Consider a system of N independent one-dimensional harmonic oscillators, where the energy for a single particle is given by (20 marks) n=(n+1/2) (a) Obtain expressions for the Helmhotlz free energy, entropy and energy on a per molecule basis. (b) What is the low temperature (kT>) limit for the energy per particle. (Hint expand around the exponential function). (d) Find an expression for the heat capacity