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)