K.1 The Schrödinger Equation
531
where ψ ε = ψ ε (ξ, η, φ). We see that the Schrödinger equation depends on two
quantities: the energy, ε, and the applied force, λ 0 .
Equation (K.5) can be solved by separation of variables. In Eq. (K.5), let
ψ ε (ξ, η, φ) = f 1 (ξ )f 2 (η)e
imφ ,
(K.6)
where m is the azimuthal quantum number and takes integer values to ensure that
ψ ε (ξ, η, φ) is single-valued on the interval 0 ≤ φ ≤ 2π . Also, multiply Eq. (K.5)
by
1
4 (ξ + η). Then Eq. (K.5) separates into two ordinary differential equations,
d
dξ
ξ
df 1
dξ
+
1
2
εξ −
m 2
4ξ
−
1
4
λ 0 ξ
2
+ β 1
f 1 = 0
( K . 7 )
and
d
dη
η
df 2
dη
+
1
2
εη −
m 2
4η
−
1
4
λ 0 η
2
+ β 2
f 2 = 0,
(K.8)
where β 1 + β 2 = 1.
K.1.2 Solution for λ 0 = 0
For the case λ 0 = 0, we know that for bound state motion (ε < 0) the energy is
ε = −
1
2n 2 , where n is the principal quantum number. The quantum number n takes
integer values in the interval 1 ≤ n ≤ ∞. We will introduce two new coordinates,
ρ 1 =
ξ
n and ρ 2 =
η
n , and let β i n = n i +
1
2 (|m| + 1) for i = 1, 2. Substituting into
Eqs. (K.7) and (K.8), we obtain
df 2
i
dρ 2
i
+
1
ρ i
df i
dρ i
+
−
1
4
+
1
ρ i
n i +
1
2
(|m| + 1)
−
m 2
4ρ 2
i
−
n 3 λ 0 ρ i
4
f i = 0,
(K.9)
for i = 1, 2. It is fairly easy to solve Eq. (K.9). Let
f i (ρ i ) = exp
−
1
2
ρ i
ρ
|m|/2
i
w i (ρ i )
(K.10)
and substitute into Eq. (K.9). We then obtain the confluent hypergeometric equation
(also called Kummer’s equation)
ρ i
d 2 w i
dρ 2
i
+ (|m| + 1 − ρ i )
dw i
dρ i
+ n i w i = 0,
(K.11)