QMGreensite_merged

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)