QMGreensite_merged

21.2. LINEARALGEBRAINBRA-KETNOTATION 319

TheHermitianconjugateM†ofalinearoperatorM isdefinedtobethatoperator
withtheproperty
<Mv|=<v|M† (21.97)

foranybravector.Thatmeans,inparticular,

Mij† = <ei|M†|ej>

= <Mei|ej>

= <ej|Mei>∗

= <ej|M|ei>∗

= Mj∗i (21.98)

ThisisthedefinitionoftheHermitianconjugateofamatrix,givenintheprevious
section. Therefore,thematrixelementsoftheoperatorM†aresimplythehermitian
conjugateofthematrixelementsoftheoperatorM. Asinthecaseofmatrices,an
HermitianOperatorisanoperatorwhichisitsownHermitianconjugate,ie.

M†=M (21.99)

AnHermitianoperatorthereforehasthepropertythat

<v|M=<Mv| (21.100)

AneigenvalueequationforthelinearoperatorMhastheform

M|vn>=λn|vn> (21.101)

Thisbecomesamatrixequation,inthebasis{|en>},bytakingtheinnerproductof
bothsidesoftheequationwith<ei|:

<ei|M|vn> = λn<ei|vn>

<ei|





∑

lj

M|el><ej|



|vn> = λnvni

∑

j

Mijvnj = λnvni (21.102)

whichisthe same as the matrixeigenvalue equation, incomponents, seen in the
previoussection.

Theorem

TheeigenvaluesofanHermitianoperatorarereal. Eigenstatescorrespondingto
differenteigenvaluesareorthogonal.