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Unformatted text preview: PHYS852 Quantum Mechanics II, Spring 2010 HOMEWORK ASSIGNMENT 2: SOLUTIONS Topics covered: Entropy, thermal states 1.  Thermalized Free Particle : In a gas of N particles, the state of particle 1 can be described by a reduced density matrix, defined in coordinate representation by ρ 1 ( r 1 , r ′ 1 ) = integraldisplay d 3 r 2 ...d 3 r N ( r 1 , r 2 ,... , r N | ρ | r ′ 1 , r 2 ,... , r ′ N ) , (1) where ρ is the full N-particle density operator. The full Hamiltonian separates as H = H 1 + H 2 + ... + H N + V 1 , 2 + V 1 , 3 + ... + V N − 1 ,N (2) where H n = P 2 n 2 M n and V n,n ′ = V ( r n − r ′ n ) are the kinetic and short-range interaction terms, respec- tively. We can assume that the interactions with the N − 1 other particles will thermalize the state of particle 1, so that ρ 1 ( r 1 , r ′ 1 ) = 1 Z ( r 1 | e − βH 1 | r ′ 1 ) , (3) (a)  In a given basis, the diagonal elements of ρ give the probabilities for the system to be in the corresponding basis states. Show that the thermalized particle is equally likely to be at any position. (b)  The off-diagonal elements of ρ measure the ‘coherence’ between the corresponding basis states. Show that there is a characteristic coherence length scale, λ c , such that the coherence between position states becomes negligible only for | r − r ′ | ≫ λ c . Give the dependence of λ c on the temperature T ....
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