QMGreensite_merged

100 CHAPTER7. OPERATORSANDOBSERVATIONS

or,in”component”notation
ψ′(x)=O ̃ψ(x) (7.12)

Alinearoperatorhasthepropertythat,foranytwofunctions|ψ 1 >andψ 2 >,and
anytwoconstantsaandb,

O{a|φ 1 >+b|ψ 2 >}=aO|ψ 1 >+bO|ψ 2 > (7.13)

Examplesoflinearoperatorsaremultiplicationbyaconstant

O ̃ψ(x)=cψ(x) (7.14)

ormultiplicationbyafixedfunctionF(x),

O ̃ψ(x)=F(x)ψ(x) (7.15)

ordifferentiation

O ̃ψ(x)= ∂

∂x

ψ(x) (7.16)

Anon-linearoperationcouldbe,e.g.,takingasquareroot

O ̃ψ(x)=

√

ψ(x) (7.17)

InLecture 4 itwasnotedthatanylinearoperatorOhasamatrixrepresentation
O(x,y)suchthateq. (7.11)canbewritten,incomponentform,

ψ′(x)=

∫

dyO(x,y)ψ(y) (7.18)

inanalogytomatrixmultiplication

%v′ = M%v

vi′ =

∑N

i=1

Mijvj (7.19)

GivenalinearoperationO ̃,wecanalwaysobtainacorrespondingmatrixrepresen-
tationO(x,y)byusingtheDiracdeltafunction

O ̃ψ(x) = O ̃

∫

dyδ(x−y)ψ(y)

=

∫

dy

[

O ̃δ(x−y)

]

ψ(y) (7.20)

sothat

O(x,y)=O ̃δ(x−y) (7.21)