CHAPTER 12
THE HYDROGEN ATOM
241
Now let us use the separation of variables again. Define:
Y(θ,φ) = Θ(θ)Φ(φ)
(db12.14)
After substituting this into the equation db 12.13 multiply by:
(db12.15)
This yields:
(db12.16)
These two equations can now be separated into:
(db12.17)
and
(db12.18)
The solutions to the equation for φ should be familiar to you already, as the equation is the
same as for the particle in a box. For this case, let us use exponential terms:
Φ(φ) = Ae
im l φ
(db12.19)
You can check that this is correct by substitution. Note that we needed to include i because
of the negative sign in the equation. The normalization constant is determined using the
condition that integration over every value of φ should yield a value of 1:
(db12.20)
Thus, the constant is:
(db12.21)
A =
1
2π
1
0
2
2
0
2
( ) ( )
=
=
=
∫
∫
−
Φ
Φ
*
d
d
π
φ
π
φ
φ φ φ
φ
A e e
im
im
l
l
A
A
2
0
2
2 2
dφ
π
π
∫ =
sin
sin
( )
(
)sin
θ θ
θ
θ
θ
θ
d
d
d
d
Θ
⎡
⎣
⎢
⎢
⎤
⎦
⎥
⎥
+
+
l l 1
2
− −
⎡
⎣
⎤
⎦
=
( )
m l
2
0
Θ θ
1
2
2
2
2
2
2
Φ
Φ
Φ
Φ
( )
( )
( )
( )
φ φ
φ
φ
φ
φ
d
d
or
d
d
= −
= −
m
m
l
l
1
2
2
Φ
Φ
Θ
( )
( )
sin
( )
sin
φ
δ
δφ
φ
θ
θ
δ
δθ
θ
δ
δθ
⎛
⎝
⎜ ⎜
⎞
⎠
⎟ ⎟ +
Θ Θ( )
(
)sin
θ
θ
⎛
⎝
⎜ ⎜
⎞
⎠
⎟ ⎟ +
+
⎡
⎣
⎢
⎢
⎤
⎦
⎥
⎥
=
l l 1
0
2
sin
( ) ( )
2
θ
θ
φ
Θ Φ
9781405124362_4_012.qxd 4/30/08 20:24 Page 241