Problem :
Problems 1 through 5 will use the following system. Suppose that we
have a two state system, in which the first state has energy
and the second, energy 3. Give the ratio
of the probability of occupancy of the first to the probability of
occupancy of the second, and simplify.
We can take the ratio of the Boltzmann factors to get the ratio of the
probabilities:
=
=
e^{2/τ}
Problem :
Calculate the partition function of the system.
Z =
e^{-/τ} =
e^{-/τ} +
e^{-3/τ}
Problem :
Calculate the absolute probability of finding the system in the state
with energy .
P(
) =
Problem :
What happens to the occupation of the state with energy
as τ→ 0 and as τ→∞?
As τ→ 0, the term of Z that is e^{-3/τ} becomes
insignificant in comparison to the term e^{-/τ}.
Therefore the absolute probability simplifies to:
P(
) =
= 1
As τ→∞, all terms go to 1, and therefore we find that:
These results make sense. If the temperature is very low in comparison
to , often stated τ, there will be
little thermal excitation that can promote the system from the first
state to the second. In that case, we can be almost certain to find the
system in the state of lower energy. If the temperature is very high,
or τ, then the gap between the states becomes
insignificant, and the system becomes about equally likely to be in
either state.
This kind of analysis, looking at the limits of your answers, is an
excellent way to check if you are on the right track. If your answers
don't make sense at the limits then you have probably made a mistake
somewhere.
Problem :
Calculate the free energy of the system.
F = - τ log Z = - τ log (e^{-/τ} + e^{-3/τ})