BioPHYSICAL chemistry

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 RHis the Rydberg constant.

ψθφnlm,,l(, , )rRrY==−nl,() ( , )mllθφwith En
mme
n

hcR

n

H

4

2

0

32 22 2^2

1

πεZ

=−

CHAPTER 12 THE HYDROGEN ATOM 245

Table 12.2

Wavefunctions for some of the lower-energy states of the hydrogen atom.

nl ml ψψ(r,θθ,ππ)

10 0

20 0

21 0

21 ± 1

30 0

31 0

31 ± 1

32 0

32 ± 1

32 ± 2

1

162

1

0

(^322)
0
2
(^30)
π a
r
a
era
⎛
⎝
⎜⎜
⎞
⎠
⎟⎟
⎛
⎝
⎜⎜
⎞
⎠
⎟⎟ −
/
/ (sin (^222) θ)e±iφ
1
81
1
0
(^322)
0
2
(^30)
π
θ
a
r
a
era
⎛
⎝
⎜⎜
⎞
⎠
⎟⎟
⎛
⎝
⎜⎜
⎞
⎠
⎟⎟ −
/
/ (sin ccos )θe±iφ
1
81 6
1
3
0
(^322)
0
2
(^30)
π a
r
a
era
⎛
⎝
⎜⎜
⎞
⎠
⎟⎟
⎛
⎝
⎜⎜
⎞
⎠
⎟⎟ −
/
/ (cos s) (^2) θ− 1
1
81
1
6
0
32
0
2
0
2
3
π a
r
a
r
a
er
⎛
⎝
⎜⎜
⎞
⎠
⎟⎟ −
⎛
⎝
⎜⎜
⎞
⎠
⎟⎟−
/
/aa (^0) (sin )θe±iφ
1
81
21
6
0
32
0
2
0
π a^2
r
a
r
a
er
⎛
⎝
⎜⎜
⎞
⎠
⎟⎟ −
⎛
⎝
⎜⎜
⎞
⎠
⎟⎟ −
/
/33a (^0) (cos )θ
1
81 3
1
27 18 2
0
32
0
2
0
π a^2
r
a
r
a
⎛
⎝
⎜⎜
⎞
⎠
⎟⎟ −+
⎛
⎝
⎜⎜
/ ⎞
⎠⎠
⎟⎟e−ra/3^0
1
8
1
0
32
0
(^20)
π
θ
a
r
a
eera
⎛
⎝
⎜⎜
⎞
⎠
⎟⎟
⎛
⎝
⎜⎜
⎞
⎠
⎟⎟ − ±
/
/ (sin ) iiφ
1
42
1
0
32
0
(^20)
π
θ
a
r
a
era
⎛
⎝
⎜⎜
⎞
⎠
⎟⎟
⎛
⎝
⎜⎜
⎞
⎠
⎟⎟ −
/
/ (cos )
1
42
1
2
0
32
0
(^20)
π a
r
a
era
⎛
⎝
⎜⎜
⎞
⎠
⎟⎟ −
⎛
⎝
⎜⎜
⎞
⎠
⎟⎟ −
/
/
11
0
32
0
π a
era
⎛
⎝
⎜⎜
⎞
⎠
⎟⎟ −
/
/