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Consider a three-dimensional gas of N indistinguishable non-interacting spin zero bosons of mass m in an external isotropic harmonic potential V(r) = (1/2)mWo² 1r12, where r = (x, y, z). This might be taken as a model for bosons in a magnetic trap. The quantized single particle energy levels are given by £(nx, ny, nz) = hwo (nx + ny + nz + 3/2), where nx, ny, nz = 0, 1, 2, are integers. a) Compute the density of states g(£). The density of states is such that g(€) de is the number of single particle states between w and &+d€. Hint: You should try to do this the same way you found g(€) in HW #7 problem 2, but generalizing to 3D. You may assume that the thermal energy is much greater than the spacing between the energy levels, i.e. kBT >> hwo. b) What is the largest value that the fugacity Z can take? c) Show that this system has Bose-Einstein condensation at sufficiently low temperature. d) Find the Bose-Einstein condensation temperature Tc as a function of the number of paricles N. e) Find how the number of particles No(T) in the condensed state varies with temperature for T=0 to Tc.

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