QMGreensite_merged

274 CHAPTER17. TIME-INDEPENDENTPERTURBATIONTHEORY

Now supposeweusethe states{φn}tospan theHilbertspace, instead oftheset
{φ(0)n }. Thenyoucaneasilyseethatinthisnewbasis,itisthematrixH,ratherthan
H 0 ,whichisdiagonal

H=











E 1 0 0 ...

0 E 2 0 ...

0 0 E 3 ...

... ...

... ...











(17.62)

because
Hij=〈φi|H|φj〉=δijEj (17.63)

ItisforthisreasonthattheprocessofsolvingtheHamiltonianeigenvalueequation
Hφn=Enφnisoftenreferredtoas“diagonalizingtheHamiltonian”.

17.4 Degenerate Perturbation Theory

TheenergyeigenvaluesoftheHydrogenatomaredegenerate.Theenergyeigenstates
φnlmdependonquantumnumbersn,l,m,buttheenergyeigenvaluesdependonlyon
n.Aswefoundbackinchapter10,thisdegeneracyistypicalwhentheHamiltonian
isinvariantwithrespecttosomesymmetryoperations(e.g.rotationaroundthex,y
orzaxes)whichdon’tcommutewitheachother.
ButnowifwetrytoapplyperturbationtheorytoaHamiltonianwithdegenerate
energyeigenvalues,wecanimmediatelyseethatthereisaproblem. Consider,e.g.
thefirst-ordercorrectiontothewavefunction

φ(1)n =

∑

i(=n

Vin

E

(0)

n −E

(0)

i

φ(0)i (17.64)

Obviously,ifthereissomeenergyEk(0)suchthatEn(0)=E(0)k ,thentheright-handside
ofthisequationisnotsmall,itisinfinite. Thatisnotexactlyasmallperturbation!
Whatcanbedoneinthiscase,whichisquitetypicalinatomicphysics?
Firstofall,tosimplifythingsalittlebit,letssupposethatthereisjustonesetof
stateswithdegenerateenergyeigenvalues,andthatthesearethestatesφ(0)n labeled
byn= 1 , 2 ,...,q,i.e.
E 1 (0)=E 2 (0)=....=Eq(0) (17.65)

Thefirst-ordercorrectiontothewavefunctionisderivedfromeq.(17.26)withN= 1
andi+=n,andrequires

(Ei(0)−E(0)n )c^1 ni=−〈φ(0)i |V|φ(0)n 〉 (17.66)

Now ifi,n≤q,thenEi(0) =En(0)andthelhs oftheequationiszero. Butthisis
impossibleunless
〈φ(0)i |V|φ(0)n 〉= 0 (17.67)