QMGreensite_merged

320 CHAPTER21. QUANTUMMECHANICSASLINEARALGEBRA

Thisis theorm provedin the last sectionfor matrices. Below we simply run
throughtheproofonceagain(itwasalsoseeninLecture7)inbra-ketnotation:

<vn|M|vm> = λm<vn|vm>

<Mvn|vm> =

<vm|Mvn>∗ =

(λn<vm|vn>)∗ =

λ∗n<vn|vm> = λm<vn|vm> (21.103)

Forn=m,thisimpliesthat λnisreal. For n+=mandλn+=λm,itimpliesthat
<vn|vm>=0;i.e. thetwovectorsareorthogonal.Thisprovestheorem.

Theorem

Ifthe eigenvaluesof alinearoperatorarenon-degenerate, thenthenormalized
eigenstatesformabasisinthelinearvectorspace.

TheeigenvaluesλnareobtainedbysolvingtheD-thorderpolynomial

det[Mij−λδij]= 0 (21.104)

AD-thorderpolynomialhasDroots,unlesstwoormorerootscoincide,inwhich
casethereisadegeneracy.Sincetheeigenvalueequationisalinearequation,itmeans
thatif|vn>isaneigenvector,sois|v′n>=N|vn>.ChooseN sothat

|v′n|= 1 (21.105)

Then,iftheeigenvaluesarenon-degenerate,bytheoremaboveitmeansthat

<vm|vn>=δnm for n,m= 1 , 2 , 3 ,...,D (21.106)

andDorthonormalvectorsformabasisforaD-dimensionallinearvectorspace.

Sincetheeigenvectors{vn}ofanHermitianmatrixMformabasisforthelinear
vectorspace, wecan representanylinearoperatorOinthe space intermsofits
matrixelementsinthe”M-basis”,i.e.

Omn=<vm|O|vn> (21.107)

and
O=

∑

mn

Omn|vm><vn| (21.108)