QMGreensite_merged

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.