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