Atomic Hydrogen

## Ground State

 Ground-State Hydrogen
 $\Large{ k }$ $\large{ \delta _{\hat{m}} }$ $\large{ \delta _{\hat{e}} }$ $\large{ \delta _{\theta} }$ $\large{ \sf{P}_{\it{k}} }$ 1 +1 0 +1 $\sf{d} \ \sf{m}\bar{\sf{m}} \ \bar{\sf{t}}\sf{b}$ 2 0 -1 +1 $\sf{\bar{d}} \ \bar{\sf{u}} \ \bar{\sf{b}}\sf{t} \ \mathrm{2}\bar{\sf{g}}$ 3 -1 0 +1 $\sf{d} \ \sf{a}\bar{\sf{a}} \ \bar{\sf{t}}\sf{b}$ 4 0 +1 +1 $\sf{\bar{d}} \ \bar{\sf{u}} \ \bar{\sf{s}}\sf{c} \ \mathrm{2}\sf{e}$ 5 +1 0 -1 $\sf{d} \ \sf{m}\bar{\sf{m}} \ \bar{\sf{t}}\sf{b}$ 6 0 -1 -1 $\sf{\bar{d}} \ \bar{\sf{u}} \ \bar{\sf{b}}\sf{t} \ \mathrm{2}\bar{\sf{g}}$ 7 -1 0 -1 $\sf{d} \ \sf{a}\bar{\sf{a}} \ \bar{\sf{t}}\sf{b}$ 8 0 +1 -1 $\sf{\bar{d}} \ \bar{\sf{u}} \ \bar{\sf{s}}\sf{c} \ \mathrm{2}\sf{e}$

Atomic hydrogen is formed by the union of a proton $\sf{p}^{+}$ with an electron $\sf{e^{-}}$, bound together by $\mathscr{F}$, a force-carrying collection of field quanta. 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 these 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

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

Then atoms of hydrogen are objectified from space-time events like the one shown in the accompanying table and movie.

 Click on this image for a look around a quark-model of hydrogen. The reference frame is suggested by shading in the background. See these spreadsheets for more detail about this and other similar models that describe hydrogen in different excited states.

# Excited States of Atomic Hydrogen

• The excited states of atomic hydrogen are modeled using components shown in the accompanying table.
• The set of quarks noted by $Ł$ is called the Lamb particle. It is the quantum of orbital angular momentum. Absorbing or emitting $Ł$ changes the azimuthal quantum number by $\Delta \ell = \pm 1$ without altering $\rm{n}$ or $j$. It is used to explain the .
• Quark-coefficients for the excited-states of hydrogen are obtained by adding together the quark-coefficients of any components. This automatically conserves quantum-numbers, energy, momemtum, etc.
• The coefficients of the Lyman photons are specified here. They include the many electrochemical quarks that are used in the models below.
• The quantum numbers $\mathrm{n}$, $\ell$, $s$ and $j$ are defined from an atom's quark coefficients

## S-states

 $\large { \mathbf{S} }$-states of Hydrogen
 $\mathbf{1S}$ $\equiv \ \ {\mathsf{p}}^{+} + \, {\mathsf{e}}^{–} \, + \, {\mathscr{F}} _{ \mathbf{H} } \, + \, {\mathscr{F \! \! \downarrow }}$ $\mathbf{1S \! \! \uparrow}$ $\equiv \ \ {\mathsf{p}}^{+} + \, {\mathsf{e}}^{–} \, + \, {\mathscr{F}} _{ \mathbf{H} } \, + \, {\mathscr{F \! \! \uparrow }}$ $\mathbf{2S}$ $\equiv \ \ \mathbf{1S \! \! \uparrow} \, + \ {\mathrm{Lyman}} \, \alpha \ - \ Ł$ $\mathbf{3S}$ $\equiv \ \ \mathbf{1S \! \! \uparrow} \, + \ {\mathrm{Lyman}} \, \beta \ - \ Ł$ $\mathbf{4S}$ $\equiv \ \ \mathbf{1S \! \! \uparrow} \, + \ {\mathrm{Lyman}} \, \gamma \ - \ Ł$ $\mathbf{5S}$ $\equiv \ \ \mathbf{1S \! \! \uparrow} \, + \ {\mathrm{Lyman}} \, \delta \ - \ Ł$ $\mathbf{6S}$ $\equiv \ \ \mathbf{1S \! \! \uparrow} \, + \ {\mathrm{Lyman}} \, \epsilon \ - \ Ł$ $\mathbf{7S}$ $\equiv \ \ \mathbf{1S \! \! \uparrow} \, + \ {\mathrm{Lyman}} \, \zeta \ - \ Ł$ $\mathbf{8S}$ $\equiv \ \ \mathbf{1S \! \! \uparrow} \, + \ {\mathrm{Lyman}} \, \eta \ - \ Ł$ $\mathbf{9S}$ $\equiv \ \ \mathbf{1S \! \! \uparrow} \, + \ {\mathrm{Lyman}} \, \theta \ - \ Ł$ $\mathbf{10S}$ $\equiv \ \ \mathbf{1S \! \! \uparrow} \, + \ {\mathrm{Lyman}} \, \kappa \ - \ Ł$ $\mathbf{11S}$ $\equiv \ \ \mathbf{1S \! \! \uparrow} \, + \ {\mathrm{Lyman}} \, \lambda \ - \ Ł$ $\mathbf{12S}$ $\equiv \ \ \mathbf{1S \! \! \uparrow} \, + \ {\mathrm{Lyman}} \, \mu \ - \ Ł$

