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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 ( , + d ) or alternitavely in the momentum interval ( k,k + dk ). Equating these two we have d 3 N s dk 3 (4 k 2 dk ) = D ( ) d (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 3 = V 8 3 (2) Inserting this expression into (1) gives D ( ) = V 2 2 k 2 dk d (3) Using the free-particle dispersion relation = ~ 2 k 2 / 2 m gives = ~ 2 k 2 2 m = d dk = ~ 2 k m = dk d = m ~ 2 k (4) Thus, (3) becomes D ( ) = V mk 2 2 ~ 2 (5) Substituting for k gives D ( ) = V 4 2 2 m ~ 2 3 / 2 (6) (b) Next, we seek to determine the mean occupation number of a single particle state h n i . The grand partition function for the system is given by Z = X ASN e- ( s- N ) / = X n =0 e- N ( - ) / ( s = N ) = 1 1- e- ( - ) / (7) 1 The expectation value of the numeber of particles in a particular mode with energy is therefore h n i = ln Z = ln 1- e- ( - ) / =...
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