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)