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.