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190 CHAPTER11. ANGULARMOMENTUM

andthenjustexpressing∇^2 insphericalcoordinates

∇^2 =(

1

r

∂^2

∂r^2

r)+

1

r^2

(

1

sinθ

∂

∂θ

sinθ

∂

∂θ

+

1

sin^2 θ

∂^2

∂φ^2

) (11.122)

Comparing∇^2 toeq. (11.64),weseethatitcontainstheoperatorL ̃^2 ,althoughthe
reasonforthisisnotsoclear. Theappearanceoftheangularmomentumoperator,
ontheotherhand,isquiteobviousfromeq.(11.112).

Havingfoundtheeigenstatesofangularmomentum,wewrite

φ(r,θ,φ)=Rklm(r)Ylm(θ,φ) (11.123)

wherethe indexk isintroducedto distinguish between statesof differentenergy,
whichhavetheeigenvaluesofL^2 andLz.SubstituteintotheSchrodingerequation

[

1

2 m

p ̃^2 r+

1

2 mr^2

L ̃^2 +V(r)

]

Rklm(r)Ylm(θ,φ) = EklmRklm(r)Ylm(θ,φ)

Ylm

[

1

2 m

p ̃^2 r+

h ̄^2

2 mr^2

l(l+1)+V(r)

]

Rklm(r) = EklmRklm(r)Ylm(θ,φ)(11.124)

CancellingYlmonbothsidesoftheequation,wenotethatneitherthepotentialnorthe
differentialoperatordependonm,sowecanwritethatRklm=RklandEklm=Ekl.
Wethenhaveanequationwhichinvolvesonlytheradialcoordinate

1

2 m

[

p ̃^2 r+

̄h^2 l(l+1)

r^2

]

Rkl(r)+V(r)Rkl(r)=EklRkl(r) (11.125)

which,using

p ̃^2 r = − ̄h^2

1

r

∂^2

∂r^2

r

= − ̄h^2

[

∂^2

∂r^2

+

2

r

∂

∂r

]

(11.126)

becomes

d^2 Rkl

dr^2

+

2

r

dRkl

dr

+

[

2 m

h ̄^2

{Ekl−V(r)}−

l(l+1)

r^2

]

Rkl(r)= 0 (11.127)

Thisistheradialequationforenergyeigenstatesinasphericalpotential.