Canonical partition function for the hydrogen atom via the Coulomb propagator

The electronic partition function for the hydrogen atom is derived by integration over the recently-available Coulomb propagator. This provides a resolution to an old paradox in statistical mechanics: the apparent divergence of the hydrogen partition function. Electronic excitation does not contribute significantly to the standard-state partition function until temperatures of the order of 5000 K. Thereafter, the continuum, with its immense density of states, makes the dominant contribution. From the discrete and continuum contributions to the partition function, a modification of the Saha-Boltzmann equation for the ionization equilibrium in atomic hydrogen is derived.