Atomic Hydrogen

Atomic hydrogen is formed by the union of a proton $\sf{p}^{+}$ with an electron $\sf{e^{-}}$, bound together by a force-carrying collection of field quanta, $\mathscr{F}$. An atom of hydrogen is noted by $\mathbf{H}$, so we write

$\mathbf{H} \equiv \left\{ \sf{p}^{+}, \sf{e}^{-}, \mathscr{F} \right\}$

The proton is represented by the quarks $\; {\sf{p^{+}}} \! \leftrightarrow 4{\sf{d}} + 4{\sf{b}} + 4{\bar{ \sf{t} }}$. The electron is modeled from this selection ${\sf{e^{-}}} \! \leftrightarrow 4{\bar{\sf{u}}} + 2{\bar{\sf{b}}} + 2{\sf{t}} + 2{\bar{\sf{s}}} + 2{\sf{c}} + 4{\bar{\sf{g}}}+ 4{\sf{e}}$. And the field for hydrogen in its spin down ground state is given by

${{\mathscr{F}}} \left( {1 {\mathbf{S}} } \right) \leftrightarrow 4\bar{\sf{d}} + 2\sf{m \bar{m}} + 2\sf{a \bar{a}} + 2\mathrm{ l \bar{l}}$

Atoms of hydrogen are objectified from space-time events as shown in the accompanying table and movie.

 Ground-State Hydrogen
 $\large{ {\mathbf{H}}_{k} }$ $\overbrace{ \hspace{220px} }$
 $k$ $\sf{p}^{+}$ $\sf{e}^{–}$ $\mathscr{F}$ 1 $\sf{d \; b\bar{t} }$ ${\sf{ \bar{d} \; m\bar{m}} }$ 2 $\sf{ \bar{u} \; c\bar{s} \; 2e }$ ${\mathrm{l}}$ 3 $\sf{d \; b\bar{t} }$ $\sf{ \bar{d} \; a \bar{a} }$ 4 $\sf{ \bar{u} \; t\bar{b} \; 2\bar{g} }$ ${\mathrm{\bar{l}}}$ 5 $\sf{d \; b\bar{t} }$ ${\sf{ \bar{d} \; m\bar{m}} }$ 6 $\sf{ \bar{u} \; c\bar{s} \; 2e }$ ${\mathrm{l}}$ 7 $\sf{d \; b\bar{t} }$ $\sf{ \bar{d} \; a \bar{a} }$ 8 $\sf{ \bar{u} \; t\bar{b} \; 2\bar{g} }$ ${\mathrm{\bar{l}}}$
 Click on this image for a look around a quark-model of hydrogen in its ground-state. Stereochemical quarks are about a million times smaller than thermodynamic quarks, so they are not shown.

## Excited States

The excited states of atomic hydrogen are built-up from the components shown in this table. The set of quarks noted by $Ł$ is a particle of orbital angular momentum. It is called the Lamb quantum. Absorbing or emitting $Ł$ changes the azimuthal atomic quantum number by $\Delta \ell = \pm 1$ without altering $\rm{n}$ or $j$. It explains the because, as shown below, excited states can be accurately constructed as
 ${ \mathrm{2S} \equiv \mathrm{1S \! \! \uparrow} + {\mathrm{Lyman}} \hspace{1px} \alpha - Ł }$ and ${ \mathrm{2P_{1/2}} \equiv \mathrm{1S \! \! \uparrow} + {\mathrm{Lyman}} \hspace{1px} \alpha }$

Quark coefficients for the Lyman photons are obtained from the gross structure of the hydrogen spectrum, they bring wet and dry quarks into the description. Quarks are indestructible, so the following models are obtained by adding together the quark-coefficients of all components. This automatically conserves charge, momentum, etc.

