In our analysis of hydrogen in Chap. 9, we discussed the period of recombination, the time in the history of the universe at which protons and electrons became bound and formed neutral hydrogen. We were able to provide an estimate for the temperature at which recombination occurred according to the ground-state energy of hydrogen. However, this estimate was very coarse because we neglected thermodynamic effects. We now have a description of such effects through the partition function, so we can provide a better estimate here.

(a) Estimate the temperature at which \(50 \%\) of hydrogen in the universe is in its ground state. To do this, you need to construct the partition function for hydrogen, which consists of a sum over all energy eigenstates. We can estimate the partition function here by simply considering just the first three energy eigenvalues of bound hydrogen. With this assumption, what is this recombination temperature?

 Don't forget about degeneracy.

20 J. Kiefer, "Optimum experimental designs," J Roy. Stat. Soc. B 21(2), 272-304 (1959); E. H. Lieb and M. B. Ruskai, "Proof of the strong subadditivity of quantum-mechanical entropy," J. Math. Phys. 14, 1938-1941 (1973).

(b) The partition function that you would naïvely write down for hydrogen has a number of problems, not the least of which is that it is infinite. This happens because the higher bound states of hydrogen have energies that are closer and closer to 0 , and so have Boltzmann factor 1 , and there are an infinite number of them. Further, if the kinetic energy of the electron is sufficiently large, then it is not bound into hydrogen, and such scattering states also need to be accounted for in the partition function.
A much more complete partition function for hydrogen addressing these shortcomings is the Planck-Larkin partition function. \({ }^{21}\) It is defined to be
\[\begin{equation*}Z_{\mathrm{P}-\mathrm{L}}=\sum_{n=1}^{\infty} n^{2}\left(e^{-\beta \frac{E_{0}}{n^{2}}}-1+\beta \frac{E_{0}}{n^{2}}\right) \tag{12.151}\end{equation*}\]
where \(E_{0}\) is the ground-state energy of hydrogen, \(n\) labels the energy eigenvalue, and \(\beta=1 / k_{B} T\), proportional to the inverse temperature. While this partition function still has some problems, \({ }^{22}\) it is at least not explicitly divergent. Actually, using the ratio test, show that this is indeed true, that the Planck-Larkin partition function converges.
(c) Additionally, show that the Planck-Larkin partition function reduces to what we expect at low temperature, as \(T \rightarrow 0\).
(d) What does the Planck-Larkin partition function reduce to in the hightemperature limit, \(T \rightarrow \infty\) ? Does this make sense?