Editing Hydrogen atom (section)

==== Features going beyond the Schrödinger solution ====
There are several important effects that are neglected by the Schrödinger equation and which are responsible for certain small but measurable deviations of the real spectral lines from the predicted ones:

* Although the mean speed of the electron in hydrogen is only 1/137th of the [[speed of light]], many modern experiments are sufficiently precise that a complete theoretical explanation requires a fully relativistic treatment of the problem. A relativistic treatment results in a momentum increase of about 1 part in 37,000 for the electron. Since the electron's wavelength is determined by its momentum, orbitals containing higher speed electrons show contraction due to smaller wavelengths.
* Even when there is no external [[magnetic field]], in the [[inertial frame]] of the moving electron, the electromagnetic field of the nucleus has a magnetic component. The spin of the electron has an associated [[magnetic moment]] which interacts with this magnetic field. This effect is also explained by special relativity, and it leads to the so-called ''[[spin–orbit interaction|spin–orbit coupling]]'', i.e., an interaction between the [[electron]]'s [[orbital motion (quantum)|orbital motion]] around the nucleus, and its [[Spin (physics)|spin]].

Both of these features (and more) are incorporated in the relativistic [[Dirac equation]], with predictions that come still closer to experiment. Again the Dirac equation may be solved analytically in the special case of a two-body system, such as the hydrogen atom. The resulting solution quantum states now must be classified by the [[Total angular momentum quantum number|total angular momentum number]] {{math|''j''}} (arising through the coupling between [[electron spin]] and [[angular momentum operator|orbital angular momentum]]). States of the same {{math|''j''}} and the same {{math|''n''}} are still degenerate. Thus, direct analytical solution of [[Dirac equation]] predicts 2S({{sfrac|1|2}}) and 2P({{sfrac|1|2}}) levels of hydrogen to have exactly the same energy, which is in a contradiction with observations ([[Lamb shift|Lamb–Retherford experiment]]).

* There are always [[quantum fluctuation|vacuum fluctuation]]s of the [[electromagnetic field]], according to quantum mechanics. Due to such fluctuations degeneracy between states of the same {{math|''j''}} but different {{math|''l''}} is lifted, giving them slightly different energies. This has been demonstrated in the famous [[Lamb shift|Lamb–Retherford experiment]] and was the starting point for the development of the theory of [[quantum electrodynamics]] (which is able to deal with these vacuum fluctuations and employs the famous [[Feynman diagram]]s for approximations using [[perturbation theory (quantum mechanics)|perturbation theory]]). This effect is now called [[Lamb shift]].

For these developments, it was essential that the solution of the Dirac equation for the hydrogen atom could be worked out exactly, such that any experimentally observed deviation had to be taken seriously as a signal of failure of the theory.