QMGreensite_merged

267

or

0 = (H 0 φ(0)n −En(0)φ(0)n )+λ(H 0 φ(1)n +Vφ(0)n −E(0)n φ(1)n −En(1)φ^0 n)+...

+λN(H 0 φ(nN)+Vφ(nN−1)−

∑N

j=0

En(j)φ(nN−j))+...

= F(0)+λF(1)+λ^2 F^2 +... (17.17)

Nowthisequationhastobetrueforeverychoiceofλ. Butsince,bydefinition,
theEn(k)andφ(nk)(x)areindependentofλ,wemusthave

F(0)=F(1)=F(2)=...= 0 (17.18)

whichgivesusaninfinitesetofcoupledequations:

(H 0 −E(0)n )φ(0)n = 0

(H 0 −E(0)n )φ(1)n = (E(1)n −V)φ(0)n

(H 0 −E(0)n )φ(2)n = −Vφ(1)n +E(1)n φ(1)n +En(2)φ(0)n

... = ...

(H 0 −En(0))φ(nN) = −Vφ(N−1)+

N∑− 1

j=1

En(j)φ(nN−j)+EnNφ(0)n

... = ... (17.19)

Wealready know the solutionof F(0) = 0, whichissimply theset of zeroth-
order(inλ)eigenstatesandeigenfunctions{φ(0)n ,En(0)}.Thentheideaisinputthese
solutionstosolve theequationF(1)= 0 fortheset {φ(1)n ,E(1)n }. Thisprovidesthe
solutiontoφn,Enuptofirstorderinλ.Wecanthencontinuetosolvetheseequations
iteratively,andobtaintheeigenstatesandeigenvaluestoanyorderinλdesired.
Butwefirsthavetodealwithaslightambiguity. Noticethatifφ(1)n isasolution
of
(H 0 −En(0))φ(1)n =(E(1)n −V)φ(0)n (17.20)

thensois
φ′n(1)=φ(1)n +aφ(0)n (17.21)

wherea isanyconstant. Thereis asimilar ambiguity forany φ(nN). Wecan get
rid ofthisambiguity isasimple way. We firstnotethat the eigenvalueequation
Hφn=Enφnisalinearequation,whichdoesn’tsettheoverallnormalization〈φn|φn〉
oftheeigenstates. Soletstemporarilychoosethenormalizationsuchthattheoverlap

〈φ^0 n|φn〉= 1 (17.22)

isunity,foranyλ. Itseasytosee,fromtheexpansion(17.12),thatthisimpliesfor
eachN,
〈φ^0 n|φ(nN)〉= 0 (17.23)