The first quantized mathematical account of hydrogen was achieved by Johann Balmer, a Swiss high-school teacher. Here is a photo of what he described; the visible part of the hydrogen spectrum. |

Photons that are absorbed or emitted by atomic hydrogen $\mathbf{H}$, are collectively known as the *spectrum* of hydrogen. They are mostly involved in the atomic and molecular interactions of everyday experience, not nuclear reactions. Energies are typically measured in (eV) rather than (MeV). All hydrogen-spectrum photons are linked to changes in the excited states of atomic hydrogen. When $\mathbf{H}$ goes from some initial state $i$, to some final state $f$, by emitting a photon $\gamma$ we write

$\begin{align} {\mathbf{H}} _{i} \to {\mathbf{H}} _{f} + \gamma \end{align}$

Atoms and photons are described by their principal quantum number $\rm{n}$ which is always conserved because $\rm{n}$ is proportional to the number of down quarks, and quarks are indestructible. So for any interaction, the conservation of down quarks guarantees that

$\begin{align} {\rm{n}} \left( {\mathbf{H}} _{i} \right) = {\rm{n}} \left( {\mathbf{H}} _{f} \right) + {\rm{n}} \left( \gamma \right) \end{align}$

Let us write ${\rm{n}}_{i} \equiv {\rm{n}} \left( {\mathbf{H}}_{i} \right)$ and ${\rm{n}}_{f} \equiv {\rm{n}} \left( {\mathbf{H}}_{f} \right)$. Then, as shown earlier, this conservation law will be automatically satisfied if

$\begin{align} {\rm{n}} _{i} = \frac{N^{\sf{D}}\left( \gamma \right) }{8} \end{align}$ | and | $\begin{align} {\rm{n}} _{f} = \frac{\Delta n^{\sf{D}}\left( \gamma \right) }{8} \end{align}$ |

where $\Delta n^{\sf{D}}$ and $N^{\sf{D}}$ note the coefficients of down quarks. Next we are going to use these relationships to formulate the photon energy, $E(\gamma)$, in terms of the principal quantum numbers of $\mathbf{H}$.

The hydrogen spectrum has many ultraviolet, visible and infrared photons. There are also some microwaves, but no gamma-rays. So overall, chemical quarks determine the range of photon energies. These chemical quarks are described by their enthalpy which is noted as $H_{\sf{chem}}$. For photons that are not gamma-rays, we can calculate $E(\gamma)$ by first assessing their momenta.

$\begin{align} E \left( {\large{\gamma}} \right) = 2 \left| \, H_{\sf{chem}}^{\mathcal{A}} \vphantom{{H_{\sf{chem}}^{\large{\gamma}}}^{9}} \right| \cdot \left[ \frac{64}{\left( \Delta n^{\sf{D}} \right)^{2} } - \frac{64}{ \left( N^{\sf{D}} \right)^{2} } \right] \end{align}$

This formula shows the strong influence of any down-quarks on these low-energy photons. Substituting-in the relationships with $\rm{n}$ discussed above gives

$\begin{align} E \left( {\large{\gamma}} \right) = 2 \left| \, H_{\sf{chem}}^{\mathcal{A}} \vphantom{{H_{\sf{chem}}^{\large{\gamma}}}^{9}} \right| \cdot \left( \frac{1}{ {\rm{n}} _{f}^{2} } - \frac{1}{ {\rm{n}} _{i}^{2} } \right) \end{align}$

This expression is used to make quark-models by adjusting the distribution of electochemical quarks so that

$\begin{align} \left| \, H_{\sf{chem}}^{\mathcal{A}} \vphantom{{H_{\sf{chem}}^{\large{\gamma}}}^{9}} \right| = \frac{hc}{2} \mathcal{R} _{\mathbf{H}} \end{align}$

where $\mathcal{R} _{\mathbf{H}}$ is the *Rydberg* constant of hydrogen.^{1} Using this constraint to eliminate $H_{\sf{chem}}$ then defines the energy of a photon for the so-called *gross* structure of hydrogen spectroscopy

$\begin{align} E_{\sf{gross}} \equiv hc \mathcal{R} _{\mathbf{H}} \left( \frac{1}{{\rm{n}}_{f}^{2}} - \frac{1}{{\rm{n}}_{i}^{2}} \right) \end{align}$

Measurements of photons report their wavelength which is related to their energy by $\lambda = hc/E$. Wavelengths may also depend on the surroundings, then the symbol $\lambda _{\sf{o}}$ indicates a wavelength where environmental effects are negligible. We assume this to write

$\begin{align} \frac{1}{ \lambda _{\sf{o}} } = \mathcal{R} _{\mathbf{H}} \left( \frac{1}{{\rm{n}}_{f}^{2}} - \frac{1}{{\rm{n}}_{i}^{2}} \right) \end{align}$

The description of hydrogen was first put in this form by Johannes Rydberg. We use it to calculate $\lambda _{\sf{o}}$ in the following tables of hydrogen-spectrum photons. Models for these photons are built exclusively from electrochemical and rotating quarks.^{2}The rotating quarks all provide an angular momentum of $𝘑 \, =1$ in various ways. But the electrochemical quarks are the same for every photon because all these photons are associated with hydrogen. Comparisons are made with experimental observations.

^{3}Some models provide excellent descriptions that are nonetheless outside of experimental uncertainty. This is indicated with an X in the following tables.

## Lyman Series

## Balmer Series

## Paschen Series

## Brackett Series

## Other Series

Results also depend on how the foregoing quarks are distributed between phase-components. For that level of detail please see the spreadsheets in the wiki-files stored here. In the models above, agreement with experiment is good to a few parts in a million. This is not perfect, but it is good enough to distinguish between competing quark-models. And so these specific quark combinations are taken to define each photon, and used later to make *fine* and *hyper-fine* models of hydrogen.