278 CHAPTER17. TIME-INDEPENDENTPERTURBATIONTHEORY
andtheφm(x)andφn(y)areone-dimensionalharmonicoscillatoreigenstates. The
totalenergyeigenvalueisthesumofEm+Em
φ(0)mn(x,y) = φm(x)φn(y)
Emn(0) = ̄hω(m+n+1) (17.89)ItseasytoseethattheenergyE(0) 00 isunique,thatE 01 (0)=E(0) 10 istwo-folddegenerate,
E(0) 20 =E 11 (0)=E 02 (0)isthree-folddegenerate,andsoon(ingeneral,theenergyEmn^0 is
(m+n+1)-folddegenerate).
NowintroduceaperturbingpotentialV′=λxy,
H=H 0 +λxy (17.90)Asanexerciseindegenerateperturbationtheory,wewillnowcomputethecorrection
totheenergiesE 10 (0)=E 01 (0)ofthefirstexcitedstates.
Thefirsttwoexcitedstateswithdegenerateenergyeigenvalues,φ 10 andφ 01 ,spana
two-dimensionalsubspaceoftheHilbertspace. Ourfirsttaskistofindtheeigenstates
ofthe 2 × 2 matrix
V =[
〈φ 10 |xy|φ 10 〉 〈φ 10 |xy|φ 01 〉
〈φ 01 |xy|φ 10 〉 〈φ 01 |xy|φ 01 〉]
(17.91)Using
xy=1
2 β^2(a+a†)(b+b†) β≡√
Mω
̄h(17.92)
weget
〈φ 10 |xy|φ 10 〉 =1
2 β^2〈φ 1 (x)|(a+a†)|φ 1 (x)〉〈φ 0 (y)|(b+b†)|φ 0 (y)〉= 0〈φ 01 |xy|φ 01 〉 =1
2 β^2〈φ 0 (x)|(a+a†)|φ 0 (x)〉〈φ 1 (y)|(b+b†)|φ 1 (y)〉= 0〈φ 10 |xy|φ 01 〉 =1
2 β^2〈φ 1 (x)|(a+a†)|φ 0 (x)〉〈φ 0 (y)|(b+b†)|φ 1 (y)〉=1
2 β^2〈φ 01 |xy|φ 10 〉 =1
2 β^2〈φ 0 (x)|(a+a†)|φ 1 (x)〉〈φ 1 (y)|(b+b†)|φ 0 (y)〉=1
2 β^2
(17.93)andtherefore
V=1
2 β^2[
0 1
1 0]
(17.94)Theproblem of findingthe eigenvectorsand eigenvalues (“diagonalizing”)the
matrixV isprettymuchthesameas solvingtheeigenvalueproblemforthePauli
matrixσx.Firstwesolvethesecularequation
det[V−EI]=E^2 −(
1
2 β^2) 2
= 0 (17.95)