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{"ops":[{"insert":"(a) Classical Einstein Solid (or \u201cBoltzmann\u201d Solid): Consider a single harmonic oscillator in three dimensions with Hamiltonian H = p 2 2m + k 2 x 2 \u22b2 Calculate the classical partition function Z = Z dp (2\u03c0~) 3 Z dx e \u2212\u03b2H(p,x) Note: in this problem p and x are three dimensional vectors (they should appear bold to indicate this unless your printer is defective). \u22b2 Using the partition function, calculate the heat capacity 3kB. \u22b2 Conclude that if you can consider a solid to consist of N atoms all in harmonic wells, then the heat capacity should be 3N kB = 3R, in agreement with the law of Dulong and Peti\n"}]}
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