 $\mathbf{S}$-states of Hydrogen $\hspace{10px} \scriptsize{ \mathrm{1S} \equiv \ {\mathsf{p}}^{+} + {\mathsf{e}}^{–} + {\mathscr{F}} { \tiny\mathsf{m} } + {\mathscr{F \! \! \downarrow }} }$ $\hspace{3px} \scriptsize{ \mathrm{1S \! \! \uparrow} \equiv \ {\mathsf{p}}^{+} + {\mathsf{e}}^{–} + {\mathscr{F}} { \tiny\mathsf{m} } + {\mathscr{F \! \! \uparrow }} }$ $\hspace{10px} \scriptsize{ \mathrm{2S} \equiv \mathrm{1S \! \! \uparrow} + {\mathrm{Lyman}} \, \alpha - Ł }$ $\hspace{10px} \scriptsize{ \mathrm{3S} \equiv \mathrm{1S \! \! \uparrow} + {\mathrm{Lyman}} \, \beta - Ł }$ $\hspace{10px} \scriptsize{ \mathrm{4S} \equiv \mathrm{1S \! \! \uparrow} + {\mathrm{Lyman}} \, \gamma - Ł }$ $\hspace{10px} \scriptsize{ \mathrm{5S} \equiv \mathrm{1S \! \! \uparrow} + {\mathrm{Lyman}} \, \delta - Ł }$ $\hspace{10px} \scriptsize{ \mathrm{6S} \equiv \mathrm{1S \! \! \uparrow} + {\mathrm{Lyman}} \, \epsilon - Ł }$ $\hspace{10px} \scriptsize{ \mathrm{7S} \equiv \mathrm{1S \! \! \uparrow} + {\mathrm{Lyman}} \, \zeta - Ł }$ $\hspace{10px} \scriptsize{ \mathrm{8S} \equiv \mathrm{1S \! \! \uparrow} + {\mathrm{Lyman}} \, \eta - Ł }$ $\hspace{10px} \scriptsize{ \mathrm{9S} \equiv \mathrm{1S \! \! \uparrow} + {\mathrm{Lyman}} \, \theta - Ł }$ $\hspace{4px} \scriptsize{ \mathrm{10S} \equiv \mathrm{1S \! \! \uparrow} + {\mathrm{Lyman}} \, \kappa - Ł }$ $\hspace{4px} \scriptsize{ \mathrm{11S} \equiv \mathrm{1S \! \! \uparrow} + {\mathrm{Lyman}} \, \lambda - Ł }$ $\hspace{4px} \scriptsize{ \mathrm{12S} \equiv \mathrm{1S \! \! \uparrow} + {\mathrm{Lyman}} \, \mu - Ł }$
 $\mathbf{{P}_{1/2}}$ States of Hydrogen $\hspace{8px} \scriptsize{ \mathrm{2P_{1/2}} \equiv \mathrm{1S \! \! \uparrow} + {\mathrm{Lyman}} \, \alpha }$ $\hspace{8px} \scriptsize{ \mathrm{3P_{1/2}} \equiv \mathrm{1S \! \! \uparrow} + {\mathrm{Lyman}} \, \beta }$ $\hspace{8px} \scriptsize{ \mathrm{4P_{1/2}} \equiv \mathrm{1S \! \! \uparrow} + {\mathrm{Lyman}} \, \gamma }$ $\hspace{8px} \scriptsize{ \mathrm{5P_{1/2}} \equiv \mathrm{1S \! \! \uparrow} + {\mathrm{Lyman}} \, \delta }$ $\hspace{8px} \scriptsize{ \mathrm{6P_{1/2}} \equiv \mathrm{1S \! \! \uparrow} + {\mathrm{Lyman}} \, \epsilon }$ $\hspace{8px} \scriptsize{ \mathrm{7P_{1/2}} \equiv \mathrm{1S \! \! \uparrow} + {\mathrm{Lyman}} \, \zeta }$ $\hspace{8px} \scriptsize{ \mathrm{8P_{1/2}} \equiv \mathrm{1S \! \! \uparrow} + {\mathrm{Lyman}} \, \eta }$ $\hspace{8px} \scriptsize{ \mathrm{9P_{1/2}} \equiv \mathrm{1S \! \! \uparrow} + {\mathrm{Lyman}} \, \theta }$ $\scriptsize{ \mathrm{10P_{1/2}} \equiv \mathrm{1S \! \! \uparrow} + {\mathrm{Lyman}} \, \kappa }$ $\scriptsize{ \mathrm{11P_{1/2}} \equiv \mathrm{1S \! \! \uparrow} + {\mathrm{Lyman}} \, \lambda }$ $\scriptsize{ \mathrm{12P_{1/2}} \equiv \mathrm{1S \! \! \uparrow} + {\mathrm{Lyman}} \, \mu }$
 $\mathbf{{P}_{3/2}}$ States of Hydrogen $\hspace{8px} \scriptsize{ \mathrm{2P_{3/2}} \equiv \mathrm{1S } + {\mathrm{Lyman}} \, \alpha - Ł }$ $\hspace{8px} \scriptsize{ \mathrm{3P_{3/2}} \equiv \mathrm{1S } + {\mathrm{Lyman}} \, \beta - Ł }$ $\hspace{8px} \scriptsize{ \mathrm{4P_{3/2}} \equiv \mathrm{1S } + {\mathrm{Lyman}} \, \gamma - Ł }$ $\hspace{8px} \scriptsize{ \mathrm{5P_{3/2}} \equiv \mathrm{1S } + {\mathrm{Lyman}} \, \delta - Ł }$ $\hspace{8px} \scriptsize{ \mathrm{6P_{3/2}} \equiv \mathrm{1S} + {\mathrm{Lyman}} \, \epsilon - Ł }$ $\hspace{8px} \scriptsize{ \mathrm{7P_{3/2}} \equiv \mathrm{1S} + {\mathrm{Lyman}} \, \zeta - Ł }$ $\hspace{8px} \scriptsize{ \mathrm{8P_{3/2}} \equiv \mathrm{1S} + {\mathrm{Lyman}} \, \eta - Ł }$ $\hspace{8px} \scriptsize{ \mathrm{9P_{3/2}} \equiv \mathrm{1S } + {\mathrm{Lyman}} \, \theta - Ł }$ $\scriptsize{ \mathrm{10P_{3/2}} \equiv \mathrm{1S } + {\mathrm{Lyman}} \, \kappa - Ł }$ $\scriptsize{ \mathrm{11P_{3/2}} \equiv \mathrm{1S } + {\mathrm{Lyman}} \, \lambda - Ł }$ $\scriptsize{ \mathrm{12P_{3/2}} \equiv \mathrm{1S } + {\mathrm{Lyman}} \, \mu - Ł }$
 $\mathbf{{D}_{3/2}}$ States of Hydrogen $\hspace{8px} \scriptsize{ \mathrm{3D_{3/2}} \equiv \mathrm{1S} + {\mathrm{Lyman}} \, \beta }$ $\hspace{8px} \scriptsize{ \mathrm{4D_{3/2}} \equiv \mathrm{1S} + {\mathrm{Lyman}} \, \gamma }$ $\hspace{8px} \scriptsize{ \mathrm{5D_{3/2}} \equiv \mathrm{1S} + {\mathrm{Lyman}} \, \delta }$ $\hspace{8px} \scriptsize{ \mathrm{6D_{3/2}} \equiv \mathrm{1S} + {\mathrm{Lyman}} \, \epsilon }$ $\hspace{8px} \scriptsize{ \mathrm{7D_{3/2}} \equiv \mathrm{1S} + {\mathrm{Lyman}} \, \zeta }$ $\hspace{8px} \scriptsize{ \mathrm{8D_{3/2}} \equiv \mathrm{1S} + {\mathrm{Lyman}} \, \eta }$ $\hspace{8px} \scriptsize{ \mathrm{9D_{3/2}} \equiv \mathrm{1S} + {\mathrm{Lyman}} \, \theta }$ $\scriptsize{ \mathrm{10D_{3/2}} \equiv \mathrm{1S} + {\mathrm{Lyman}} \, \kappa }$ $\scriptsize{ \mathrm{11D_{3/2}} \equiv \mathrm{1S} + {\mathrm{Lyman}} \, \lambda }$ $\scriptsize{ \mathrm{12D_{3/2}} \equiv \mathrm{1S} + {\mathrm{Lyman}} \, \mu }$
