Hand written solution required

Let us condider N independent quantum harmonic oscillators subject to a Halmiltonian IN 1=1 where n; = 0, 1, 2, ... is the quantum occupation number for the it oscillator. 1. Calculate the partition function, the Free energy, the mean energy, the entropy and the heat capacity as a function of temperature T and the total number of particle N. 2. Comment on the difference between heat capacities for classical and quantum oscilla- tors. 3. Compute the energie fluctuation (AE) = (E?) + (E) and the relative error AE E