Editing Hydrogen atom (section)

=====Energy levels=====
The energy levels of hydrogen, including [[fine structure]] (excluding [[Lamb shift]] and [[hyperfine structure]]), are given by the [[Fine-structure constant|Sommerfeld fine-structure]] expression:<ref name="Sommerfeld">{{cite book | first=Arnold |last= Sommerfeld |title=Atombau und Spektrallinien | trans-title=Atomic Structure and Spectral Lines | publisher=Friedrich Vieweg und Sohn| location=Braunschweig|year=1919| isbn=3-87144-484-7}} [https://archive.org/stream/atombauundspekt00sommgoog German] [https://archive.org/details/AtomicStructureAndSpectralLines English]</ref>

<math display="block">\begin{align}
E_{j \, n} = {} & -\mu c^2 \left[ 1 - \left( 1 + \left[ \frac{\alpha}{n - j - \frac{1}{2} + \sqrt{\left( j + \frac{1}{2} \right)^2 - \alpha^2}} \right]^2 \right)^{-1 / 2} \right] \\
\approx {} & -\frac{\mu c^2 \alpha^2}{2 n^2} \left[ 1 + \frac{\alpha^2}{n^2} \left( \frac{n}{j + \frac{1}{2}} - \frac{3}{4} \right) \right],
\end{align}</math>

where <math>\alpha</math> is the [[fine-structure constant]] and <math>j</math> is the [[total angular momentum quantum number]], which is equal to <math>\left| \ell \pm \tfrac{1}{2} \right|</math>, depending on the orientation of the electron spin relative to the orbital angular momentum.<ref>{{cite book |last1=Atkins |first1=Peter |last2=de Paula |first2=Julio |title=Physical Chemistry |date=2006 |publisher=W. H. Freeman |isbn=0-7167-8759-8 |page=[https://archive.org/details/atkinsphysicalch00pwat/page/349 349] |edition=8th |url=https://archive.org/details/atkinsphysicalch00pwat/page/349 }}</ref> This formula represents a small correction to the energy obtained by Bohr and Schrödinger as given above. The factor in square brackets in the last expression is nearly one; the extra term arises from relativistic effects (for details, see [[#Features going beyond the Schrödinger solution]]). It is worth noting that this expression was first obtained by [[Arnold Sommerfeld|A. Sommerfeld]] in 1916 based on the relativistic version of the [[Old quantum theory|old Bohr theory]]. Sommerfeld has however used different notation for the quantum numbers.