166
3 Quantum Mechanics – II
Following the same procedure
g(ϕ)
sin θ
d
dθ
sin θ
d f (θ )
dθ
+
f (θ )
sin
2
θ
d
2 g(ϕ)
dϕ 2 + λ f (θ )g(ϕ) = 0
sin θ
f (θ )
d
dθ
sin θ
d f (θ )
dθ
+ λ sin
2
θ =
−1
g(ϕ)
d
2 g(ϕ)
dϕ 2 = m
2
(7)
where m
2 is a positive constant
d
2 g
dϕ 2 = −m
2
ϕ
(8)
gives the normalized function
g = (1/
√
2π) e
imϕ
(9)
m is an integer since g(ϕ + 2π) = g(ϕ)
Dividing (6) by sin
2
θ and multiplying by f , and rearranging
1
sin θ
d
dθ
sin θ
d f
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)
and l is an integer.
With the change of variable ψ r (r ) = χ(r )/r
The first term in (5) becomes
d
dr
r
2 ψ r
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 of λ from (11), (5) becomes upon rearrangement
−
2
2m
d
2
χ
dr 2 +
V (r ) +
l(l + 1)
2
2mr 2
χ = Eχ
(12)
Thus, the radial motion is similar to one dimensional motion of a particle
in a potential
V e = V (r ) +
l(l + 1)
2
2mr 2
(13)
where V e is the effective potential. The additional “potential energy” is
interpreted to arise physically from the angular momentum. A classical
particle that has angular momentum L about the axis through the origin
perpendicular to the plane of its path has the angular velocity ω = L/mr
2
where its radial distance from the origin is r . An inward force mω
2 r =
mL
2
/ωr
3 is required to keep the particle in the path. This “centripetal