 $\mathbf{{D}_{5/2}}$ States of Hydrogen $\hspace{8px} \scriptsize{\mathrm{3D_{5/2}} \equiv \mathrm{1S } + {\mathrm{Lyman}} \, \beta - {\scriptsize{\mathsf{2}}} {\mathscr{F \! \! \uparrow }} }$ $\hspace{8px} \scriptsize{ \mathrm{4D_{5/2}} \equiv \mathrm{1S } + {\mathrm{Lyman}} \, \gamma - {\scriptsize{\mathsf{2}}} {\mathscr{F \! \! \uparrow }} }$ $\hspace{8px} \scriptsize{ \mathrm{5D_{5/2}} \equiv \mathrm{1S } + {\mathrm{Lyman}} \, \delta - {\scriptsize{\mathsf{2}}} {\mathscr{F \! \! \uparrow }} }$ $\hspace{8px} \scriptsize{\mathrm{6D_{5/2}} \equiv \mathrm{1S} + {\mathrm{Lyman}} \, \epsilon - {\scriptsize{\mathsf{2}}} {\mathscr{F \! \! \uparrow }} }$ $\hspace{8px} \scriptsize{ \mathrm{7D_{5/2}} \equiv \mathrm{1S} + {\mathrm{Lyman}} \, \zeta - {\scriptsize{\mathsf{2}}} {\mathscr{F \! \! \uparrow }} }$ $\hspace{8px} \scriptsize{\mathrm{8D_{5/2}} \equiv \mathrm{1S} + {\mathrm{Lyman}} \, \eta - {\scriptsize{\mathsf{2}}} {\mathscr{F \! \! \uparrow }} }$ $\hspace{8px} \scriptsize{ \mathrm{9D_{5/2}} \equiv \mathrm{1S } + {\mathrm{Lyman}} \, \theta - {\scriptsize{\mathsf{2}}} {\mathscr{F \! \! \uparrow }} }$ $\scriptsize{ \mathrm{10D_{5/2}} \equiv \mathrm{1S } + {\mathrm{Lyman}} \, \kappa - {\scriptsize{\mathsf{2}}} {\mathscr{F \! \! \uparrow }} }$ $\scriptsize{ \mathrm{11D_{5/2}} \equiv \mathrm{1S } + {\mathrm{Lyman}} \, \lambda - {\scriptsize{\mathsf{2}}} {\mathscr{F \! \! \uparrow }} }$ $\scriptsize{ \mathrm{12D_{5/2}} \equiv \mathrm{1S } + {\mathrm{Lyman}} \, \mu - {\scriptsize{\mathsf{2}}} {\mathscr{F \! \! \uparrow }} }$
 $\mathbf{F}$ States of Hydrogen $\scriptsize{ \mathrm{4F_{5/2}} \equiv \mathrm{1S } + {\mathrm{Lyman}} \, \gamma - {\mathscr{F \! \! \uparrow }} + {\mathscr{F \! \! \downarrow }} }$ $\scriptsize{ \mathrm{4F_{7/2}} \equiv \mathrm{1S } + {\mathrm{Lyman}} \, \gamma - {\scriptsize{\mathsf{3}}} {\mathscr{F \! \! \uparrow }} + {\mathscr{F \! \! \downarrow }} }$ $\scriptsize{ \mathrm{5F_{7/2}} \equiv \mathrm{1S } + {\mathrm{Lyman}} \, \delta - {\scriptsize{\mathsf{3}}} {\mathscr{F \! \! \uparrow }} + {\mathscr{F \! \! \downarrow }} }$

