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?
Best Answer
This is quite a subtle problem:
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: