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hw05soln

# hw05soln - 1 Homework 5(due November 15 Problem 5-1(3 pts...

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1 Homework 5 (due November 15) Problem 5-1 (3 pts.) Let the dispersion relationship for an ideal Bose gas particles be given by the following scaling law: ε ( p ) p α What should be the dimensionality of the physical space d for this system to be capable of Bose condensation? Solution: Bose condensation is possible if µ = 0 can be reached at a fi nite density, i.e. when the following integral converges: ρ c = 4 π (2 π ~ ) 3 Z 0 exp μ ± ( p ) T 1 ¸ 1 p 2 dp Divergency is only possible at small p. in this limit, [exp ( ± ( p ) /T ) 1] p α . Hence, the following integral should converge: P Z 0 p d α 1 dp We conclude that d > α Problem 5-2 (4 pts.) Find the energy density U of a photonic gas in d dimensions, as a function of temperature T (assume all the fundamental constants to be the same as in our world). By using the standard thermodynamic relationships, fi nd the corresponding entropy density s , free energy density f , and radiation pressure P .

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hw05soln - 1 Homework 5(due November 15 Problem 5-1(3 pts...

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