17.4. DEGENERATEPERTURBATIONTHEORY 281
wherea 0 = ̄h^2 /me^2 istheBohrradius. Thesecularequationisthen
det[V −EI]
= det
〈 200 |z| 200 〉−E 〈 200 |z| 211 〉 〈 200 |z| 210 〉 〈 200 |z| 21 − 1 〉
〈 211 |z| 200 〉 〈 211 |z| 211 〉−E 〈 211 |z| 210 〉 〈 211 |z| 21 − 1 〉
〈 210 |z| 200 〉 〈 210 |z| 211 〉 〈 210 |z| 210 〉−E 〈 210 |z| 21 − 1 〉
〈 21 − 1 |z| 200 〉 〈 21 − 1 |z| 211 〉 〈 21 − 1 |z| 210 〉 〈 21 − 1 |z| 21 − 1 〉−E
= det
−E 0 3 a 0 0
0 −E 0 0
3 a 0 0 −E 0
0 0 0 −E
= E^4 −(3a 0 )^2 E^2
= 0 (17.109)Therootsofthesecularequationare
E= 0 , 3 a 0 , − 3 a 0 (17.110)Thereforethe4-folddegenerateE 2 energylevelsplitsintothree(notfour)levels
E 2 =⇒
E 2 + 3 a 0 eEz
E 2
E 2 − 3 a 0 eEz(17.111)
Becausethesubspaceisfour-dimensional,buttheperturbationresultsonlyinthree
distinctenergies,thedegeneracyisnotentirelylifted;asubsetofstatesinthis4D
subspacestillhavedegenerateenergies.
Nextwefigureouttheeigenstatescorrespondingtothen= 2 energyeigenvalues
atfirstorder.Wecanalwaysexpressageneralstateinthesubspacespannedbythe
zeroth-ordern= 2 stateasasuperposition
|ψ〉=a|φ 200 〉+b|φ 211 〉+c|φ 210 〉+d|φ 20 − 1 〉 (17.112)Invectornotation,wecanwrite
ψ%=
a
b
c
d
(17.113)
andtheeigenvalueequation
Vψ=Eψ (17.114)
becomes
0 0 3 a 0 0
0 0 0 0
3 a 0 0 0 0
0 0 0 0
a
b
c
d
=E
a
b
c
d
(17.115)