 2006 Midterm : ends 11 a.m., Thursday, 11/15/06 Directions : The allotted time is 80 minutes. The 5 problems count equally. No books or notes are allowed, and please ask for help only if a question’s meaning is unclear. 1. Mean-field theories: (a) (10 points) Use Landau theory to obtain the mean-field critical exponent δ for a tricritical point: recall that m H 1 . (b) (10 points) An ordinary critical point in mean-field theory has δ = 3 and β = 1 / 2. Draw three plots of m versus H at constant temperature, one for T slightly above T c , one for T = T c , and one for T slightly below T c . Indicate how any interesting behaviors related to δ and β appear in your plots. 2. Suppose that for t 0, a single s = 1 / 2 spin is in the thermal equilibrium state at temperature T of the Hamiltonian H 0 = - h z σ z , σ z = 1 0 0 - 1 . (1) (a) (3 points) What is the value at t = 0 of S x = ¯ h 2 σ x ? Recall that σ x = 0 1 1 0 , σ y = 0 - i i 0 (2) (b) (3 points) What is the value at t = 0 of S z ?
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