## Energy Levels

Let $\, E^{\prime} {\mathsf{(}} {\mathrm{H}^{\ast}} {\mathsf{)}}$ note the energy of some excited state $\mathrm{H}^{\ast}$, on a scale where the null-value of $E^{\prime}\! =0$ is obtained when the electron is very far from the proton. For this case1, the ground-state has a value of $E^{\prime} ( {\mathrm{1S}} ) = \small{\sf{\text{-13.6 (eV)}}}$. There are some challenges to making accurate measurements in this context. So we also specify another quantity $E ( {\mathrm{H} ^{\ast}} )$ as the energy on a scale where the ground-state always has a value of exactly zero by definition

$E ( {\mathrm{H}^{\ast}} ) \equiv E^{\prime} ( {\mathrm{H}^{\ast}} ) - E^{\prime} ( {\mathrm{1S}} )$

Then the excited states of hydrogen are described by the sum

$E^{\prime} \equiv \, E_{\sf{\, fine}} + E_{\sf{\, hyperfine}} + E_{\sf{\, Lamb}}$

The terms in this expression depend strongly on the principal quantum number $\mathrm{n}$. But there is also a weaker dependence on $j$, $\, \ell$, and $s$. These atomic quantum numbers are defined from quark coefficients. So the energy of an atomic state is a function of its quark coefficients, and can be formulated as follows. The fine structure of the hydrogen spectrum is given by

 \begin{align} E_{\sf{\, fine}} \equiv - \frac{hc \, {\mathcal{R}}_{\mathrm{H}} }{ {\mathrm{n}}^{2} } \left( 1+ \, f_{\sf{fine}} \right) \end{align} where \begin{align} f_{\sf{fine}} \equiv \frac{\alpha^{2}}{\mathrm{n}} \left[ \frac{1}{ \, j + {\scriptsize{1/2}} } - \frac{3}{4 {\mathrm{n}} } \right] \end{align}
These expressions were developed by and . The number $\alpha$ is called Sommerfeld's constant and ${\mathcal{R}}_{\mathrm{H}}$ is another constant called the Rydberg number. The hyperfine energy term is given by

\begin{align} E_{\sf{ hyperfine}} \equiv \frac{ 1+ \delta_{z}}{ \, {\mathrm{n}}^{3} } k_{\sf{ hyperfine}} \end{align}

where $\delta_{z}$ is the helicity. Describing the hyperfine splitting is easy for WikiMechanics because we explicitly retain accounts of both up-quarks and down-quarks in our atomic models.2 For $E_{\sf{\, Lamb}}$, the assessment has been strongly influenced by as

\begin{align} E_{\sf{\, Lamb}} \simeq E_{\sf{\, Bethe}} \equiv \frac{\alpha^{5} m^{\sf{e}} c^{2} }{4 {\mathrm{n}}^{3} } \begin{cases} \hspace{65px} k_{{\sf{Lamb}}} \; &\sf{\text{if} } &\ell =0 \\ \\ \; \; \frac{-2 s}{\pi \left( \, j + {\scriptsize{1/2}} \right) \left( \ell + {\scriptsize{1/2}} \right) } + k_{{\sf{Bethe}}} &\sf{\text{if} } &\ell \ne 0 \end{cases} \end{align}

This formula is based on his work3. It can de facto represent energy levels, but transition energies are mostly outside of experimental uncertainty. So to improve accuracy we include another summand

 \begin{align} E_{\sf{\, Lamb}} \equiv E_{\sf{\, Bethe}} + E_{\sf{\, chiral}} \end{align} where \begin{align} E_{\sf{\, chiral}} \equiv \frac{ \; s N^{\large{Ⓛ}} }{ \, {\mathrm{n}}^{6} } k_{\sf{chiral}} \end{align}

and $N ^{\large{Ⓛ}}$ notes the total number of levo quarks in the atom. All the foregoing equations, together with formulae for the atomic quantum numbers, are combined with the quark-models to describe the energy levels of atomic hydrogen. Results are compared with experimental observations4,5,6
below.

The WikiMechanics quark-models of hydrogen thus reproduce the quantum-numbers of excited states perfectly. And calculated energy levels for all 28 measured states are within experimental uncertainty. So the description of an inert hydrogen atom is complete. But transition energies are not fully in agreement with observation, so this analysis is continued later for some finer details in the hydrogen spectrum.

## Stability of Atomic Hydrogen

The stability of a particle is described by its mean life which is a function of its thermodynamic temperature, $T$. And the temperature of a hydrogen atom in its ground state is supposed to be very close to zero. Quark models have been carefully adjusted to obtain this. But, despite much effort, the closest to be had for ${\mathbf{H}} (1 {\mathrm{S}} )$ is $\small{ \; T=-8.9 \times 10^{-6} \; \sf{\text{(K)}} }$. We doubt that this number has physical significance. Rather, it shows the limit of our computing techniques. Temperature calculations depend on small differences between large numbers. Some rounding errors are inevitable. And if the temperature is near zero, then these errors can be significant. The non-zero result for hydrogen suggests that our temperature calculations are questionable for any result more exact than a few parts in a million. This seems to be near the limit of what we can obtain from our present computing arrangements7 which are ordinary office equipment.

 Next step: molecular hydrogen.

Related WikiMechanics articles.

page revision: 1223, last edited: 13 Jul 2020 22:40