Problem 4 A simple model for a quantum one-dimensional solid consists of M indepen- dent oscillators, each with energies En=hw|n+ L " 2 where w is the angular frequency. a) Write the partition function for a single quantum oscillator. (Hint: Y > jy = ) ( ( ) b) Write the partition function for a system of M oscillators. (c) Compute the internal energy for this system. ) (d) Compute the heat capacity for this system