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 coupling, i.e., an interaction between the electron's orbital motion around the nucleus, and its spin.
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 cOupliNg, i.E., aN InteracTIon beTween tHe eLectRon'S oRBITal mOtiON arounD the NUcLeuS, ANd its spIn.