11.5. THERADIALEQUATIONFORCENTRALPOTENTIALS 189
Uponquantization,r,%pand%Lbecomeoperators,
r ̃F(r,θ,φ) = rF(r,θ,φ)
%pF ̃ (r,θ,φ) = −ih ̄∇F(r,θ,φ)
%LF ̃ (r,θ,φ) = −ih ̄r%×∇F(r,θ,φ) (11.114)
andwedefinetheoperatorcorrespondingtopras
p ̃r =
1
2
(
%r
r
·%p ̃+%p ̃·
%r
r
)
(11.115)
Thisorderingofrandp ̃ischosensothatp ̃r isanHermitianoperator. Ifwehad
simplysubstituted%p→%p ̃in(11.113),thenp ̃rwouldnotbeHermitian,andneither
wouldtheHamiltonian,whichinvolvesp ̃^2 r.Intermsofdifferentialoperators
p ̃rF = −
1
2
i ̄h
[
1
r
%r·∇+∇·
%r
r
]
F
= −
1
2
i ̄h
[
∂
∂r
+
%r
r
·∇+
1
r
(∇·%r)+%r·(∇
1
r
)
]
F
= −
1
2
i ̄h
[
2
∂
∂r
+
3
r
−
1
r
]
F
= −i ̄h
[
∂
∂r
+
1
r
]
F
= −i ̄h
1
r
∂
∂r
rF (11.116)
sothat
p ̃r=−i ̄h
1
r
∂
∂r
r (11.117)
Intermsofthisoperator,theSchrodingerequation
H ̃φ(r,θ,φ)=Eφ(r,θ,φ) (11.118)
is [
1
2 m
(
p ̃^2 r+
1
r^2
L ̃^2
)
+V(r)
]
φ(r,θ,φ)=Eφ(r,θ,φ) (11.119)
or,explicitly
[
−
̄h^2
2 m
(
(
1
r
∂^2
∂r^2
r)+
1
r^2
(
1
sinθ
∂
∂θ
sinθ
∂
∂θ
+
1
sin^2 θ
∂^2
∂φ^2
)
)
+V(r)
]
φ=Eφ(11.120)
Thisequationcouldbeobtainedmorequicklybystartingfrom
[
−
̄h^2
2 m
∇^2 +V(r)
]
φ=Eφ (11.121)