QMGreensite_merged

4.2. HILBERTSPACE 51

f(x),−∞<x<∞representthe”components”of|f>,andf∗(x)representsthe
componentsofthecorrespondingbra<f|.
Thereisalinearoperationcalledmatrixmultiplicationwhichturnsavectorinto
anothervector
%v′=M%v (4.24)

or
|v′>=M|v> (4.25)

inournewnotation.Intermsofcomponents,matrixmultiplicationisdefinedas

v′i=

∑N

j=1

Mijvj (4.26)

andithasthelinearitypropertythat

M(a|u>+b|v>)=aM|u>+bM|v> (4.27)

whereaandbareconstants. Thereisasimilarlinearoperationwhichturnsfunctions
intootherfunctions
|f′>=O|f> (4.28)

havingthelinearityproperty

O(a|f>+b|g>)=aO|f>+bO|g> (4.29)

Intermsof”components,”thisoperationiswritten

f′(x)=

∫∞

−∞

dyO(x,y)f(y) (4.30)

whereO(x,y)issomefunctionoftwovariablesxandy,incompleteanalogytothe
ruleformatrixmultiplication(4.26). Finally,theexpressioninlinearalgebra

<u|M|v> = %u·M%v

=

∑N

i=1

∑N

j=1

u∗iMijvj (4.31)

corresponds,inthecaseoffunctions,totheexpression

<g|O|f>=

∫∞

−∞

dx

∫∞

−∞

dyg∗(x)O(x,y)f(y) (4.32)

Afunction,therefore, isjustavectorwithacontinuousindex. Sincethereare
aninfinitenumberof”components”(onecomponentf(x)foreachvalueofx),the
vectorisamemberofaninfinite-dimensionalspaceknownas”HilbertSpace.”Stated
anotherway: Hilbert Space is theinfinite-dimensional vectorspace of all
square-integrablefunctions.