272 CHAPTER17. TIME-INDEPENDENTPERTURBATIONTHEORY
Then,tofirstorderinλ
En=h ̄ω(n+1
2
)+ 3 λ(
̄h
2 mω) 2
[1+ 2 n(n+1)] (17.49)whichwecanalsoexpressas
En= ̄h(ω+δω)(n+1
2
)+n^2 ̄hδω (17.50)where
δω= 6 λ̄h
4 m^2 ω^3(17.51)
Sotheeffectofthex ̇^4 perturbationisessentiallyashiftδω intheangular fre-
quencyoftheoscillator,togetherwithanincreaseinthespacingofenergylevels.Its
interestingtonotethatnomatterhowsmallλmaybe,theperturbativeexpansion
fortheenergymustbreakdowncompletelywhennislargeenough.Thisisbecause
thezeroth-order energyincreasesonlylinearlywithn, whereasthefirst-ordercor-
rectionincreasesquadratically. Thisbreakdownofperturbationtheoryhasasimple
physicalexplanation. Whateverthevalueofλ, itisalwaystruethat λx^4 >^12 kx^2
whenxislargeenough. Buthighly excitedharmonic oscillatorstates,whichhave
largen,spreadoutfarfromtheoriginx=0,andthereforeprobetheregionwhere
theperturbingpotentialislargerthanthezeroth-orderpotential. Insuchregions,
onecannotexpectperturbationtheorytowork,andinfactitdoesn’t.
17.3 Perturbation Theory in Matrix Notation
Letusdefinethematrixelementsofanoperator,inthebasisofHilbertspacespanned
by{φ(0)n },as
Oij=〈φ
(0)
i |O|φ(0)
j 〉 (17.52)Inthisbasis,H 0 isadiagonalmatrix:
[H 0 ]ij = 〈φ(0)i |H 0 |φ(0)j 〉
= δijEj(0)H 0 =
E 10 0 0 ...
0 E 20 0 ...
0 0 E 30 ...
... ...
... ...
(17.53)
Fordiagonalmatrices,solvingtheeigenvalueproblem
H 0 φ(0)n =En(0)φ(0)n (17.54)