QMGreensite_merged

17.4. DEGENERATEPERTURBATIONTHEORY 275

foralli,n<qwithi+=n,whichisgenerallynotthecase.
Thewayoutofthisdilemmaistofirstnoticethatthesetof{φ(0)n ,n= 1 , 2 ,...,q}
spansafinite,q-dimensionalsubspaceofthefullHilbertspace,andanystateinthis
subspaceisaneigenstateofH 0 ,withthesamedegenerateenergyE 1 (0)=E(0) 2 =....
ThenletV betheHermitian,q×qmatrix,whosematrixelementsare

Vij=〈φ(0)i |V|φ(0)j 〉 i,j≤q (17.68)

Bythe usualtheorems aboutHermitianoperators(whichalsoapply toHermitian
matrices),theeigenstates{φn,n= 1 , 2 ,...,q}ofV

Vφn=Enφn (17.69)

spanthesameq-dimensionalHilbertspaceasthe{φ(0)n ,n= 1 , 2 ,...,q},andmoreover,
fori+=n
〈φ(0)i |V|φ(0)n 〉= 0 (17.70)

providingalltheEnaredifferent(whichwewillassume).
Thismeansthatifwereplacetheinitialsetofeigenstates{φ(0)n }ofH 0 withanew
setofzeroth-ordereigenstates

φ′n(0)=

{

φn n≤q

φ(0)n n>q

(17.71)

theneq.(17.26)isautomaticallysatisfiedfori,n≤q

0 =(Ei(0)−En(0))c^1 ni=−〈φ′i(0)|V|φ′n(0)〉= 0 (17.72)

Wecanthenconsistentlytake

c^1 ni= 0 for i,n≤q (17.73)

Given the newset of zeroth-order wavefunctions{φ′n(0)}, perturbation theorycan
be appliedexactly as inthe non-degenerate case, and wefind forthe first order
correctionsthat

φ(1)n =

∑

i>q

〈φn|V|φ(0)i 〉

En(0)−Ei(0)

φ(0)i for n≤q

φ(1)n =

∑

i(=n

〈φ(0)n |V|φ′i(0)〉

E(0)n −Ei(0)

φ′i(0) for n>q

En(1) = 〈φ′n(0)|V|φ′n(0)〉

=

{

En n≤q

〈φn|V|φn〉 n>q

(17.74)