QMGreensite_merged

11.5. THERADIALEQUATIONFORCENTRALPOTENTIALS 191

• TheFreeParticle ThesimplestpossiblecentralpotentialisV(r)=0. Ifwe
  write

E=

̄h^2 k^2

2 m

(11.128)

Thentheradialequationis

d^2 Rkl

dr^2

+

2

r

dRkl

dr

+

[

k^2 −

l(l+1)

r^2

]

Rkl(r)= 0 (11.129)

Rescalingthercoordinate,
u=kr (11.130)

thisbecomes [
d^2
du^2

+

2

u

d

du

+

(

1 −

l(l+1)

u^2

)]

Rkl(r)= 0 (11.131)

Writing
Rkl(r)=Wl(u) (11.132)

wehaveasecond-orderdifferentialequation

d^2 Wl

du^2

+

2

u

dWl

du

+

(

1 −

l(l+1)

u^2

)

Wl= 0 (11.133)

whichisknown as theSpherical Bessel Equation. Solutionsof this equation,
whicharefiniteanddifferentiableatr = 0 arethespherical Bessel functions
Wl(u)=jl(u),thefirstfewofwhicharelistedbelow:

j 0 (u) =

sin(u)

u

j 1 (u) =

sin(u)

u^2

−

cos(u)

u

j 2 (u) =

( 3

u^3

−

1

u

)

sin(u)−

3

u^2

cos(u) (11.134)

Puttingeverythingtogether,theeigenstatesofenergyandangularmomentum

{H ̃, L ̃^2 ,L ̃z} (11.135)

forV = 0 are
φklm(r,θ,φ)=jl(kr)Ylm(θ,φ) (11.136)

witheigenvalues

Ek=

̄h^2 k^2

2 m

L^2 =l(l+1) ̄h^2 Lz=m ̄h (11.137)

Sincethereisoneandonlyoneeigenstatecorrespondingtoagivensetofeigenvalues
of{H ̃, L ̃^2 ,L ̃z},itfollowsthatE, L^2 , Lzisacompletesetofobservables.