28 Excited States of Hydrogen • EthnoPhysics

Outline

Hydrogen Ground State

Atomic hydrogen is found in a few different varieties. The most abundant isotope is called protium. It is formed by the union of a proton $\mathsf{p}^{+}$ with an electron $\mathsf{e^{-}},$ bound together by a force-carrying collection of field quanta that are noted by $\mathscr{F} .$ An atom of hydrogen is noted by $\mathbf{H} ,$ so we write

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

The proton is represented by the quarks

$\mathsf{p^{+}} \leftrightarrow 4\mathsf{d} + 4\mathsf{b} + 4\overline{ \mathsf{t} }$

The electron is modeled from this selection

$\mathsf{e^{-}} \leftrightarrow 4\overline{\mathsf{u}} + 2\overline{\mathsf{b}} + 2\mathsf{t} + 2\overline{\mathsf{s}} + 2\mathsf{c} + 4\overline{\mathsf{g}}+ 4{\mathsf{e}}$

And the field for hydrogen in its spin-down ground-state is given by this collection of field quanta

$\mathscr{F} \! \left( 1\mathbf{S} \right) \leftrightarrow 2\overline{\mathsf{d}} \overline{\mathsf{d}} + 2\mathsf{m \overline{m}} + 2\mathsf{a \overline{a}} + 2\mathrm{ l \overline{l}}$

Click on this image for a look around a quark-model of hydrogen in its ground-state.

Other excited forms of hydrogen are defined by different fields that involve more quarks. But this ground-state is tiny, it has just 6 quarks per octant.

Protium does not contain any neutrons. But there are two naturally occurring isotopes called deuterium and tritium which are discussed later. The rest of this article is devoted to protium, and from here on we just refer to it as hydrogen. Thus atoms of ground-state hydrogen are objectified from space-time events as shown in the accompanying table and movie.

Ground State Hydrogen
	$\mathbf{H}_{k}$
	$\overbrace{ \hspace{140px} }$
$\mathit{k}$	$\mathsf{p}^{+}$	$\mathsf{e}^{-}$	$\mathscr{ F }$
1	$\mathsf{d \; b\overline{t} }$		$\mathsf{ \overline{d} \; m\overline{m}}$
2		$\mathsf{ \overline{u} \; c\overline{s} \; 2e }$	$\mathrm{l}$
3	$\mathsf{d \; b\overline{t} }$		$\mathsf{ \overline{d} \; a\overline{a} }$
4		$\mathsf{ \overline{u} \; t\overline{b} \; 2\overline{g} }$	$\overline{ \mathrm{l} }$
5	$\mathsf{d \; b\overline{t} }$		$\mathsf{ \overline{d} \; m\overline{m}}$
6		$\mathsf{ \overline{u} \; c\overline{s} \; 2e }$	$\mathrm{l}$
7	$\mathsf{d \; b\overline{t} }$		$\mathsf{ \overline{d} \; a\overline{a} }$
8		$\mathsf{ \overline{u} \; t\overline{b} \; 2\overline{g} }$	$\overline{ \mathrm{l} }$

Stereochemical quarks are about a million times smaller than thermodynamic quarks, so they are not shown in the movie. Note that the foregoing model uses only levo quarks as stereochemical components. So we call this case levorotatory hydrogen. If all of the levo quarks are replaced by dextro quarks then the model is of dextrorotatory hydrogen.

The energy difference between dextrorotatory and levorotatory hydrogen is small enough to usually be ignored. But the distinction is logically important when we use hydrogen to define the handedness of a coordinate system. For the rest of this article we only discuss the levorotatory case, and simply call it hydrogen.

Excited States of Hydrogen

The excited states of atomic hydrogen are built-up from the components shown in the following table. Some rotating quarks are grouped together as spin-up or spin-down field quanta. And a magnetic field is specified using a few muonic quarks.

