338 CHAPTER21. QUANTUMMECHANICSASLINEARALGEBRA
weget
<ψ|∂H
∂P
|ψ>=<ψ|1
i ̄h[X,H]|ψ> (21.204)Likewise,thesecondEhrenfestequation
<−
∂H
∂x>=∂t<p> (21.205)leadsto
<ψ|−∂H
∂X
|ψ>=<ψ|1
i ̄h[P,H]|ψ> (21.206)Comparing the rhsand lhs of eqs. (21.204)and (21.206)leads us to require, as
operatoridentities,
[X,H] = i ̄h∂H
∂P
[P,H] = −i ̄h∂H
∂X
(21.207)
LetusassumethatH[X,P]canbeexpressedasasumofproductsofoperatorsX
andP. OfcourseXcommuteswithpowersofX,andP commuteswithpowersof
P.^2 Nowsupposethat[X,P]=i ̄hI.Then
[X,Pn] = XPP...P−PP...PX
= PXP...P+i ̄hPn−^1 −PP...PX
= PPX...P+ 2 i ̄hPn−^1 −PP...PX
.
.
= ni ̄hPn−^1
= i ̄h∂
∂P
Pn (21.208)Inthesameway,
[P,Xn] = PXX...X−XX...XP
= XPX...X−i ̄hXn−^1 −XX...XP
.
.
= −ni ̄hPn−^1
= −i ̄h∂
∂X
Xn (21.209)(^2) ThisisbecauseXandXnhavethesamesetofeigenstates,andlikewiseforPandPn.