p847ps7 - Physics 847: Problem Set 7 Due Thursday, May 27,...

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Unformatted text preview: Physics 847: Problem Set 7 Due Thursday, May 27, 2010 at 11:59 P. M. Each problem is worth 10 points unless specified otherwise. 1. Consider the spin-1/2 Heisenberg model in an applied magnetic field B = B ˆ z . The Hamiltonian may be written H = − J ¯ h 2 summationdisplay ( ij ) bracketleftbigg 1 2 ( S i + S j − + S i − S j + ) + S iz S jz bracketrightbigg − geB 2 mc summationdisplay i S iz . (1) Here S = ( S ix ,S iy ,S iz ) is a spin-1/2 operator as defined in class, S i + = S ix + iS iy , S i − = S ix − iS iy , and ge/ (2 mc ) is a positive constant. (a). What is the ground state energy and spin configuration? If B negationslash = 0, is this state degenerate? (b). Calculate the spin wave spectrum for this Hamiltonian, following the approach used in class. Show, in particular, that the spectrum has a gap , i. e., the lowest spin wave excitation has an energy of Δ negationslash = 0, and find Δ. What is the temperature dependence of the spin wave specific heat in this case at low temperatures (...
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This note was uploaded on 07/15/2011 for the course PHYSICS 847 taught by Professor Stroud during the Spring '10 term at Ohio State.

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p847ps7 - Physics 847: Problem Set 7 Due Thursday, May 27,...

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