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.