a certain point in space in ψ*ψ dτ that exponentially
decreases as the radius increases (Figure 12.3).
We normally want to know the probability of finding the electron at a certain radius, not at a specific
point, so we can average over the volume of a thin
radial shell, giving the probability:
ψ*ψ(4πr
2
) dr
(12.16)
The probability at a given radius has a very different dependence on the radius (Figure 12.4). For
small values of the radius, the probability increases
approximately as r
2 until the exponential term
becomes dominant. As a result the function reaches
a maximum value at an intermediate radius.
The most probable radial position of the electron
is simply the peak position of this term. We can
find the peak by finding the value for which the
derivative is zero:
(12.17)
For the ground state this can be calculated to give:
(12.18)
(12.19)
What is the probability of finding the electron
within this radius? We can calculate it.
(12.20)
To solve this let x = r/a 0 .
(12.21)
4
4
0
3
0
2
0
1
2
0
2 2
0
1
a
a x e a x
x e
x
x
x
( )
(
)
∫
∫
−
−
=
d
d
ψ ψ τ
π
π
* d
d
(
)
/
=
=
∫
∫
−
1
4
4
0
3
0
0
2
2
0
3
0
0
0
a
e
r r
a
a
a
r a
r r e
r
r a
a
2 2
0
0
0
−
∫
/ d
0
8
1
0
3
2
0
0
0
/
=
−
⎛
⎝
⎜ ⎜
⎞
⎠
⎟ ⎟
=
−
a
e
r
r
a
r a
r a
or
0
4
1
4
2
0
3
2
0
3
0
(
/
=
⎛
⎝
⎜ ⎜
⎞
⎠
⎟ ⎟ =
−
d
d
d
d
r
r
a
e
a r
r
r a
π π
2 2 2 0
e
r a
− / )
0
4
2
(
)
=
d
d
*
r
r
π ψ ψ
CHAPTER 12
THE HYDROGEN ATOM
249
Probability
ψ*ψδτ
r
r0
r0
Figure 12.3 The dependence of the
probability of the 1s state at a point
located at a radius r.
Radial probability (4πr
2
)ψ*ψ
4
2
0
0
0.2
0.4
0.6
r/a 0
Figure 12.4 The radial distribution of the
probability of finding 1s state within a
shell of radius r.
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