Editing Positronium (section)

== Energy levels ==
{{main|Bohr model#Electron energy levels}}

While precise calculation of positronium energy levels uses the [[Bethe–Salpeter equation]] or the [[Breit equation]], the similarity between positronium and hydrogen allows a rough estimate.  In this approximation, the energy levels are different because of a different effective mass, ''μ'', in the energy equation (see [[Bohr model#Electron energy levels|electron energy levels]] for a derivation):
<math display="block">E_n = -\frac{\mu q_\mathrm{e}^4}{8 h^2 \varepsilon_0^2} \frac{1}{n^2},</math>
where:

* {{math|''q''<sub>e</sub>}} is the [[Elementary charge|charge magnitude]] of the electron (same as the positron),
* {{mvar|h}} is the [[Planck constant]],
* {{math|''ε''<sub>0</sub>}} is the [[electric constant]] (otherwise known as the permittivity of free space),
* {{mvar|μ}} is the [[reduced mass]]: <math display="block">\mu = \frac{m_\mathrm{e} m_\mathrm{p}}{m_\mathrm{e} + m_\mathrm{p}} = \frac{m_\mathrm{e}^2}{2m_\mathrm{e}} = \frac{m_\mathrm{e}}{2},</math> where {{math|''m''<sub>e</sub>}} and {{math|''m''<sub>p</sub>}} are, respectively, the mass of the electron and the positron (which are ''the same'' by definition as antiparticles).

Thus, for positronium, its reduced mass only differs from the electron by a factor of 2. This causes the energy levels to also roughly be half of what they are for the hydrogen atom.

So finally, the energy levels of positronium are given by
<math display="block"> E_n = -\frac{1}{2} \frac{m_\mathrm{e} q_\mathrm{e}^4}{8 h^2 \varepsilon_0^2} \frac{1}{n^2} = \frac{-6.8~\mathrm{eV}}{n^2}.</math>

The lowest energy level of positronium ({{math|1=''n'' = 1}}) is {{val|-6.8|u=eV}}. The next level is {{val|-1.7|u=eV}}. The negative sign is a convention that implies a [[bound state]].  Positronium can also be considered by a particular form of the [[Two-body Dirac equations|two-body Dirac equation]]; Two particles with a [[Coulomb's law|Coulomb interaction]] can be exactly separated in the (relativistic) [[center-of-momentum frame]] and the resulting ground-state energy has been obtained very accurately using [[finite element method]]s of [[Janine Shertzer]].<ref name="Shertzer"/>  Their results lead to the discovery of anomalous states.<ref>
{{cite journal
 |last=Patterson |first=Chris W.
 |date=2019
 |title=Anomalous states of Positronium
 |journal=[[Physical Review A]]
 |volume=100 |issue=6|pages=062128
 |doi=10.1103/PhysRevA.100.062128
 |arxiv=2004.06108
 |bibcode=2019PhRvA.100f2128P
 |s2cid=214017953
}}</ref><ref>
{{cite journal
 |last=Patterson |first=Chris W.
 |date=2023
 |title=Properties of the anomalous states of Positronium
 |journal=[[Physical Review A]]
 |volume=107 |issue=4|pages=042816
 |doi=10.1103/PhysRevA.107.042816
 |arxiv=2207.05725
|bibcode=2023PhRvA.107d2816P
 }}</ref>
The Dirac equation whose Hamiltonian comprises two Dirac particles and a static Coulomb potential is not relativistically invariant.  But if one adds the {{math|{{sfrac|''c''<sup>2''n''</sup>}}}} (or {{math|''α''<sup>2''n''</sup>}}, where {{mvar|α}} is the [[fine-structure constant]]) terms, where {{math|''n'' {{=}} 1,2...}}, then the result is relativistically invariant.  Only the leading term is included.  The {{math|''α''<sup>2</sup>}} contribution is the Breit term; workers rarely go to {{math|''α''<sup>4</sup>}} because at {{math|''α''<sup>3</sup>}} one has the Lamb shift, which requires quantum electrodynamics.<ref name="Shertzer">
{{cite journal
 |last1=Scott |first1=T.C. |last2=Shertzer |first2=J. |author2-link= Janine Shertzer |last3=Moore |first3=R.A.
 |date=1992
 |title=Accurate finite element solutions of the two-body Dirac equation
 |journal=[[Physical Review A]]
 |volume=45 |pages=4393–4398
 |doi=10.1103/PhysRevA.45.4393
 |bibcode=1992PhRvA..45.4393S |pmid=9907514 |issue=7
}}</ref>