## P1/2 – states

 $\mathbf{ {\large{P}}_{1/2}}$ – states of Hydrogen
 $\mathbf{2P_{1/2}}$ $\equiv \ \ \mathbf{1S \! \! \uparrow} \, + \ {\mathrm{Lyman}} \, \alpha$ $\mathbf{3P_{1/2}}$ $\equiv \ \ \mathbf{1S \! \! \uparrow} \, + \ {\mathrm{Lyman}} \, \beta$ $\mathbf{4P_{1/2}}$ $\equiv \ \ \mathbf{1S \! \! \uparrow} \, + \ {\mathrm{Lyman}} \, \gamma$ $\mathbf{5P_{1/2}}$ $\equiv \ \ \mathbf{1S \! \! \uparrow} \, + \ {\mathrm{Lyman}} \, \delta$ $\mathbf{6P_{1/2}}$ $\equiv \ \ \mathbf{1S \! \! \uparrow} \, + \ {\mathrm{Lyman}} \, \epsilon$ $\mathbf{7P_{1/2}}$ $\equiv \ \ \mathbf{1S \! \! \uparrow} \, + \ {\mathrm{Lyman}} \, \zeta$ $\mathbf{8P_{1/2}}$ $\equiv \ \ \mathbf{1S \! \! \uparrow} \, + \ {\mathrm{Lyman}} \, \eta$ $\mathbf{9P_{1/2}}$ $\equiv \ \ \mathbf{1S \! \! \uparrow} \, + \ {\mathrm{Lyman}} \, \theta$ $\mathbf{10P_{1/2}}$ $\equiv \ \ \mathbf{1S \! \! \uparrow} \, + \ {\mathrm{Lyman}} \, \kappa$ $\mathbf{11P_{1/2}}$ $\equiv \ \ \mathbf{1S \! \! \uparrow} \, + \ {\mathrm{Lyman}} \, \lambda$ $\mathbf{12P_{1/2}}$ $\equiv \ \ \mathbf{1S \! \! \uparrow} \, + \ {\mathrm{Lyman}} \, \mu$

## P3/2 – states

 $\mathbf{ {\large{P}}_{3/2}}$ – states of Hydrogen
 $\mathbf{2P_{3/2}}$ $\equiv \ \ \mathbf{1S } \, + \ {\mathrm{Lyman}} \, \alpha \ - \ Ł$ $\mathbf{3P_{3/2}}$ $\equiv \ \ \mathbf{1S } \, + \ {\mathrm{Lyman}} \, \beta \ - \ Ł$ $\mathbf{4P_{3/2}}$ $\equiv \ \ \mathbf{1S } \, + \ {\mathrm{Lyman}} \, \gamma \ - \ Ł$ $\mathbf{5P_{3/2}}$ $\equiv \ \ \mathbf{1S } \, + \ {\mathrm{Lyman}} \, \delta \ - \ Ł$ $\mathbf{6P_{3/2}}$ $\equiv \ \ \mathbf{1S} \, + \ {\mathrm{Lyman}} \, \epsilon \ - \ Ł$ $\mathbf{7P_{3/2}}$ $\equiv \ \ \mathbf{1S} \, + \ {\mathrm{Lyman}} \, \zeta \ - \ Ł$ $\mathbf{8P_{3/2}}$ $\equiv \ \ \mathbf{1S} \, + \ {\mathrm{Lyman}} \, \eta \ - \ Ł$ $\mathbf{9P_{3/2}}$ $\equiv \ \ \mathbf{1S } \, + \ {\mathrm{Lyman}} \, \theta \ - \ Ł$ $\mathbf{10P_{3/2}}$ $\equiv \ \ \mathbf{1S } \, + \ {\mathrm{Lyman}} \, \kappa \ - \ Ł$ $\mathbf{11P_{3/2}}$ $\equiv \ \ \mathbf{1S } \, + \ {\mathrm{Lyman}} \, \lambda \ - \ Ł$ $\mathbf{12P_{3/2}}$ $\equiv \ \ \mathbf{1S } \, + \ {\mathrm{Lyman}} \, \mu \ - \ Ł$

