solid consider a three-dimensional simple harmonic os- cillator with mass m and spring constant k (i. e., the mass is attracted to the origin with the same spring constant in all three directions). the hamiltonian is given in the usual way by 2m 2 d calculate the classical partition function note: in this exercise p and x are three-dimensional vectors. d using the partition function, calculate the heat capacity 3kb d conclude that if you can consider a solid to con- sist of n atoms all in harmonic wells, then the heat capacity should be 3nkb 3r, in agreement with the law of dulong and petit. (b) quantum einstein solid now consider the same hamiltonian quantum calculate the quantum partition function where the sum over j is a sum over all eigenstates. d explain the relationship with bose statistics. d find an expression for the heat capacity d show that the high-temperature limit agrees with the law of dulong and petit d sketch the heat capacity as a function of tem- perature (see also exercise 2.7 for more on the same topic)