The set of quarks noted by $\text{\L}$ is called a Lamb quantum. This particle $\text{\L}$ is like a little bit of orbital angular momentum because absorbing or emitting it changes the azimuthal quantum number by $\Delta \ell = \pm 1$ without altering $\rm{n}$ or $j$ . The Lamb quantum is used to explain the Lamb shift because, as shown below, excited states can be accurately constructed as

$2\mathrm{S} \equiv \mathrm{1S \! \! \uparrow} + \mathrm{Lyman} \hspace{1px} \alpha - \text{\L}$

and

$2\mathrm{P_{1/2}} \equiv \mathrm{1S \! \! \uparrow} + \mathrm{Lyman} \hspace{1px} \alpha$

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

S-states

Quark models of excited states are presented in formulae as combinations of protons, electrons, photons and field quanta. They are also shown explicitly in lists of quark coefficients. Specific excited states are identified using standard atomic spectroscopic notation . Here are some models for the $\mathbf{S}$ -states. They all have an azimuthal quantum number of $\ell=0$ .

P-states

Here are some models for the excited $\mathbf{P}$ -states of hydrogen. They have an azimuthal quantum number of $\ell=1$ .

D-states

Here are models for the $\mathbf{D}$ -states. They are characterized by an azimuthal quantum number of $\ell=2$ . Not all these states have been directly measured. But we make models of them anyway because they are used later in calculations about fine structure in the hydrogen spectrum.

F-states

And finally, here are models for the $\mathbf{F}$ -states of excited hydrogen which all have an azimuthal quantum number of $\ell=3$ .

Energy Levels

Let $E^{\prime} \mathsf{(} \mathrm{H}^{\ast} \mathsf{)}$ note the mechanical 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 away from the proton. For this case¹Hydrogen Atom – Chemistry WebBook NIST Standard Reference Database Number 69. National Institute of Standards and Technology, Gaithersburg MD, USA. the ground-state has a value of $E^{\prime} ( {\mathrm{1S}} ) = -13.61$ (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, by definition, the ground-state always has a value of exactly zero

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

Then the excited states of hydrogen are described by the sum

$E^{\prime} \equiv \, E_{\mathsf{\, fine}} + E_{\mathsf{\, hyperfine}} + E_{\mathsf{\, Bethe}} + E_{\mathsf{\, chiral}}$

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 content, and can be formulated as follows. The fine structure of the hydrogen spectrum is given by

$E_\mathsf{\, fine} \equiv - \dfrac{hc \, \mathcal{R}_{\mathrm{H}}}{\mathrm{n}^{2}} \left(1+ \, f_{\mathsf{fine}} \right)$

where

$f_{\mathsf{fine}} \equiv \dfrac{\alpha^{2}}{\mathrm{n}} \left[ \dfrac{1}{\, j + \text{½} } - \dfrac{3}{4\mathrm{n}} \right]$

These expressions were developed by Arnold Sommerfeld and Paul Dirac . The number $\alpha$ is named the fine structure constant . And $\mathcal{R}_{\mathrm{H}}$ is another number called the Rydberg constant . Recall that $\mathcal{R}_{\mathrm{H}}$ was introduced earlier to analyze the gross spectrum of hydrogen. The hyperfine energy term is given by

$E_{\mathsf{hyperfine}} \equiv \dfrac{ h \! \left( 1 + \delta_{z} \right) }{2} \, \nu^{\mathbf{H}}$

where $\delta_{z}$ is the helicity of the excited state, and $\nu^{\mathbf{H}}$ is the hydrogen hyperfine transition frequency. Describing the hyperfine splitting is simple for EthnoPhysics because we explicitly retain accounts of both up-quarks and down-quarks in our atomic models. The transition frequency has an observed²Helmut Hellwig, Robert F. C. Vessot, Martin W. Levine, Paul W. Zitzewitz, David W. Allan, and David J. Glaze. Measurement of the Unperturbed Hydrogen Hyperfine Transition Frequency IEEE Transactions on Instrumentation and Measurement, Volume IM-19, Number 4, November 1970. value of $\nu^{\mathbf{H}} = 1 \, 420 \, 405 \, 751.768 \, \pm \, 0.002$ (Hz).

The assessment of $E_{\mathsf{\, Bethe}}$ has been strongly influenced by Hans Bethe . This formula is broadly based on his work³Hans A. Bethe and Edwin E. Salpeter, Quantum Mechanics of One- and Two-Electron Atoms, Springer-Verlag OHG, Berlin 1957.

