set8 - Department of Physics The Ohio State University...

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Department of Physics Prof. R. Bundschuh The Ohio State University Eighth Problem Set for Physics 847 (Statistical Physics II) Winter quarter 2004 Important date: Mar 16 9:30am-11:18am Fnal exam Due date: Tuesday, Mar 2 21. Density operator of a free particle 10 points A free particle is described by the Hamiltonian ˆ H = ˆ ~ p 2 / 2 m . We assume that the particle is in a cubic box of volume V = L 3 . a) Write the canonical density operator in the momentum base | ~ k i , i.e., calculate h ~ k 0 | ˆ ρ | ~ k i . When calculating the partition function you may replace the sum by an integral. b) Write the canonical density operator in the coordinate base | ~ r i , i.e., calculate h ~ r 0 | ˆ ρ | ~ r i . You may again replace sums by integrals. (Hint: use h ~ k | ~ r i = V - 1 / 2 exp( - i ~ k~ r ).) 22. Single energy level 10 points We consider one energy level of a large quantum system as the subsystem which we want to describe while we summarize all the other levels of the system as the particle bath. The
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set8 - Department of Physics The Ohio State University...

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