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)