set6 - M and chemical potential μ d The surface is exposed...

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Department of Physics Prof. R. Bundschuh The Ohio State University Sixth 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 17 16. Adsorption to a surface 10 points Consider a solid surface to be a two-dimensional lattice with M sites. Each site can be either empty or occupied with a single adsorbed atom. An adsorbed atom has a binding energy - ε and we neglect any interactions between the atoms. a) Calculate the grand canonical partition function of the adsorbed atoms as a function of temperature T , lattice size M , and chemical potential μ 0 . Use variables n i ∈ { 0 , 1 } for each i = 1 , . . . , M to describe if site i is empty or occupied. b) Calculate the grand canonical potential of the adsorbed atoms as a function of temper- ature T , lattice size M , and chemical potential μ 0 . c) Calculate the average number of adsorbed atoms N as a function of temperature T , lattice size
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Unformatted text preview: M , and chemical potential μ . d) The surface is exposed to an ideal gas of the atoms at some pressure P and the same temperature T as the surface. Calculate the fraction N/M of adsorbed atoms as a function of the pressure P of the ideal gas and the temperature T of the system. (Hint: in thermodynamic equilibrium the chemical potentials of the adsorbed atoms and the atoms in the ideal gas have to be equal.) 17. Density operator 10 points a) In a two-dimensional Hilbert space an operator ˆ ρ is given by the matrix ˆ ρ = 1 2 ˆ 1 + a 1 a 2 a 3 1-a 1 ! . Determine for which values of the three complex parameters a 1 , a 2 , and a 3 this operator is a density operator. ±or which values of the three parameters is it a pure state? b) Prove that for a hermitian Hamiltonian ˆ H the operator ˆ ρ ≡ e-β ˆ H / tr e-β ˆ H is a density operator. 1...
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