852_2010hw2_Solutions - PHYS852 Quantum Mechanics II Spring...

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PHYS852 Quantum Mechanics II, Spring 2010 HOMEWORK ASSIGNMENT 2: SOLUTIONS Topics covered: Entropy, thermal states 1. [20] 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) [10] 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) [10] 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 . The probability to find the thermalized particle at position r is: P 1 ( r ) ρ 1 ( r , r ) ( r | e P 2 2 Mn b T | r ) integraldisplay d 3 p ( r | e P 2 2 Mk b T | p )( p | r ) integraldisplay dp x dp y dp z e p 2 x + p 2 y + p 2 z 2 Mk b T e i p · r / planckover2pi1 e
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