## D3/2 – states

 $\mathbf{ {\large{D}}_{3/2}}$ – states of Hydrogen
 $\mathbf{3D_{3/2}}$ $\equiv \ \ \mathbf{1S} \, + \ {\mathrm{Lyman}} \, \beta$ $\mathbf{4D_{3/2}}$ $\equiv \ \ \mathbf{1S} \, + \ {\mathrm{Lyman}} \, \gamma$ $\mathbf{5D_{3/2}}$ $\equiv \ \ \mathbf{1S} \, + \ {\mathrm{Lyman}} \, \delta$ $\mathbf{6D_{3/2}}$ $\equiv \ \ \mathbf{1S} \, + \ {\mathrm{Lyman}} \, \epsilon$ $\mathbf{7D_{3/2}}$ $\equiv \ \ \mathbf{1S} \, + \ {\mathrm{Lyman}} \, \zeta$ $\mathbf{8D_{3/2}}$ $\equiv \ \ \mathbf{1S} \, + \ {\mathrm{Lyman}} \, \eta$ $\mathbf{9D_{3/2}}$ $\equiv \ \ \mathbf{1S} \, + \ {\mathrm{Lyman}} \, \theta$ $\mathbf{10D_{3/2}}$ $\equiv \ \ \mathbf{1S} \, + \ {\mathrm{Lyman}} \, \kappa$ $\mathbf{11D_{3/2}}$ $\equiv \ \ \mathbf{1S} \, + \ {\mathrm{Lyman}} \, \lambda$ $\mathbf{12D_{3/2}}$ $\equiv \ \ \mathbf{1S} \, + \ {\mathrm{Lyman}} \, \mu$

## D5/2 – states

 $\mathbf{ {\large{D}}_{5/2}}$ – states of Hydrogen
 $\mathbf{3D_{5/2}}$ $\equiv \ \ \mathbf{1S } \, + \ {\mathrm{Lyman}} \, \beta \ - \, 2{\mathscr{F \! \! \uparrow }}$ $\mathbf{4D_{5/2}}$ $\equiv \ \ \mathbf{1S } \, + \ {\mathrm{Lyman}} \, \gamma \ - \, 2{\mathscr{F \! \! \uparrow }}$ $\mathbf{5D_{5/2}}$ $\equiv \ \ \mathbf{1S } \, + \ {\mathrm{Lyman}} \, \delta \ - \, 2{\mathscr{F \! \! \uparrow }}$ $\mathbf{6D_{5/2}}$ $\equiv \ \ \mathbf{1S} \, + \ {\mathrm{Lyman}} \, \epsilon \ - \, 2{\mathscr{F \! \! \uparrow }}$ $\mathbf{7D_{5/2}}$ $\equiv \ \ \mathbf{1S} \, + \ {\mathrm{Lyman}} \, \zeta \ - \, 2{\mathscr{F \! \! \uparrow }}$ $\mathbf{8D_{5/2}}$ $\equiv \ \ \mathbf{1S} \, + \ {\mathrm{Lyman}} \, \eta \ - \, 2{\mathscr{F \! \! \uparrow }}$ $\mathbf{9D_{5/2}}$ $\equiv \ \ \mathbf{1S } \, + \ {\mathrm{Lyman}} \, \theta \ - \, 2{\mathscr{F \! \! \uparrow }}$ $\mathbf{10D_{5/2}}$ $\equiv \ \ \mathbf{1S } \, + \ {\mathrm{Lyman}} \, \kappa \ - \, 2{\mathscr{F \! \! \uparrow }}$ $\mathbf{11D_{5/2}}$ $\equiv \ \ \mathbf{1S } \, + \ {\mathrm{Lyman}} \, \lambda \ - \, 2{\mathscr{F \! \! \uparrow }}$ $\mathbf{12D_{5/2}}$ $\equiv \ \ \mathbf{1S } \, + \ {\mathrm{Lyman}} \, \mu \ - \, 2{\mathscr{F \! \! \uparrow }}$

