54 CHAPTER4. THEQUANTUMSTATE
Theresult(4.45)establishesthatthesequenceofgaussians(4.40)isadeltasequence:
δ(x−y)= lim
L→∞
√
L
π
e−(x−y)
(^2) /L
(4.47)
Inasimilarway,wecanprovethat(4.41)isadeltasequence
lim
L→∞
∫∞
−∞
dyf(y)
∫L
−L
dk
2 π
eik(x−y)
= lim
L→∞
∫∞
−∞
dyf(y)
sin[L(x−y)]
π(x−y)
(4.48)
Changevariablestoz=L(x−y),andthisbecomes
lim
L→∞
∫∞
−∞
dyf(y)
∫L
−L
dk
2 π
eik(x−y)
= lim
L→∞
∫∞
−∞
dyf(x−
z
L
)
sin(z)
πz
= f(x)
∫∞
−∞
dz
sin(z)
πz
= f(x) (4.49)
whichestablishesthat
δ(x−y)=
∫∞
−∞
dk
2 π
eik(x−y) (4.50)
AnumberofusefulidentitiesfortheDiracdeltafunctionarelistedbelow:
f(x)δ(x−a) = f(a)δ(x−a) (4.51)
f(x)δ[c(x−a)] = f(x)
1
|c|
δ(x−a) (4.52)
f(x)
d
dx
δ(x−a) = −
df
dx
δ(x−a) (4.53)
f(x)δ[g(x)] = f(x)
∣∣
∣∣
∣
dg
dx
∣∣
∣∣
∣
− 1
δ(x−x 0 ) (4.54)
where,inthelastline,g(x)isafunctionwithazeroatg(x 0 )= 0
Sinceδ(x−y)isnota function, theseidentities should not be interpretedas
meaning:foragivenvalueofx,thenumberontheright-handsideequalsthenumber
ontheleft-handside.Instead,theymeanthat
∫∞
−∞
dx”left-handside”=
∫∞
−∞
dx”right-handside” (4.55)