322 CHAPTER21. QUANTUMMECHANICSASLINEARALGEBRA
Thex-representationreferstothebasisinHilbertSpacespannedbytheeigen-
states ofthepositionoperatorX. The eigenvaluesof X arealltherealnumbers
x∈[−∞,∞],andwewilldenotethe(normalized)ketvectorcorrespondingtoan
eigenvaluex 0 simplyas|x 0 >,i.e.
X|x 0 >=x 0 |x 0 > (21.115)Sincetheketvectorsformanorthonormalbasis,itmeansthat
<x|y>=δ(x−y) (21.116)andalsothat
I=
∫
dy|y><y| (21.117)isthe identity operator. To gofrom thebra-ketform (21.114)of the eigenvalue
equationtothematrixform(21.112),weusetheidentityoperatorI
O|ψn> = λn|ψn>
OI|ψn> = λn|ψn>
∫
dyO|y><y|ψn> = λn|ψn> (21.118)andtaketheinner-productofbothsidesoftheequationwiththebra<x|
∫
dy <x|O|y><y|ψn> = λn<x|ψn> (21.119)
Comparingtheforms(21.119)with(21.112)showsthatwemustidentify
O(x,y)≡<x|O|y> (21.120)astheMatrixElementofOintheX-Representation,and
ψn(x)≡<x|ψn> (21.121)astheWavefunctionofState|ψn>intheX-Representation. Ifwegoonestep
further,andwrite
O(x,y)=O ̃δ(x−y) (21.122)
whereO ̃isanoperatoractingonfunctionsofthevariablex,andthensubstitutethis
expressionintothematrixformoftheeigenvalueeq. (21.112),wegettheoriginal
form(21.111).
Mostofthecalculationsdoneinelementaryquantummechanicsarecarriedout
inthepositionbasis,i.e. usingthefirst form(21.111)ofthe eigenvalueequation,
andsolvingforwavefunctionsinthex-representation.Nevertheless,thereisnothing
sacredaboutthex-representation,andotherrepresentations aresometimesuseful.
Amongthesearethep-representation,the harmonic-oscillator representation, and