## F-states

 $\large { \mathbf{F} }$-states of Hydrogen
 $\mathbf{4F_{5/2}}$ $\equiv \ \ \mathbf{1S } \, + \ {\mathrm{Lyman}} \, \gamma \ - \, {\mathscr{F \! \! \uparrow }} \ + \, {\mathscr{F \! \! \downarrow }}$ $\mathbf{4F_{7/2}}$ $\equiv \ \ \mathbf{1S } \, + \ {\mathrm{Lyman}} \, \gamma \ - \, 3{\mathscr{F \! \! \uparrow }} \ + \, {\mathscr{F \! \! \downarrow }}$ $\mathbf{5F_{7/2}}$ $\equiv \ \ \mathbf{1S} \, + \ {\mathrm{Lyman}} \, \delta \ - \, 3{\mathscr{F \! \! \uparrow }} \ + \, {\mathscr{F \! \! \downarrow }}$

## Fine Structure

\begin{align} E_{\sf{fine}} \equiv - \frac{hc \, {\mathcal{R}}_{\mathrm{H}} }{ {\mathrm{n}}^{2} } \left( 1+ f_{\sf{fine}} \right) \end{align}

where
$\rm{n}$ is the principal quantum number
$h$ and $c$ are constants
${\mathcal{R}}_{\mathrm{H}}$ is the Rydberg number for hydrogen

The fine structure factor is influenced by and

\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}

where
$\alpha$ is the fine structure constant

## Hyperfine Energy

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

where
$\delta_{z}$ is the helicity

$k_{\sf{down}}$ is a constant set by the observed frequency of the 21cm line in the hydrogen spectrum. It has a value of about 3 (µeV), and is related to the small internal energy of down-quarks $U^{\sf{d}}$.

## Lamb Shift

The post-war approach is influenced by

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

• $s$ is the spin angular momentum quantum number
• $k_{\ell}$ is a dimensionless constant with a value near one.
• $k_{{\sf{Bethe}}}$ is a dimensionless constant with a value near fourteen.
• depends on assumptions that WikiMechanics does not accept uncritically, including continuous space, Schrödinger equation, unitary proton, …
• gives a logical connection to classical interpretations of angular momenta
• perfectly describes quantum numbers, and atomic energy levels to within experimental uncertainty, for all 28 measured states of excited hydrogen
• describes transition energies such that 49 (of 87) measured photons are outside of experimental uncertainty
• Lyman-alpha (and its observation of 15 significant figures) are outside experimental uncertainty
• explains Lamb shift using complicated devices such as; perturbations, renormalization, zitterbewegung, vacuum fluctuations, …

The WikiMechanics approach uses the Lamb quantum, $\mathrm{Ł}$, to explain Lamb shift, by adding another energy term

$E_{\sf{chiral}} \equiv N^{\mathbf{D}} \, k_{\sf{dextro}} + N^{\mathbf{L}} \, k_{\sf{levo}}$

• $k_{{\sf{dextro}}}$ and $k_{{\sf{levo}}}$ are constants with a values of a few nano electronvolts.
• improves the description of transition energies so that just 41 observed photons are outside of experimental uncertainty
• Lyman-alpha is inside of experimental uncertainty
• confirms Balmer's pristine vision of $\rm{n}$ as the principal quantum number

$E^{\prime}$ is the mechanical energy where a value of zero is obtained far from the nucleus

$E^{\prime} \equiv E_{\sf{fine}} +E_{\sf{hyperfine}} + E_{\ell} + E_{\sf{chiral}}$

For this case, the $\mathrm{1S}$ ground-state has a value of about -13.6 (eV). There are some experimental challenges with this arrangement. So we also define $E ( {\mathrm{H ^{\ast}}} )$ as the energy of some excited state ${\mathrm{H ^{\ast}}}$ on a scale where the ground-state is always zero

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

## Stability of Atomic Hydrogen

The stability of a particle is described by its mean life. And a quantitative analysis of particle lifetimes is set by a customary understanding that the temperature of a hydrogen atom in its ground state is $\ T=0 \ \sf{\text{(K)}}$. Quark models for all nuclear particles have been carefully adjusted to try to get this. But, despite much effort, the closest obtained from current models is $\ T=-8.9 \times 10^{-6} \ \sf{\text{(K)}}$. We doubt that this number has physical significance. Rather, it shows the limits of our calculation technique.

The stability of a nuclear particle depends on a balance of competing effects. And so temperature calculations are based on a few small differences between large numbers. Some are inevitable. And if the temperature is near zero, then these errors can add-up to being a significant problem.

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 arrangements1 which are ordinary office equipment.

 Next step: length.

Related WikiMechanics articles.

page revision: 835, last edited: 29 Aug 2019 00:44