# Partition functions

Hello people,

Somebody asked me the following. Anybody want to give it a go?

"Consider the following.

We know, from elementary quantum mechanics, that the Bohr energies
of the hydrogen atom go as (-E_0 / n^2), where n is, of course, the
principal quantum number.

We also know, from elementary statistical mechanics, that the
partition function of a system is the sum of exp (-E / k T), over all states.

Taken together, one can easily demonstrate that the partition
function of the hydrogen atom actually diverges.

How can this be? What is the resolution to this apparent paradox?"

## Answers and Replies

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Bystander
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1/n^2 diverges?

For the actual hydrogen atom, shouldn't you have Es = constant * n^2. For the atom's electrons, you have this 1/n^2 thing. I don't think the grand partition function wouldn't diverge in this case, unless you had and infinite number of electrons which isn't possible. Maybe I'm wrong, but it might be that you have to use the grand canonical ensemble for this system instead of the canonical ensemble.

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I think you guys mix up some things here....

Everything you write is OK except the combination of them. Indeed the energy levels of Hydrogen go like 1/n^2; and indeed the partition function is a sum over states, but it is a sum over OCCUPIED states. There is only one electron occupying a state at any time here, so in this case the partition function consists of a single term. There really is no point in using statistical methods, when you have a single particle (or two for that matter) instead of a near infinite amount like 10^23 (as is the case in solid matter).