QMGreensite_merged

268 CHAPTER17. TIME-INDEPENDENTPERTURBATIONTHEORY

andthisconditioneliminatestheambiguitymentionedabove. Ofcourse,afterhaving
computedφn(x)bytheprocessofiterativelysolvingthesetofF(N)= 0 equations,
wewillfinallyhavetonormalizeφn(x)intheusualwaytoobtainaphysicalstate.
Letsseehowthisiterativeprocessworks. SupposewehavesolvedalltheF(k)= 0
equationsuptok=N−1.Thek=N-thequation,inbra-ketnotation,is

(H 0 −En(0))|φ(nN)〉=−V|φ(nN−1)〉+

N∑− 1

j=1

En(j)|φn(N−j)〉+EnN|φ(0)n 〉 (17.24)

Tosolveforφ(nN),firstexpressitintermsofacompletesetofstatesspanning the
Hilbertspace,namely,thezero-thordereigenstatesofH 0

φ(nN)=

∑

i(=n

cNniφ(0)i (17.25)

wherewecanneglecti=nbecauseof (17.23). Nowmultiply eq.(17.24)onboth

sidesbythebravector〈φ(0)i |,withi+=n

〈φ(0)i |(H 0 −En(0))|φ(nN)〉 = −〈φ(0)i |V|φ(nN−1)〉+

N∑− 1

j=1

En(j)〈φi(0)|φ(nN−j)〉+ENn〈φ(0)i |φ(0)n 〉

(Ei(0)−En(0))〈φi(0)|φ(nN)〉 = −〈φ(0)i |V|φ(nN−1)〉+

N∑− 1

j=1

En(j)〈φ(0)i |φ(nN−j)〉

(E(0)i −En(0))cNni = −〈φ(0)i |V|φ(nN−1)〉+

N∑− 1

j=1

En(j)cNni−^1 (17.26)

sothat

cNni=

1

En(0)−Ei(0)



〈φ(0)i |V|φ(nN−1)〉−

N∑− 1

j=1

En(j)cNni−^1



 (17.27)

Therefore

φ(nN)=

∑

i(=n

1

En(0)−E(0)i



〈φ(0)i |V|φ(nN−1)〉−

N∑− 1

j=1

En(j)cNni−^1



φ(0)n (17.28)

istheN-thordercorrectiontothezeroth-orderwavefunction,expressedintermsof
thelowerorderEn(j)andcNni−^1.
Nextwehave togetthe N-th ordercorrectionto theenergy, andforthiswe
multiplyeq.(17.24)onbothsidesbythebravector〈φ(0)n |

〈φ(0)n |(H 0 −En(0))|φ(nN)〉 = −〈φ(0)n |V|φ(nN−1)〉+

N∑− 1

j=1

En(j)〈φn(0)|φ(nN−j)〉+EnN〈φ(0)n |φ(0)n 〉

0 = −〈φ(0)n |V|φ(nN−1)〉+

N∑− 1

j=1

En(j)〈φn(0)|φ(nN−j)〉+EnN〈φ(0)n |φ(0)n 〉

(17.29)