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1. (1/2)kT is

*defined*as the average kinetic energy of the molecules of a substance at temperature T, right?

2. You can derive the Boltzmann distribution/Boltzmann factors using (1/2)kT as the kinetic energy, making an argument about kinetic energy being converted into potential energy, and solving a differential equation. Feynman does this.

3. Then, we arrive at the ultraviolet catastrophe. We get a power spectrum that goes as the square of the frequency, times kT. If we then say that kT is the classical average energy of the SHO, and replace it with the correct average energy (a sum of hw*(Boltzmann Factors)), we get the right result. However, we found those Boltzmann factors using the wrong (classical) harmonic oscillator energy. What gives?

Thanks for any help! This question has been driving me crazy for the past couple days.