quantum mechanics - Partition function of a hydrogen gas

Hi I have a doubt (I'm not very expert in statistical mechanics, so sorry for this question). We consider a gas of hydrogen atoms with no interactions between them. The partition function is: $$ Z=\frac{Z_s^N}{N!} $$ where $Z_s$ is the partition function of one atom. So we write $$ Z_s =Tr\{e^{\beta \hat H}\} $$ and we must consider sum to discrete and continous spectrum. So I'd write for the contribute of the discrete spectrum: $$ Z_{s_{disc}}=\sum_{n=1}^{+\infty}n^2 e^{\beta \frac{E_0}{n^2}} $$ but this serie doesn't converge. For continuum spectrum, I'm not be able to write the contribute to the sum, because I have infinite degeneration, so where's my mistake?

I have thought that the spectrum tha I considered for energy values is for free atom in the space, and partition function maybe is defined for systems with finite volume.

So, this doubt however would me ask a problem. Can whe prove that operator $e^{-\beta \hat H}$ always has got finite trace for halll phsical systems?

Answer

This is quite a subtle problem:

• Notice that you have the same problem if you have only a single hydrogen atom.


• You've ignored all the unbound states of the hydrogen atom, which make thing even worse (they are higher in energy, and have even larger degeneracies).


• Divergent partition function is actually not really a problem --- all physical properties will depend on derivatives of the logarithm of it.


• However, you will find that at any finite temperature things like entropy (which are physical) does indeed diverge.


• This is again not so bad if you realise that you've effectively got an infinite system --- the entropy per volume should still be finite.

Concretely, you should regularise things by considering a finite box of volume V, with N particles. This will modify the relevant states (and their energies) in correct (but incredibly hard to compute) ways to render all physical properties finite.

Intuitively the effect is due to the n'th bound state having a radius $O(n^2)$. Thus you can approximate the effects of a finite boundary by simply removing all states above a certain size --- the divergences are so slow that in practice physical quantities will only depend very weakly on the cut-off.

References:

1. http://scienceblogs.com/builtonfacts/2011/01/the_hydrogen_partition_functio.php


2. http://pubs.acs.org/doi/abs/10.1021/ed043p364