than one nucleus? For the hydrogen atom, the solutions are given by
the product of the three terms, yielding the solutions listed in Table 12.2
with the form:
(12.3)
where R H is the Rydberg constant.
ψ
θ φ
θ φ
n l m
n l
l
m
n
l
l
r
R rY
E
, ,
,
( , , )
( ) ( , )
=
= −
with
m me
n
hcR
n
H
4
2
0
2 2
2
2
32
1
π ε Z
= −
CHAPTER 12
THE HYDROGEN ATOM
245
Table 12.2
Wavefunctions for some of the lower-energy states of the hydrogen atom.
n
l
m l
ψ ψ(r,θ θ,π π)
1
0
0
2
0
0
2
1
0
2
1
±1
3
0
0
3
1
0
3
1
±1
3
2
0
3
2
±1
3
2
±2
1
162
1
0
3 2
2
0
2
3 0
π a
r
a
e
r a
⎛
⎝
⎜ ⎜
⎞
⎠
⎟ ⎟
⎛
⎝
⎜ ⎜
⎞
⎠
⎟ ⎟
−
/
/ (sin
2 2
2
θ
φ
)e
i
±
1
81
1
0
3 2
2
0
2
3 0
π
θ
a
r
a
e
r a
⎛
⎝
⎜ ⎜
⎞
⎠
⎟ ⎟
⎛
⎝
⎜ ⎜
⎞
⎠
⎟ ⎟
−
/
/ (sin c cos )
θ
φ
e
i
±
1
81 6
1
3
0
3 2
2
0
2
3 0
π a
r
a
e
r a
⎛
⎝
⎜ ⎜
⎞
⎠
⎟ ⎟
⎛
⎝
⎜ ⎜
⎞
⎠
⎟ ⎟
−
/
/ ( cos s
)
2
1
θ −
1
81
1
6
0
3 2
0
2
0
2
3
π a
r
a
r
a
e
r
⎛
⎝
⎜ ⎜
⎞
⎠
⎟ ⎟
−
⎛
⎝
⎜ ⎜
⎞
⎠
⎟ ⎟
−
/
/ a a
i
e
0
(sin )
θ
φ
±
1
81
2 1
6
0
3 2
0
2
0
2
π a
r
a
r
a
e
r
⎛
⎝
⎜ ⎜
⎞
⎠
⎟ ⎟
−
⎛
⎝
⎜ ⎜
⎞
⎠
⎟ ⎟
−
/
/3 3 0
a (cos )
θ
1
81 3
1
27 18
2
0
3 2
0
2
0
2
π a
r
a
r
a
⎛
⎝
⎜ ⎜
⎞
⎠
⎟ ⎟
−
+
⎛
⎝
⎜ ⎜
⎞
/
⎠ ⎠
⎟ ⎟
−
e
r a
/3 0
1
8
1
0
3 2
0
2 0
π
θ
a
r
a
e
e
r a
⎛
⎝
⎜ ⎜
⎞
⎠
⎟ ⎟
⎛
⎝
⎜ ⎜
⎞
⎠
⎟ ⎟
−
±
/
/ (sin )
i iφ
1
4 2
1
0
3 2
0
2 0
π
θ
a
r
a
e
r a
⎛
⎝
⎜ ⎜
⎞
⎠
⎟ ⎟
⎛
⎝
⎜ ⎜
⎞
⎠
⎟ ⎟
−
/
/ (cos )
1
4 2
1
2
0
3 2
0
2 0
π a
r
a
e
r a
⎛
⎝
⎜ ⎜
⎞
⎠
⎟ ⎟
−
⎛
⎝
⎜ ⎜
⎞
⎠
⎟ ⎟
−
/
/
1 1
0
3 2
0
π a
e
r a
⎛
⎝
⎜ ⎜
⎞
⎠
⎟ ⎟
−
/
/
9781405124362_4_012.qxd 4/30/08 20:25 Page 245