This preview shows page 1. Sign up to view the full content.
(b) The extrema of
ψ
2
(
r
) for 0 <
r
<
∞
may be found by squaring the given function,
differentiating with respect to
r
, and setting the result equal to zero:
−
−
−
=
−
1
32
24
0
6
()
/
ra
a
e
π
which has roots at
r
= 2
a
and
r
= 4
a
. We can verify directly from the plot above that
r
=
4
a
is indeed a local maximum of
200
2
.
r
As discussed in part (a), the other root (
r
= 2
a
)
is a local minimum.
(c) Using Eq. 3943 and Eq. 3941, the radial probability is
Pr
r
r
r
a
r
a
e
200
2
200
2
2
3
2
4
8
2
.
/
==
−
F
H
G
I
K
J
−
π
(d) Let
x = r
/
a
. Then
2
2
/2
2
4
3
2
200
3
00
0
0
1
2
(2
)
(
4
4 )
88
1
[4! 4(3!) 4(2!)] 1
8
x
x
rr
P
r dr
e
dr
x
x e dx
x
x
x e dx
aa
∞∞
∞
∞
−−
−
⎛⎞
=−
=
−
+
⎜⎟
⎝⎠
=−+
=
∫∫
∫
∫
where we have used the integral formula
0
∞
−
z
=
xe dx n
nx
!
.
54. (a) The plot shown below for 
200
(
r
)
2
is to be compared with the dot plot of Fig.
3922. We note that the horizontal axis of our graph is labeled “
This is the end of the preview. Sign up
to
access the rest of the document.
 Spring '08
 Any
 Physics

Click to edit the document details