QMGreensite_merged

13.1. SPINWAVEFUNCTIONS 219

However,wereallywanttheinteractionenergyintherestframeofthelaboratory,
whichisalso (approximately)the restframe of the proton. Intransforming this
energybackto theprotonrestframethereisaslightsubtlety: theelectronisnot
movingwithauniformvelocity%v, butisratherinaccelleratedmotionas itcircles
aroundtheproton. Totakethisaccellerationcarefullyintoaccountwouldtakeus
toofarafield, butsufficeittosaythattheinteractionenergyH′aboveismodified
byafactorof^12 ,calledtheThomasprecessionfactor^1

H′ = −

1

2

%μ·B%

= −

%μ

2 Mc

·(E%×%p)

= −

e

2 Mc

%μ·

(

%r

r^3

×%p

)

= −

e

2 Mc

1

r^3

L%·%μ (13.87)

Using

%μ = −

eg

2 Mc

S%

≈ −

e

Mc

S% (13.88)

weget

H′=

e^2

2 M^2 c^2

1

r^3

L%·S% (13.89)

Thisexpressionisknownasthe”spin-orbit”coupling,becauseitinvolvesacoupling
oftheelectronspinangularmomentumwiththeelectronorbitalangularmomentum.
ThefullHydrogenatomHamiltonianshouldcontainthisspin-dependentterm.
Nowthespin-orbitterminvolvesallthex,y,zcomponentsofangularmomentum,
andweknowthatthereisnophysicalstatewhichisaneigenstateofalloftheseterms
simultaneously. However,letusdefinethetotalelectronangularmomentum

J%=%L+S% (13.90)

Then
J^2 =L^2 + 2 %L·S%+S^2 (13.91)

or
%L·S%=^1
2

(J^2 −L^2 −S^2 ) (13.92)

ThetotalHamiltonianisthen

H=H 0 +

e^2

4 M^2 c^2

1

r^3

(J^2 −L^2 −S^2 ) (13.93)

(^1) Aderivationcanbefoundin,e.g.,Jackson’sElectrodynamics.