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)