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276 CHAPTER17. TIME-INDEPENDENTPERTURBATIONTHEORY

17.4.1 Diagonalizing theSubmatrix

Equations(17.74)givetheperturbativesolutionstoHφn =Enφnto firstorderin
λ,when thefirstq energyeigenvalues aredegenerate, providing onecan solvethe
eigenvalueequation

V φn=Enφn (17.75)

Sincethefirstq eigenstates{φ(0)n , n= 1 , 2 ,...,q}areabasisofthe q-dimensional
subspace,wecanwrite

φn=

∑q

i=0

aniφ(0)i (17.76)

Writingthe{φ(0)n ,n= 1 , 2 ,..q}asbasisvectors,asineq.(17.55),wecanwritethe
eigenvectorsφnintheq-dimensionalsubspaceintheform

φ%n=















an 1

an 2

an 3

.

.

.

anq















(17.77)

(In thefullHilbertspace φnis aninfinite-componentvector,withzerosfollowing
anq.)Then,inmatrixnotation,theeigenvalueproblem(17.75)issimply













V 11 V 12 V 13.. V 1 q

V 21 V 22 V 23.. V 2 q

V 31 V 32 V 33.. V 3 q

......
......
Vq 1 Vq 2 Vq 3.. Vqq













φ%n=Enφ%n (17.78)

Wehavealreadydiscussedhowtosolvetheeigenvalueproblemfora 2 × 2 matrix,
forexampleinsolvingfortheeigenstatesofthePaulispinmatrixσyinChapter13.
Ingeneral,onefirstsolvesfortheeigenvaluesEnbysolvingthesecularequation

det[V−EI]= 0 (17.79)

whereIistheq×q unitmatrix. Thesecularequationisaq-thorderpolynomial
which(ingeneral)hasqroots. Theserootsarethesetofeigenvaluesofthematrix
V,whichwehavedenoted{En}. Then,foreacheigenvalue,wesolveforthevector
φ%nineq.(17.78)algebraically(asillustratedfortheeigenstatesofσyinChapter13).