14.6 The hydrogen atom 403
To separate the angular variables we now write
Y(θ, φ) 1 = 1 Θ(θ) 1 × 1 Φ(φ) (14.57)
Substitution of this product and its derivatives in the angular equation, division by
Y 1 = 1 ΘΦ, and multiplication by sin
21 θgives
(14.58)
so that, with separation constant−m
2(14.59)
(14.60)
The separation of the variables is complete. It is now only necessary to solve the three
boundary value problems represented by equation (14.55), (14.59), and (14.60), with
appropriate boundary conditions.
The Φequation
(14.59)
The function Φis defined in the interval 01 ≤ 1 φ 1 ≤ 12 πand must satisfy the condition
Φ(2π) 1 = 1 Φ(0) (14.61)
for continuity round the circle. We therefore have the same boundary value problem
as that discussed in Section 12.7 for the particle in a ring and in 14.5 for angular
motion of the particle in a circular box. The normalized solutions are
m 1 = 1 0, ±1, ±2, = (14.62)
or, in real form (equations (12.71)),
(14.63)
form 1 = 1 0, 1, 2, =. The functions form an orthonormal set, with property
(14.64)
Z
02 πΦΦ
mm mm*( ) ( )φφφδd
′ , ′=
1
2
1
2
1
2
1
2
()cos ()sinΦΦ ΦΦ
mm mmm
i
+= , −= m
−−ππ
φφ
Φ
mim()φ e
φ=,
1
2 π
d
d
m
222Φ
Φ
()
()
φ
φ
=− φ
1
1
22sin
sin ( )
sin
θθ
θ
θ
θd
d
d
d
ll
Θ m
++−
Θ= 0
d
d
m
222Φ
Φ
()
()
φ
φ
=− φ
sin
sin ( ) sin
θ
θ
θ
θ
θ
Θ
d Θ
d
d
d
ll
++
1 +
211
0
22Φ
dΦ
dφ
=