342 CHAPTER21. QUANTUMMECHANICSASLINEARALGEBRA
butofcourseonecouldalsoexpresstheLagrangianinsphericalcoordinatesr,θ,φ,
andobtainconjugatemomenta
pr=
∂L
∂r ̇
pθ=
∂L
∂θ ̇
pφ=
∂L
∂φ ̇
(21.230)
ItisatheoremofclassicalmechanicsthatthePoissonBracketsofanytwoquantities
donotdependofthechoiceofcanonicalcoordinatesandmomenta. Therefore,any
equationexpressedintermsofPoissonBracketsholdstrueforeverypossiblechoice
of{qa,pa}.
Now,fromtheSchrodingerequationwehave
∂t<Q> = ∂t<ψ|Q|ψ>
=
1
i ̄h
<ψ|(QH−HQ)|ψ>
= <ψ|
1
i ̄h
[Q,H]|ψ> (21.231)
Comparingthisequationtotheclassicalequationofmotion
∂tQ=[Q,H]PB (21.232)
suggeststhattheclassical-to-quantumtransitionisobtainedbyassociatingthePois-
sonbracketsofclassicalobservableswiththecommutatorsofcorrespondingoperators,
accordingtotherule
[A,B]PB→
1
i ̄h
[A,B] (21.233)
Makingthisidentification forthePoissonbrackets(21.228)bringsusagaintothe
basicoperatorcommutatorequations
[Qa,Qb] = 0
[Pa,Pb] = 0
[Qa,Pb] = i ̄hδabI (21.234)
of canonical quantization. Thisbeautifuland deep connection between thePois-
sonbrackets ofclassicalmechanics, andthecommutators ofoperatorsinquantum
mechanics,wasfirstpointedoutbyDirac.