166 3 Quantum Mechanics – II
Following the same procedure
g(φ)
sinθ
d
dθ
sinθ
df(θ)
dθ
+
f(θ)
sin^2 θ
d^2 g(φ)
dφ^2
+λf(θ)g(φ)= 0
sinθ
f(θ)
d
dθ
(
sinθ
df(θ)
dθ
)
+λsin^2 θ=
− 1
g(φ)
d^2 g(φ)
dφ^2
=m^2 (7)
wherem^2 is a positive constant
d^2 g
dφ^2
=−m^2 φ (8)
gives the normalized function
g=(1/
√
2 π)eimφ (9)
mis an integer sinceg(φ+ 2 π)=g(φ)
Dividing (6) by sin^2 θand multiplying byf, and rearranging
1
sinθ
d
dθ
(
sinθ
df
dθ
)
+
(
λ−
m^2
sin^2 θ
)
f= 0 (10)
(c) The physically accepted solution of (10) is Legendre polynomials when
λ=l(l+1) (11)
andlis an integer.
With the change of variableψr(r)=χ(r)/r
The first term in (5) becomes
d
dr
(
r^2
ψr
dr
)
=
d
dr
[
r^2
(
−
χ
r^2
+
1
r
dχ
dr
)]
=
d
dr
[
r
dχ
dr
−χ
]
=r
d^2 χ
dr^2
+
dχ
dr
−
dχ
dr
=r
d^2 χ
dr^2
With the substitution ofλfrom (11), (5) becomes upon rearrangement
(
−
^2
2 m
)
d^2 χ
dr^2
+
[
V(r)+
l(l+1)^2
2 mr^2
]
χ=Eχ (12)
Thus, the radial motion is similar to one dimensional motion of a particle
in a potential
Ve=V(r)+
l(l+1)^2
2 mr^2
(13)
whereVeis the effective potential. The additional “potential energy” is
interpreted to arise physically from the angular momentum. A classical
particle that has angular momentumLabout the axis through the origin
perpendicular to the plane of its path has the angular velocityω=L/mr^2
where its radial distance from the origin isr. An inward forcemω^2 r=
mL^2 /ωr^3 is required to keep the particle in the path. This “centripetal