# Divergence of a partition function

Let us consider a collection of non-interacting hydrogen atoms at a certain temperature T.
The energy levels of the hydrogen atom and their degeneracy are:

En = -R/n²
gn = n²​

The partition function in statistical physics is given by:

Z = Sum(gn Exp(-En/kT), n=1 to Inf)​

This function is the "generating function" for all thermodynamic quantities.
The free energy is a simple function of Z.

For the spectrum of hydrogen, this partition function does not converge.

We can consider other systems:
For a particle in a box, the levels are proportional to n².
For an harmonic oscillator, the level are proportional to (n+1/2).

In both cases, the partition function converges.

The particle-in-a-box corresponds to a confined system, this might explain the convergence.
However, the harmonic oscillator is not a confined system, but the partition function does converge.

Could you help me to understand the meaning of all that?

Thanks

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Galileo
Homework Helper
The contribution from the excited states after the ground state are neglegibly low, it is only for very very high n that the divergent terms come into play. What
would the Bohr radius be for those hydrogen atoms?
IIRC, they would be so large that if the hydrogen atoms would be confined the energy levels would eventually go like n^2 as for a particle in a box.
http://www.iop.org/EJ/abstract/0143-0807/22/5/303

Galileo,

But where does that matter in the fundamentals of statistical mechanics?
There must be an essential hypothesis that I violated in calculating Z in this (naïve) way.

Thanks,

Michel

PS: I have no access to IOP, and it costs 30\$ ! I will have to figure out the details by myself, as a real hobbyist ...

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Hi,

It seems also that the most popular answer is based on the size of the hydrogen atom as a function of the occupied level. Another aspect is that of ionised hydrogen that should also be included in the partition function, ... but this brings us to plasma physics.

But I still even don't understand why at low temperatures (say ambient) hydrogen would be in a low-n states instead of in any Rydberg state. I still need help ...

Finally, I had a further annoying idea. Actually, the well-know energy levels of the hydrogen atoms are not stable, they have a finite lifetime. I hoped to decrease the weight of high-n states in the partition function by this way, but I don't see how this could occur. So I now have a further question: how can we include the finite lifetime of the eigenstates in the evaluation of the partition function, ... and does that even make some sense.

Any idea?

Michel

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Non-analyticities in a partition function usually corresponds to a phase transition at some temperature. In this case, at any temperature, a finite amount of energy is enough to kick the system out of a bound state, which of course could be a "phase transition". Any partition function which is for a system in a bound state that also has a free state will diverge for the bound state, this is pretty easy to prove. What I gave you is just my interpretation of what's up.

reilly
There's no reason to suppose that your partition function should converge. You neglect kinetic energy, which bound states generally have; and you have completely neglected the positive energy or scattering states for the hydrogen system. With a little KE you are home free with a convergent partition function, but not a particularly interesting one.

However, toss in some photons and things change. Mainly because photons interact with hydrogen and change the internal states, without much recoil. In fact, you have posed one version of the problem first solved by Planck, and made into a statistical equilibrium problem by Einstein.

Regards,
Reilly Atkinson

reilly,

Unfortunately this does not help really:
With a little KE you are home free with a convergent partition function, but not a particularly interesting one.
simply because the partition function factors in an electronic part and a translational part.
The translational part is very wel known, but the electronics, or internal, part does not converge!

However, you are right to suggest that making the model more realistic can change drastically the situation.
For example, beyond a certain size of the atom, VanderWaals forces could be introduced and remove the divergence.

To be realistic, one should also realize that all the levels except the fundamental are unstable: these are not really energy level !!! (in a sense)
Therefore the equilbrium of monoatomic hydrogen is a purely abstract problem: the reality is more like the equilibrium between atoms and radiations, in mixed states.

Therefore this divergence could be the source of many other interresting questions and ideas.

Like: is there a general and simple way to include unstables levels in a partition function. (together with the EM field here)

Another question: if we want to analyse further this abstract monoatomic hydrogen question: what is wrong with it, and why this divergence, what is the meaning?

Or: what are the conditions for the convergence of the partition function, and their physical meaning?

Michel

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reilly
Several more points;the partition function describes a system in thermal equilibrium, which only ocurs via interactions. And, the upper levels of hydrogen are perfectly stable with no photons.

A big-picture approach might use a thermal bath in which to embed to atom so as to get equilibrium. In such an approach there is a non-zero prbability for a bound state to make a transition to a + energy scattering state. So in this approach both bound and free states must be used.

Another approach is to realize that the appropriate partition function must use the photon + hydrogen Hamiltonian. That brings us to perturbation theory, which is discussed in many books -- Feynman's Stat Mech book is excellent. Also, similar problems are discussed generally and by example by Cohen-Tannoudji et al's Atom-Photon Processes and Applications.

lalbatross, most of what you are asking is answered in the literature of statistical mechanics and quantum optics.

Regards,
Reilly Atkinson

reilly,

lalbatross, most of what you are asking is answered in the literature of statistical mechanics and quantum optics.

That's quite possible. I will check Cohen-Tannoudji, and Landau-Lifchitz which is mentioned by Peierls.

However this will never replace some good discussion.

Michel

reilly