An oscillator with frequency f = 8×1011Hz (about that of a diatomic molecule) is in equilibrium with a thermal reservoir at temperature T. The spacing between the energy levels of the oscillator is given by ϵ=hf, where h=6.626×10−34 Js.
a) For what temperature does P1/P0=1/2, where P1 is the probability that the oscillator has E=ϵ (the first excited state) and P0 is the probability that it has E=0 (the lowest energy state)?
b) If T is 10% of the value you calculated in part a), what is the ratio P1/P0?
c) At the temperature of part b), what is the ratio P2/P1, where P2 is the probability that the oscillator is in the second excited state (E=2ϵ)?
d) Now suppose you have a large number of these oscillators (such as in a solid). At the temperature of parts b) and c), what is the average thermal energy per oscillator divided by kT? That is, calculate /kT. (Equipartition would have this ratio be 1, but that's not the answer here.)