Editing Hydrogen atom (section)

== Alternatives to the Schrödinger theory ==
In the language of [[Werner Heisenberg|Heisenberg's]] [[matrix mechanics]], the hydrogen atom was first solved by [[Wolfgang Pauli]]<ref name="pauli_1926">
{{cite journal
 | last = Pauli | first = W | author-link = Wolfgang Pauli
 | date = 1926
 | title = Über das Wasserstoffspektrum vom Standpunkt der neuen Quantenmechanik
 | journal = Zeitschrift für Physik
 | volume = 36 | issue = 5 | pages = 336–363
 | doi = 10.1007/BF01450175
|bibcode = 1926ZPhy...36..336P | s2cid = 128132824 }}</ref> using a [[rotational symmetry]] in four dimensions [O(4)-symmetry] generated by the [[angular momentum]]
and the [[Laplace–Runge–Lenz vector]]. By extending the symmetry group O(4) to the [[dynamical group]] O(4,2),
the entire spectrum and all transitions were embedded in a single irreducible group representation.<ref>
{{cite journal
| title = Group Dynamics of the Hydrogen Atom
| author = Kleinert H.
| journal =  Lectures in Theoretical Physics, Edited by W.E. Brittin and A.O. Barut, Gordon and Breach, N.Y. 1968
| pages = 427–482
| date = 1968
| url = http://www.physik.fu-berlin.de/~kleinert/kleiner_re4/4.pdf
}}</ref>

In 1979 the (non-relativistic) hydrogen atom was solved for the first time within [[R.P. Feynman|Feynman's]] [[path integral formulation]]
of [[quantum mechanics]] by Duru and Kleinert.<ref>
{{cite journal
| title = Solution of the path integral for the H-atom
| author = Duru I.H., Kleinert H.
| journal = Physics Letters B
| volume = 84
| issue = 2
| pages = 185–188
| date = 1979
| doi = 10.1016/0370-2693(79)90280-6
| url = http://www.physik.fu-berlin.de/~kleinert/kleiner_re65/65.pdf
|bibcode = 1979PhLB...84..185D }}</ref><ref>
{{cite journal
| title = Quantum Mechanics of H-Atom from Path Integrals
| author = Duru I.H., Kleinert H.
| journal = Fortschr. Phys.
| volume = 30
| issue = 2
| pages = 401–435
| date = 1982
| doi = 10.1002/prop.19820300802
| url = http://www.physik.fu-berlin.de/~kleinert/kleiner_re83/83.pdf
|bibcode = 1982ForPh..30..401D }}</ref> This work greatly extended the range of applicability of [[R.P. Feynman|Feynman's]] method.

Further alternative models are [[de Broglie–Bohm theory|Bohm mechanics]] and the [[Quantum potential#Quantum Hamilton–Jacobi equation|complex Hamilton-Jacobi formulation of quantum mechanics]].