# Boltzmann Distribution

## 1 Definition

Degeneracy, often denoted ${\displaystyle g_{i}}$, is how many distinct configurations of an system map to one statistical "outcome."

## 2 Degeneracies of the hydrogen atom

The energy degeneracies corresponding to each E_n for the hydrogen atom are given by:

${\displaystyle g_{n}=\sum _{l=0}^{n-1}(2l+1)=n^{2}\,\!}$

Taking the two electron spin states and the two proton spin states of hydrogen into account, our degeneracies then ultimately become

${\displaystyle g_{n}=4n^{2}.\,\!}$

Using this expression for our degeneracies, we can then show the ratios of excited states of hydrogen to the ground state as a function of temperature:

Ratios of hydrogen excited states to the ground state as a function of temperature