21.5. POISSONBRACKETS 341
and
Qa[{q},{q′}] = qaδD(q−q′)Pa[{q},{q′}] = −i ̄h∂
∂qaδD(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 ̇