Page 193 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

184 The hydrogen atom
3
Z
D 10 (r) ˆ 4 r e
2 ÿ2 Zr=a 0
a 0
3 2
1 Z 2 Zr
D 20 (r) ˆ r 2 ÿ e ÿ Zr=a 0 (6:66)
8 a 0 a 0
5
1 Z
4 ÿ Zr=a 0
D 21 (r) ˆ r e
24 a 0
Higher-order functions are readily determined from Table 6.1. The radial
distribution functions for the 1s, 2s, 2p, 3s, 3p, and 3d states are shown in
Figure 6.5.
The most probable value r mp of r for the 1s state is found by setting the
derivative of D 10 (r) equal to zero
3

dD 10 (r) Z Zr
ˆ 8 r 1 ÿ e ÿ2 Zr=a 0 ˆ 0
dr a 0 a 0
which gives
r mp ˆ a 0 =Z (6:67)

Thus, for the hydrogen atom (Z ˆ 1) the most probable distance of the electron
from the nucleus is equal to the radius of the ®rst Bohr orbit.
The radial distribution functions may be used to calculate expectation values
of functions of the radial variable r. For example, the average distance of the
electron from the nucleus for the 1s state is given by

1 Z 3 1 3a 0
3 ÿ2 Zr=a 0
hri 1s ˆ rD 10 (r)dr ˆ 4 r e dr ˆ (6:68)
0 a 0 0 2Z
where equations (A.26) and (A.28) were used to evaluate the integral. By the
same method, we ®nd
6a 0 5a 0
hri 2s ˆ , hri 2p ˆ
Z Z
The expectation values of powers and inverse powers of r for any arbitrary
state of the hydrogen-like atom are de®ned by

1 1
k k k 2 2
hr i nl ˆ r D nl (r)dr ˆ r [R nl (r)] r dr (6:69)
0 0
In Appendix H we show that these expectation values obey the recurrence
relation

k ‡ 1 k a 0 kÿ1 1 ÿ k 2 a 2 kÿ2
hr i nl ÿ (2k ‡ 1) hr i nl ‡ kl(l ‡ 1) ‡ 0 hr i nl ˆ 0
n 2 Z 4 Z 2
(6:70)