QMGreensite_merged

21.4. CANONICALQUANTIZATION 337

and,inaddition,wemakethecrucialidentificationofx ̃andp ̃asthefollowingoper-
ators,actingonfunctionsofx

x ̃f(x) = xf(x)

pf ̃ (x) = −i ̄h

∂

∂x

f(x) (21.198)

All of this was deduced fromthe wave equationfor De Broglie waves, theBorn
interpretation,andtheEhrenfestPrinciple. Fromthediscussioninthischapter,we
seethatthewavefunctionψ(x)isthewavefunctionofastateinthex-representation,
theoperatorsx ̃andp ̃operateexclusivelyonsuchwavefunctions.
Now,althoughthe x-representationisextremely usefulinquantummechanics,
itisno more fundamentalthan,say, achoiceof cartesian coordinates inclassical
mechanics.Thepointofvectornotationinclassicalmechanicsistowritedynamical
equationswhichholdinanyreferenceframe. Nowwewouldliketodothesamefor
quantummechanics. Uptoapoint,thisiseasy;itsjustaquestionofreplacingthe
wavefunctionψ(x)bytheketvector|ψ>,i.e.

Classical Quantum

−−−−−−−−− −−−−−−−−−

(x,p) |ψ>

O(x,p) O(X,P)

∂tx=∂∂Hp, ∂tq=−∂∂Hx i ̄h∂t|ψ>=H(X,P)|ψ>

cc (21.199)

But whatdo wenow sayaboutoperatorsX andP, withoutreferringdirectlyto
the x-representation? The answeris that the correct, representation-independent
statementisjustthecommutator

[X,P]=i ̄hI (21.200)

whichcanbededucedfromtheEhrenfestprincipleandtheSchrodingerequationas
follows: BeginningfromtheEhrenfestcondition

<

∂H

∂p

>=∂t<x> (21.201)

or

<ψ|

∂H

∂P

|ψ>=∂t<ψ|X|ψ> (21.202)

andapplyingtheSchrodingerequationinbra-ketform

∂t|ψ> =

1

i ̄h

H|ψ>

∂t<ψ| = −

1

i ̄h

<ψ|H (21.203)