(c)
(d) The region beyond L/2 is classically forbidden and so the
probability of finding the particle in this region is very small,
although non-zero.
(e) The wavefunctions will become those for the simple harmonic
oscillator.
11.22 (a)
(b)
(c)
(d) F = −k(x n − x n−1 ) + k(x n+1 − x n )
(e)
CHAPTER 12
12.1
12.2 (a) The ground state is n = 1
(b) The first excited state is n = 2
12.3
12.4
2
0
2
1 058 10
0
0
10
.
−
=
=
=
×
−
r
a
r
a
so
m
ψ
π
200
0
3
0
2
1
4
1
2
2
0
/
=
−
⎛
⎝
⎜
⎜
⎞
⎠
⎟
⎟
−
a
r
a
e
r a
E
hcR h
.
= −
= −
3
1 5
2
eV
E
hcR h
.
= −
= −
2
3 4
2
eV
E
hcR h
.
= −
= −
1
13 6
2
eV
V r
e
r
( ) =
−
2
0
4πε
−
+
−
+
−
−
+
Z
2
2
2
1
2
1
2
2
2
m x
x
k x x
x
x
n
n
n
n
n
n
d
d
ψ ( )
[(
) (
) ] ] ( )
( )
ψ
ψ
n
n
x
E x
=
ψ
ψ
α
*
d
d f o rt h e
2
0
2
2
2
2
Å
Å
x
x x x
N e
x
∞
∞
−
∫
∫
=
( ) ( )
/
g ground state
−
+
=
Z
2
2
2
2
2
2
μ
ψ
ψ
ψ
d
dx
x
k x x
E x
( )
( )
( )
μ
ρ
ρ
( )( )
=
+
=
+
=
m m
m m
m
m
1 2
1
2
14 14
14 14
7
Δ
Δ
9
Z
.
( .
=
=
×
×
−
−
2
1 054 10
2 1 054 10
34
34
m x
Js
k kg
m
m s
)( .
)
.
0 5 10
1 0 10
11
11
1
×
=
×
−
−
466
ANSWERS TO PROBLEMS
9781405124362_5_end.qxd 4/29/08 9:17 Page 466