17.4. DEGENERATEPERTURBATIONTHEORY 283
stillgivesa= 1 /
√
2,leadingtotheeigenstate|φ〉E 2 −=1
√
2
[
|φ 200 〉−|φ 210 〉]
(17.124)forenergyeigenvalue
E 2 −=E 2 (0)− 3 ea 0 Ez (17.125)
CaseIII: E= 0
Finally,weconsiderE=0,andeq.(17.115)becomes3 a 0
c
0
a
0
= 0 (17.126)Thistime,wecanonlyconcludethata=c=0,andthatanynormalizedeigenstate
oftheform
|φ〉E 2 =a|φ 211 〉+b|φ 21 − 1 〉 (17.127)
hasanenergyeigenvalue
E 2 =E 2 (0) (17.128)
Thenormalizationconditioninthiscaseonlytellsusthat
a^2 +b^2 = 1 (17.129)andthereforethelinearlyindependenteigenstateswithenergyE 2 spanatwo-dimensional
subspaceofthefullHilbertspace. Thustheperturbationdoesnotremoveallofthe
degeneracyinenergyeigenvalues;thereisstillsomeleft.
Thereasonthatthereisstillsomedegeneracyleft,asseenintheE = 0 case,is
thatthereisstillsomesymmetryleftaftertheexternalelectricfieldisapplied.Foran
electricfielddirectedalongthez-axis,theHamiltonianisstillinvariantwithrespect
torotationsaroundthez-axis,aswellasreflectionsalongthexandy-axes;theresult
issomeremnantdegeneracyintheenergyeigenvalues.