bei48482_FM

214 Chapter Six

As and are normalized functions, the actual probability P(r) drof finding the elec-

tron in a hydrogen atom somewhere in the spherical shell between rand rdrfrom

the nucleus (Fig. 6.10) is

P(r) drr^2 R^2 dr



0

 ^2 sin d

2 

0

 ^2 d

r^2 R^2 dr (6.25)

Equation (6.25) is plotted in Fig. 6.11 for the same states whose radial functions R

were shown in Fig. 6.8. The curves are quite different as a rule. We note immediately

that Pis not a maximum at the nucleus for sstates, as Ritself is, but has its maximum

a definite distance from it.

The most probable value of rfor a 1selectron turns out to be exactly a 0 , the or-

bital radius of a ground-state electron in the Bohr model. However, the averagevalue

of rfor a 1selectron is 1.5a 0 , which is puzzling at first sight because the energy lev-

els are the same in both the quantum-mechanical and Bohr atomic models. This

apparent discrepancy is removed when we recall that the electron energy depends

upon 1 rrather than upon rdirectly, and the average value of 1 rfor a 1selectron

is exactly 1 a 0.

Example 6.2

Verify that the average value of 1 rfor a 1selectron in the hydrogen atom is 1 a 0.

Solution

The wave function of a 1selectron is, from Table 6.1,



er^ a^0



a 0

3   2

0

P(

r)

dr

=

(^2) r
|R
|nl
2 dr
5 a 0 10 a 0 15 a 0 20 a 0 25 a 0
r
1 s
2 p
2 s
3 d 3 p
3 s
Figure 6.11The probability of finding the electron in a hydrogen atom at a distance between rand
rdrfrom the nucleus for the quantum states of Fig. 6.8.
Nucleus
r
dr
Figure 6.10The probability of
finding the electron in a hydrogen
atom in the spherical shell be-
tween rand rdrfrom the nu-
cleus is P(r) dr.
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