QMGreensite_merged

172 CHAPTER11. ANGULARMOMENTUM

11.1 The Angular MomentumCommutators

Angularmomentuminclassicalphysicsisdefinedas

L%=%r×%p (11.3)

or,incomponents,

Lx = ypz−zpy

Ly = zpx−xpz

Lz = xpy−ypx (11.4)

Thecorrespondingquantum-mechanicaloperatorsareobtainedbyreplacingpwith
p ̃,i.e.

L ̃x = yp ̃z−zp ̃y=−i ̄h

{

y

∂

∂z

−z

∂

∂y

}

L ̃y = zp ̃x−xp ̃z=−ih ̄

{

z

∂

∂x

−x

∂

∂z

}

L ̃z = xp ̃y−yp ̃z=−i ̄h

{

x

∂

∂y

−y

∂

∂x

}

(11.5)

Sphericalcoordinatesarerelatedtocartesiancoordinatesby

z = rcosθ

x = rsinθcosφ

y = rsinθsinφ (11.6)

andasphericallysymmetricpotential,alsoknownasacentralpotential,isafunc-
tionwhichisindependentoftheangularcoordinatesθ, φ,i.e.

V(r,θ,φ)=V(r) (11.7)

Wehavealready deducedthat theHamiltonian shouldcommutewiththeangular
momentumoperators,simplybecausetheHamiltonianofaparticleinacentralpo-
tentialisinvariantunderarotationofcoordinates,andtheoperatorswhichgenerate
rotationsaretheexponentialoftheangularmomentumoperators. Thesecommuta-
torscanofcoursebecheckeddirectly,e.ginthecaseofthez-componentofangular
momentumwehave

[L ̃z,H ̃] = [L ̃z,

1

2 m

p ̃^2 +V(r)]

=

1

2 m

[L ̃z,p ̃^2 ]+[Lz,V(r)] (11.8)