PHYS 427_HE02_B_OpenBook_Solutions

PHYS 427_HE02_B_OpenBook_Solutions - UIUC Physics 427...

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Unformatted text preview: UIUC Physics 427 Midterm Part B (Open Book) Solutions Philip Powell November 22, 2008 1. Problem 1 (a) We consider a gas of N spinless bosons, each of mass m and in a volume V at temperature T . We can determine the density of states (in 3 dimensions) by counting the number of states in an energy interval ( ε,ε + dε ) or alternitavely in the momentum interval ( k,k + dk ). Equating these two we have d 3 N s dk 3 (4 πk 2 dk ) = D ( ε ) dε (1) Where d 3 N s /dk 3 is the density of states in momentum space. Applying periodic boundary conditions gives d 3 N s dk 3 = L 2 π 3 = V 8 π 3 (2) Inserting this expression into (1) gives D ( ε ) = V 2 π 2 k 2 dk dε (3) Using the free-particle dispersion relation ε = ~ 2 k 2 / 2 m gives ε = ~ 2 k 2 2 m = ⇒ dε dk = ~ 2 k m = ⇒ dk dε = m ~ 2 k (4) Thus, (3) becomes D ( ε ) = V mk 2 π 2 ~ 2 (5) Substituting for k gives D ( ε ) = V 4 π 2 2 m ~ 2 3 / 2 √ ε (6) (b) Next, we seek to determine the mean occupation number of a single particle state h n ε i . The grand partition function for the system is given by Z = X ASN e- ( ε s- μN ) /τ = ∞ X n =0 e- N ( ε- μ ) /τ ( ε s = Nε ) = 1 1- e- ( ε- μ ) /τ (7) 1 The expectation value of the numeber of particles in a particular mode with energy ε is therefore h n ε i = τ ∂ ∂μ ln Z = τ ∂ ∂μ ln 1- e- ( ε- μ ) /τ = τ...
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This note was uploaded on 12/06/2009 for the course PHYS 427 taught by Professor Flynn during the Fall '08 term at University of Illinois, Urbana Champaign.

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PHYS 427_HE02_B_OpenBook_Solutions - UIUC Physics 427...

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