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50 CHAPTER4. THEQUANTUMSTATE

oftheindexiinsomerangei= 1 , 2 ,...,N.Butafunctionfisalsoasetofnumbers
labeledbyanindex.Thereisonenumber,denotedf(x),foreachreal-numbervalueof
theindexxintherange−∞<x<∞.Theargumentofafunction,x,iscompletely
analogoustotheindexiofavector;thevalueofthefunctionf(x)correspondsto
thecomponentviofthevector.
Thenotationofvectoralgebrahastheadvantagethatwhenwewanttoreferto
thevectoraswhole,andnotanyparticularcomponentofthevector,wecanwrite
%v, (or|v >for thecolumn vectorin ournew”bra-ket”notation). Onthe other
hand,whenwedenoteafunctionbyf(x),itissometimesunclearwhetherweare
referringtothefunctionasawhole,orjusttheparticularvalueatthepointx. Let
usadoptthenotationthat|f>referstotheentirefunction,andf(x)justthevalue
(”component”)at theargument(”index”)x. Then itseasyto seethat functions
behaveineverywaylikevectorswithacontinuousindex.Forexample,onecanadd
vectors
%v=a%u+bw% (4.16)

whichmeans,intermsofcomponents

vi=aui+bwi (4.17)

andonecanalsoaddfunctions

|f>=a|g>+b|h> (4.18)

whichmeans,intermsof”components”

f(x)=ag(x)+bh(x) (4.19)

Vectorshaveinnerproductswitheachother

<u|v>=%u·%v=

∑N

i=1

u∗ivi (4.20)

aswellasanormwiththemselves

|v|^2 =<v|v>=

∑N

i=1

vi∗vi (4.21)

andsodofunctions:
<g|f>=

∫∞

−∞

dxg∗(x)f(x) (4.22)

and
|f|^2 =<f|f>=

∫∞

−∞

dxf∗(x)f(x) (4.23)

If{vi},i= 1 , 2 ,...,N arethecomponentsof thecolumn(”ket”)vector|v>,then
{v∗i}arethe componentsof thecorrespondingrow (”bra”)vector <v|. Likewise,