QMGreensite_merged

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)

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