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# set7 - Department of Physics The Ohio State University Prof...

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Department of Physics Prof. R. Bundschuh The Ohio State University Seventh Problem Set for Physics 847 (Statistical Physics II) Winter quarter 2004 Important date: Mar 16 9:30am-11:18am final exam Due date: Tuesday, Feb 24 18. Spin 1 atom 12 points An atom with spin 1 has a Hamiltonian ˆ H = A ˆ S 2 z + B ( ˆ S 2 x - ˆ S 2 y ), where ˆ S x , ˆ S y , and ˆ S z are the x , y , and z component of the spin angular momentum operator. In the basis of eigenstates of the operator ˆ S z these three operators have the matrix representations ˆ S z = ¯ h 1 0 0 0 0 0 0 0 - 1 , ˆ S x = ¯ h 2 0 1 0 1 0 1 0 1 0 , and ˆ S y = ¯ h i 2 0 1 0 - 1 0 1 0 - 1 0 . At time t = 0 the atom is initially in an eigenstate of ˆ S z with eigenvalue +¯ h . a) Write the density matrix (in the basis of eigenstates of ˆ S z ) at t = 0. b) Compute the density matrix at time t in the basis of eigenstates of ˆ S z . c) Compute the average z component of the spin at time t .

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