QMGreensite_merged

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