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## Main Question or Discussion Point

**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

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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