227
asmalleffect,thereisapowerfulmethodknownastime-independentperturbation
theorywhichallowsoneto”sneakup”ontheeigenstatesandeigenvaluesofH,given
theeigenstatesandeigenvaluesofH 0. Thismethodwillbepresentedinallitsglory
inLecture17,butletusborrow,aheadoftime,oneoftheresults,whichisthat,if
|njjzls>isaneigenstateofH 0 ,theenergyeigenvalueofHisapproximately
Enj′ jzls≈En+<njjzls|H′|njjzls> (14.39)
Thenthespin-orbitcouplingintroducesacorrectiontotheatomicenergieswhichcan
becalculated:
∆E = <njjzls|H′|njjzls>
=
e^2
4 M^2 c^2
∫
drr^2 Rnl(r)
1
r^3
Rnl(r)<Φjjz|(J^2 −L^2 −S^2 )|Φjjz>
=
e^2
4 M^2 c^2
∫
drr^2 Rnl(r)
1
r^3
Rnl(r)×[j(j+1)−l(l+1)−
1
2
(
1
2
+1)] ̄h^2
= [j(j+1)−l(l+1)−
1
2
(
1
2
+1)] ̄h^2
e^2
M^2 c^2 a^30 n^3 l(l+1)(2l+1)
(14.40)
wherea 0 istheBohrradius.Foragivenl,wehaveseenthatthejquantumnumber
canhaveoneoftwovalues,j=l+^12 orj=l−^12 ,inwhichcase
j(j+1)−l(l+1)−
3
4
=
{
l for j=l+^12
−(l+1) for j=l−^12
(14.41)
UsingalsotheexpressionfortheBohrenergy
En=−
Me^4
2 ̄h^2 n^2
(14.42)
anddefiningthe”FineStructureConstant”
α≡
e^2
̄hc
≈
1
137
(14.43)
thefinalresultfortheenergiesE′ofatomicorbitsis
E′n,j=l+ 12 = En+
1
(2l+1)(l+1)
|En|
α^2
n
E′n,j=l− 1
2
= En−
1
(2l+1)l
|En|
α^2
n
(14.44)
Thefactthattheelectronenergylevelsnowdepend(slightly)onlandj=l±^12 ,
inadditiontotheprincipalquantumnumbern,causesasmallsplittingoftheatomic
spectrallinesassociated withtransitionsbetweenthevariouselectronstates. The