Consider N = Zi=o Ni >> 1 ideal bosons with mass m are confined within a volume of V and ni is the population in the i-th eigenstate. Given a grand canonical ensemble with chemical potential u and temperature T, the probability to find a = 0,1,2 ... bosons in the state is pa o e-B(aEi-an) where B = KBT BEC occurs when a sizable fraction of the particles occupies the ground state No = O (N). In this case the excited state population is given by Nex = Zi=1 Mi. Show that the BEC transition occurs when u increases toward E. Comment: Here No = O(N) means limN-yoo N No > 0.) Now consider N > 1 ideal bosons in a 2D box, show that BEC occurs only at T=0. Now consider N bosons in a 3D isotropic harmonic potential H p2 ma 272 + with 2m 2 eigenenergy E = (nx + ny + nz)hw, nx, ny, nz = 0, 1, 2 ... Determine the BEC critical temperature T and the population in the ground state and first 3 excited states at T = Tc