QMGreensite_merged

21.4. CANONICALQUANTIZATION 339

GiventhatH=H(P,X)canbeexpressedasapolynomialinXandP,therelation
(21.207)follows.
Tofindtheoperatorsx ̃andp ̃inx-representation(whichisalsosometimesref-
ered to as the Schrodinger Representation), we use the definition of the X-
representationasthebasisconsistingofXeigenstates

X|x 0 >=x 0 |x 0 > (21.210)

sothat

X(x,y) = <x|X|y>

= <Xx|y> (hermiticityofX)

= x<x|y> (|x> aneigenstateofX)

= xδ(x−y) (orthonormality) (21.211)

ThenP(x,y)isdeterminedbytheconditionthat

i ̄hδ(x−y) = <x|[X,P]|y>

=

∫

dz[X(x,z)P(z,y)−P(x,z)X(z,y)

= (x−y)P(x,y) (21.212)

andthisconditionissatisfiedby

P(x,y)=i ̄h

∂

∂x

δ(x−y) (21.213)

aswasshownabove. Thenwriting,asbefore

O(x,y)=O ̃δ(x−y) (21.214)

sothat
<x|O|ψ>=O ̃ψ(x) (21.215)

wehaveshown, simplyfromthecommutationrelation(21.200)andtheeigenvalue
equation(21.210)that

x ̃ψ(x) = xψ(x)

p ̃ψ(x) = −i ̄h

∂

∂x

ψ(x) (21.216)

Wearenowreadytopresenttheprocedure,firstenunciatedbyDirac,forquantiz-
inganymechanicalsystem. TheprocedureisknownasCanonicalQuantization,
andgoesasfollows:

• CanonicalQuantization,or,QuantumMechanicsinaNutshell