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52 CHAPTER4. THEQUANTUMSTATE

4.3 The Dirac Delta Function

Ifthelinearoperation

f′(x)=

∫∞

−∞

dyO(x,y)f(y) (4.33)

isanalogoustomatrixmultiplicationinordinarylinearalgebra,thenwhatchoiceof
O(x,y)corresponds,inparticular,tomultiplicationbytheunitmatrix?
Inlinearalgebra,multiplicationofanyvectorbytheunitmatrixIisanoperation
thattakesthevectorintoitself:

|v>=I|v> (4.34)

Incomponentform, theunitmatrixisknownas theKronecker DeltaIij =δij,
whichobviouslymusthavethepropertythat

vi=

∑

i

δijvj (4.35)

TheKroneckerdeltasatisfyingthisequation,foranyvector|v>whatever,isgiven
bythediagonalmatrix

δij=

{

1 if i=j

0 if i+=j

(4.36)

Thecorrespondingoperationforfunctions

|f>=I|f> (4.37)

iswrittenincomponentform

f(x)=

∫∞

−∞

dyδ(x−y)f(y) (4.38)

andthefunctionδ(x−y)whichfulfillsthisequation,foranyfunctionf(x)whatever,
isknownastheDiracDeltafunction.Itisusefulnotjustinquantummechanics,
butthroughoutmathematicalphysics.
TheDiracdeltafunctionδ(x−y)isdefinedasthelimitofasequenceoffunctions
δL(x−y),knownasadeltasequence,whichhavethepropertythat

f(x)= lim

L→∞

∫∞

−∞

dyδL(x−y)f(y) (4.39)

foranyfunctionf(x).Twosuchsequenceswhichcanbeshowntohavethisproperty
are

δL(x−y)=

√

L

π

e−L(x−y)

2

(4.40)

and

δL(x−y)=

∫L

−L

dk

2 π

eik(x−y) (4.41)