A simple harmonic oscillator with frequency Hz is allowed to come to thermal equilibrium with a heat bath at temperature T. The spacing between energy levels for a SHO is hf, where h is Planck's constant.

a) For what T is the probability that the S.H.O. is in the first excited state P₁ just the probability that it is in the ground (lowest energy) state, P₀?
T = K

b) If T is only the value in part a), what is the ratio P₁/P₀?
P₁/P₀ =

c) At the T of part (b), what is the ratio of the probability of the oscillator being in the second excited state to the probability of it being in the first excited state?
P₂/P₁=

d) Now suppose you have an Einstein solid made up of a very large number of these oscillators. At the T of parts (b) and (c), what is the average thermal energy per oscillator?
U per SHO = J

e) If the 3-D solid has atoms, what is its total heat capacity for T >> hf/k?
C = J/K

Comments:

• This problem shows you how the chance of being in even the first excited state vanishes very rapidly when kT is less than the energy spacing of the states. The probability of being in higher excited states falls even more rapidly. The thermal energy drops far below the equipartition value at these low T's.