Introduction: Two-level systems, that is systems with essentially only two energy levels are
important kind of systems, as at low enough temperatures, only the two lowest energy levels
will be involved. Especially important are solids where each atom has two levels with different
energies depending on whether the electron of the atom has spin up or down.
Consider a set of N gas atoms each with two energy levels. The atoms in a gas are
indistinguishable, as they are not located in fixed places and identical. The energy of these two
levels are $\varepsilon_1$ and $\varepsilon_2 = \varepsilon_1 + \varepsilon$ where $\varepsilon > 0$. The two levels have the same statistical weight g. The
system is in equilibrium with a heat reservoir at a temperature T. Let $N_1$ and $N_2$ represent the
numbers of molecules having energies $\varepsilon_1$ and $\varepsilon_2$, respectively.
1. Write down the partition function for an atom.
2. Find the probabilities $p_1$ and $p_2$ for an atom to have energy $\varepsilon_1$ and $\varepsilon_2$, respectively.
3. Determine $N_1 - N_2$.
4. We designate $\theta = \varepsilon/k_B$ the so-called scale temperature which is an important physical
quantity for the system. Find the limits for $N_1$, $N_2$, and $N_1 - N_2$ in the two cases $T << \theta$
and $T >> \theta$. Interpret the obtained results.
5. Give the expression of the internal energy U of the system.
6. Assuming that $\varepsilon_1$ and $\varepsilon_2$ do not depend on T at constant volume, calculate the molar
specific heat $c_v$.
