In quantum mechanics we learn that the average energy of a simple harmonic oscillator is given by ε = (n + 1/2) hω, with n = 0, 1, 2,... Classically, the energy is given by ε = (1/2m)p²x + (1/2)κx².

(a) What would be the classical value of the heat capacity for N of these oscillators? (Use equipartition.)

(b) If the heat capacity fell well below the classical value for temperatures below 190 K, what would you estimate the frequency ω to be?

(c) In real solids having Na atoms, we might expect 6Na degrees of freedom, each with an average energy (1/2)kT , giving a total thermal energy of 3NakT . But at temperatures below the Debye temperature, the internal energy is lower than this. Does this violate the equipartition theorem? Explain.