QMGreensite_merged

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)