Statistical Mechanics - Physics at Oregon State University

40 CHAPTER 2. THE CANONICAL ENSEMBLE
A system has only two quantum states available. State number one has
energy zero and state number two has energy ɛ > 0.
(a) Calculate the partition function for this system.
(b) Calculate U(T) and S(T).
(c) Calculate the heat capacity as a function of temperature.
The heat capacity as a function of temperature peaks at a certain value of
the temperature. This peak is called a Schottky anomaly.
(d) Is this peak associated with a phase transition? Give an argument justifying
your answer.
Problem 4.
(From Reif) A sample of mineral oil is placed in an external magnetic field
H. Each proton has spin 1
2 and a magnetic moment µ; it can, therefore, have two
possible energies ɛ = ∓µH, corresponding to the two possible orientations of
its spin. An applied radio frequency field can induce transitions between these
two energy levels if its frequency ν satisfies the Bohr condition hν = 2µH. The
power absorbed from this radiation field is then proportional to the difference
in the number of nuclei in these two energy levels. Assume that the protons in
the mineral oil are in thermal equilibrium at a temperature T which is so high
that µH ≪ kT . How does the absorbed power depend on the temperature T of
the sample?
Problem 5.
(From Reif) A system consists of N weakly interacting particles, each of
which can be in either of two states with respective energies ɛ1 and ɛ2, where
ɛ1 < ɛ2.
(a) Without explicit calculation, make a qualitative plot of the mean energy
Ē of the system as a function of its temperature T. What is Ē in the limit of
very low and very high temperatures? Roughly near what temperature does Ē
change from its low to its high temperature limiting values?
(b) Using the result of (a), make a qualitative plot of the heat capacity CV
(at constant volume) as a function of the temperature T.
(c) Calculate explicitly the mean energy Ē(T ) and heat capacity CV (T )
of this system. Verify that your expressions exhibit the qualitative features
discussed in (a) and (b).
Problem 6.