$E_{\mathsf{\, Bethe}} \equiv \dfrac{\alpha^{5} m^{\mathsf{e}} c^{2}}{4\mathrm{n}^{3}} \begin{cases} \hspace{35px} k_{\mathsf{\, S}} &\mathsf{\text{if}} \; \ell=0 \\ \\ \; \dfrac{-2s}{\pi \left(\, j + \text{½} \right) \left( \ell + \text{½} \right)} + k_{\mathsf{Bethe}} &\mathsf{\text{if}} \; \ell \ne 0 \end{cases}$

The numbers $k_{\mathsf{Bethe}}$ and $k_{\mathsf{\, S}}$ are treated as adjustable parameters. The value of $k_{\mathsf{Bethe}}$ is approximately equal to $\pi \! / 2.$

The chiral energy term $E_{\mathsf{\, chiral}}$ depends on an atom’s stereochemical quarks. Let $\pmb{\chi }$ note a generic stereochemical quark. Then the total number of stereochemical quarks in an atom is

$n^{\pmb{ \chi }} \equiv n^{\mathbf{d}} + n^{\overline{\mathbf{d}}} + n^{\mathbf{\, l}} + n^{\overline{\mathbf{\,l}}}$

Recall that $s$ notes the spin-angular-momentum quantum number. Then the energy associated with chirality is given⁴It is possible to make an accurate description of inert hydrogen without this chiral term. It is included here because it is required for making a precise description of transition energies later. This chiral formula does not distinguish between right or left handedness because that level of detail is unnecessary for atomic hydrogen. However, we may later employ more elaborate expressions to describe hydrogen molecules with their more complicated vibrational and rotational spectra. by

$E_{\mathsf{\, chiral}} \equiv \dfrac{\; n^{\pmb{\chi}} \! s }{\, \mathrm{n}^{6}} \, k_{\mathsf{chiral}}$

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 observations⁵Kramida, A., Ralchenko, Yu., Reader, J., and the NIST ASD Team (2018). NIST Atomic Spectra Database (version. 5.6.1). National Institute of Standards and Technology, Gaithersburg, MD, USA. in the table below.

The EthnoPhysics quark-models of hydrogen thus reproduce the quantum-numbers of excited states correctly. And calculated energy levels for all 28 measured states are within experimental uncertainty. So the description of an inert hydrogen atom is okay. 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

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 $T=-8.9 \times 10^{-6}$ (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 if 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 arrangements.⁶For more detail about any of the foregoing calculations please see the Atoms & Photons spreadsheet. As of January 2019 calculations are being done using; Microsoft Excel for Mac, Version 15.16, running on an iMac Model 16.2 with an Intel Core i5, 3.1 GHz processor and 8 GB of memory.

Length

Length measurements and Cartesian coordinates are defined. Some atomic shapes and motions are discussed. The Euclidean metric is derived.

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References
1	Hydrogen Atom – Chemistry WebBook NIST Standard Reference Database Number 69. National Institute of Standards and Technology, Gaithersburg MD, USA.
2	Helmut Hellwig, Robert F. C. Vessot, Martin W. Levine, Paul W. Zitzewitz, David W. Allan, and David J. Glaze. Measurement of the Unperturbed Hydrogen Hyperfine Transition Frequency IEEE Transactions on Instrumentation and Measurement, Volume IM-19, Number 4, November 1970.
3	Hans A. Bethe and Edwin E. Salpeter, Quantum Mechanics of One- and Two-Electron Atoms, Springer-Verlag OHG, Berlin 1957.
4	It is possible to make an accurate description of inert hydrogen without this chiral term. It is included here because it is required for making a precise description of transition energies later. This chiral formula does not distinguish between right or left handedness because that level of detail is unnecessary for atomic hydrogen. However, we may later employ more elaborate expressions to describe hydrogen molecules with their more complicated vibrational and rotational spectra.
5	Kramida, A., Ralchenko, Yu., Reader, J., and the NIST ASD Team (2018). NIST Atomic Spectra Database (version. 5.6.1). National Institute of Standards and Technology, Gaithersburg, MD, USA.
6	For more detail about any of the foregoing calculations please see the Atoms & Photons spreadsheet. As of January 2019 calculations are being done using; Microsoft Excel for Mac, Version 15.16, running on an iMac Model 16.2 with an Intel Core i5, 3.1 GHz processor and 8 GB of memory.

Hydrogen Gets Excited