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.