Chapter 13
Electron Spin
Letsgoback,foramoment,totheclassicalpictureoftheHydrogenatom,withan
electronmovinginacircularorbitaroundtheproton. Achargemovinginacircle
constitutesacurrentloop,and, accordingtotheusuallawsofelectromagnetisma
currentloopisassociatedwithamagneticmomentaccordingtotherule
μ=
1
c
IA (13.1)
whereμisthemagneticmoment,Iisthecurrent,andAistheareaenclosedbythe
circularorbit. Thecurrentisthechargeperunitlengthofloop,timesthevelocity,
sointhiscase,foracircularorbitofradiusr
I = −
e
2 πr
v
v =
p
M
=
L
Mr
A = πr^2 (13.2)
Puttingitalltogether,andnotingthatasvectorquantitiesboth%μand%Larenormal
totheloop,weget
%μ=−
e
2 Mc
%L (13.3)
(notethattheradiusrdropsoutoftheexpression). Finally,inanexternalmagnetic
fieldB%,theinteractionenergybetweenthemagneticdipoleandthefieldisgivenby
Emag = −B%·%μ
=
e
2 Mc
B%·L% (13.4)
Supposethenthatahydrogenatomissittinginanexternalconstantmagnetic
field,directedalongthez-axis.Inthatcase,theHamiltonianrelevantforthemotion
oftheelectronis
H=H 0 +
e
2 Mc
BzLz (13.5)