192 CHAPTER11. ANGULARMOMENTUM
Forthefreeparticleinonedimension,wefoundthatE,Pwasacompletesetof
observables,wherePisparity,andalsothatpisacompletesetofobservables,where
pismomentum. Similarly,inthreedimensions,thethreecomponentsofmomentum
{px,py,pz} area complete set of observables, because there is oneand onlyone
eigenstateof{p ̃x,p ̃y,p ̃z}withagivensetofeigenvalues
−ih ̄
∂
∂x
ψp(x,y,z) = pxψp(x,y,z)
−i ̄h
∂
∂y
ψp(x,y,z) = pyψp(x,y,z)
−i ̄h
∂
∂z
ψp(x,y,z) = pzψp(x,y,z) (11.138)
namely
ψp(x,y,z)=Nexp[i(pxx+pyy+pzz)/ ̄h] (11.139)
ThiswavefunctionisalsoaneigenstateoftheHamiltonian
H ̃ψp=− ̄h
2
2 m
∇^2 ψp=Epψp (11.140)
withenergy
Ep=
1
2 m
(p^2 x+p^2 y+p^2 z) (11.141)
TheHamiltonianofafreeparticleisinvariantundertranslations
%r→%r+%a (11.142)
andalsounderrotationsofthecoordinates. Theenergyeigenstates{ψp}areeigen-
statesoftheHamiltonianandthetranslationoperator
T=ei"a·
"p ̃
(11.143)
whiletheenergyeigenstates{jl(kr)Ylm(θ,φ)}areeigenstatesoftheHamiltonianand
therotationoperatoralongthez-axis
R=eiδφ
L ̃z
(11.144)
aswellasthetotalangularmomentumoperatorL ̃^2.