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 union of a spin-up field with a spin-down field

${\mathit{\Omega}} \equiv \left\{ \, {\mathscr{F}} \! \! \uparrow, \ {\mathscr{F}} \! \! \downarrow \vphantom{H^H} \right\}$.

is called a quantum of orbital angular momentum. Absorbing or emitting ${\mathit{\Omega}}$ changes the azimuthal quantum number by ±1 without altering the principal quantum number, or total angular momentum. Models of hydrogen, do not contain stereochemical quarks.

## S-states

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

## 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
 $\mathbf{2P_{3/2}}$ $\equiv \ \ \mathbf{1S } \, + \ {\mathrm{Lyman}} \, \alpha \ - \, 𝜴$ $\mathbf{3P_{3/2}}$ $\equiv \ \ \mathbf{1S } \, + \ {\mathrm{Lyman}} \, \beta \ - \, 𝜴$ $\mathbf{4P_{3/2}}$ $\equiv \ \ \mathbf{1S } \, + \ {\mathrm{Lyman}} \, \gamma \ - \, 𝜴$ $\mathbf{5P_{3/2}}$ $\equiv \ \ \mathbf{1S } \, + \ {\mathrm{Lyman}} \, \delta \ - \, 𝜴$ $\mathbf{6P_{3/2}}$ $\equiv \ \ \mathbf{1S} \, + \ {\mathrm{Lyman}} \, \epsilon \ - \, 𝜴$ $\mathbf{7P_{3/2}}$ $\equiv \ \ \mathbf{1S} \, + \ {\mathrm{Lyman}} \, \zeta \ - \, 𝜴$ $\mathbf{8P_{3/2}}$ $\equiv \ \ \mathbf{1S} \, + \ {\mathrm{Lyman}} \, \eta \ - \, 𝜴$ $\mathbf{9P_{3/2}}$ $\equiv \ \ \mathbf{1S } \, + \ {\mathrm{Lyman}} \, \theta \ - \, 𝜴$ $\mathbf{10P_{3/2}}$ $\equiv \ \ \mathbf{1S } \, + \ {\mathrm{Lyman}} \, \kappa \ - \, 𝜴$ $\mathbf{11P_{3/2}}$ $\equiv \ \ \mathbf{1S } \, + \ {\mathrm{Lyman}} \, \lambda \ - \, 𝜴$ $\mathbf{12P_{3/2}}$ $\equiv \ \ \mathbf{1S } \, + \ {\mathrm{Lyman}} \, \mu \ - \, 𝜴$

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

Quark-coefficients for the excited-states of hydrogen are obtained by adding together the quark-coefficients of any components. This method automatically conserves quantum numbers, internal energy, momemtum, etc. The coefficients of photons in the hydrogen spectrum are discussed here.

## Energy

$E_{\sf{gross}}$ is the energy of an excited state of hydrogen, from and

\begin{align} E_{\sf{gross}} \equiv \frac{hc \, {\mathcal{R}}_{\mathrm{H}} }{ {\mathrm{n}}^{2} } \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, given1 by

\begin{align} {\mathcal{R}}_{\mathrm{H}} \equiv {\mathcal{R}}_{\infty} \frac{m_{\sf{p}}}{m_{\sf{p}} + m_{\sf{e}}} \end{align}

where
${\mathcal{R}}_{\infty}$ is a constant
$m_{\sf{p}}$ is the rest mass of a proton
$m_{\sf{e}}$ is the rest mass of a electron

## Fine Structure

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 a constant

## Hyperfine Energy

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

where
$\delta_{z}$ is the helicity

$k_{\sf{hyperfine}}$ 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

Influenced by

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

where
$s$ is the spin angular momentum quantum number
$k_{\sf{Lamb}}$ is a dimensionless constant with a value near fourteen.

## Total Energy

$E^{\prime} \! \left( {\mathrm{H}} \vphantom{H^H} \right)$ is the mechanical energy on a scale where the value of zero is obtained at a distance very far from the nucleus.

$E^{\prime} \! \left( {\mathrm{H}} \vphantom{H^H} \right) \equiv \left( 1 + f_{\sf{fine}} \vphantom{H^H} \right) E_{\sf{gross}} +E_{\sf{hyperfine}} + E_{\sf{Lamb}}$

$E \left( {\mathrm{H}} \vphantom{H^H} \right)$ is the mechanical energy of $\mathrm{H}$, on a scale where $E \left( \mathrm{1S} \vphantom{H^H} \right) \equiv 0$.

$E \left( {\mathrm{H}} \vphantom{H^H} \right) \ \equiv \ E^{\prime} \! \left( {\mathrm{H}} \vphantom{H^H} \right) - E^{\prime} \! \left( \mathrm{1S} \vphantom{H^H} \right)$

## 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 arrangements2 which are ordinary office equipment.

 Next step: length.

Related WikiMechanics articles.

page revision: 746, last edited: 18 Jul 2019 18:22