Consider a three-dimensional simple harmonic oscillator 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

H(p, x) = p2 /2m + (1/2) k x2

(a) Calculate the classical partition function

Z = (2 )-3 d3p d3x exp[H(p,x)]

Note: in this exercise p and x are three-dimensional vectors.

(b) Using the partition function, calculate the heat capacity

(c) Conclude that if you can consider a solid to consist of N atoms all in harmonic wells, then the molar heat capacity should be 3NA kB = 3R, in agreement with the law of Dulong and Petit.