21.5. POISSONBRACKETS 341
and
Qa[{q},{q′}] = qaδD(q−q′)
Pa[{q},{q′}] = −i ̄h
∂
∂qa
δD(q−q′) (21.223)
UsingthesematrixelementsintheSchrodingerequation
i ̄h<{q}|ψ>=<{q}|H|ψ> (21.224)
leadstotheSchrodingerequationinSchrodingerrepresentation
i ̄hψ(q)=H
[
{−i ̄h
∂
∂qa
},{qa}
]
ψ(q) (21.225)
whichisnormallythemostusefulformofthisequation.
21.5 Poisson Brackets
LetQ =Q(q,p)besome observabledependingon thecanonical coordinatesand
momenta{qa,pa}.UsingHamilton’sequationsandthechainrulefordifferentiation,
thevariationintimeofthisquantityis
∂tQ =
∑
a
[
∂Q
∂qa
∂qa
∂t
+
∂Q
∂pa
∂pa
∂t
]
=
∑
a
[
∂Q
∂qa
∂H
∂pa
−
∂Q
∂pa
∂H
∂qa
]
= [Q,H]PB (21.226)
wherewe have defined thePoisson Bracketof any two observables A(p,q)and
B(p,q)as
[A,B]PB=
∑
a
[
∂A
∂qa
∂B
∂pa
−
∂A
∂pa
∂B
∂qa
]
(21.227)
Note,inparticular,that
[qa,qb]PB= 0 [pa,pb]PB= 0 [qa,pb]PB=δab (21.228)
AnimportantfeatureofthePoissonbracketoftwoobservablesisthatitalways
hasthesame value,no matterwhichsetofconjugatecoordinatesandmomentum
areused. Normallythere aremanypossible choicesof coordinatesandmomenta.
Forexample,formotionofbaseballinthreedimensions,wecouldusethecartesian
coordinatesx,y,z,andthenobtaintheconjugatemomenta
px=
∂L
∂x ̇
py=
∂L
∂y ̇
pz=
∂L
∂z ̇