CHAPTER 10
10.1 For the first excited state, n = 2 so λ =
= L.
10.2
10.3
10.4
10.5
10.6
10.7
=
−
⎛
⎝
⎜ ⎜
⎞
⎠
⎟ ⎟
⎛
⎝
⎜ ⎜
⎞
⎠
⎟ ⎟
⎡
⎣
⎢
⎢
⎤
⎦
⎥
sin
2 1
2
2
2
L
x
x
L
L
π
π ⎥ ⎥
= + =
l
l
/
/
.
4
3 4
1
2
1 0 82
π
ψ
ψ
π
*
d
( ) ( )
sin
/
/
/
x
x x
L
x
L
l
l
l
4
3 4
2
4
3
2
∫
=
⎛
⎝
⎜ ⎜
⎞
⎠
⎟ ⎟
l l
x
/4
∫
d
E
m L
h
mL
=
⎛
⎝
⎜ ⎜
⎞
⎠
⎟ ⎟ =
Z
2
2
2 2
2
2
8
π
π
−
= −
−
⎛
⎝
⎜ ⎜
⎞
⎠
⎟ ⎟
⎛
⎝
⎜
Z
Z
2
2
2
2
2
2
2
2
m x
x
m
L L
x
L
d
d
ψ
π
π
( )
sin
⎜ ⎜
⎞
⎠
⎟
⎟
=
⎛
⎝
⎜ ⎜
⎞
⎠
⎟ ⎟
=
( )
( )
Z
2
2
2m L
x
E x
π ψ
ψ
d
d
2
2
2
2
2
x
L
x
L
L L
x
sin
sin
π
π
π
⎛
⎝
⎜
⎜
⎞
⎠
⎟
⎟
= −
⎛
⎝
⎜ ⎜
⎞
⎠
⎟ ⎟
L L
ψ
π
1
2
( )
sin
x
L
x
L
=
⎛
⎝
⎜ ⎜
⎞
⎠
⎟ ⎟
−
=
Z
2
2
2
2m x
x
E x
d
d
ψ
ψ
( )
( )
ψ
ψ
π
*
d
( ) ( )
sin
/
x x x x
L
x
n x
L
L
L
2
0
4
2
2
0
2
=
⎛
⎝
⎜ ⎜
⎞
⎠
⎟ ⎟
∫
/ /4
∫
dx
ψ
ψ
π
*
d
( ) ( )
sin
/
/
x x x
L
n x
L
L
L
=
⎛
⎝
⎜ ⎜
⎞
⎠
⎟ ⎟
∫
∫
2
0
4
2
0
4
d dx
2
0
2
L
x
L i x
x
L
x
L
sin
sin
π
π
Z d
d
d
⎛
⎝
⎜ ⎜
⎞
⎠
⎟ ⎟
∫
p
L
x
L i x
x
L
L
=
⎛
⎝
⎜ ⎜
⎞
⎠
⎟ ⎟
⎛
⎝
⎜ ⎜
⎞
⎠
⎟ ⎟
⎛
⎝
⎜
∫
2
0
sin
sin
π
∂
∂
π
Z
⎜ ⎜
⎞
⎠
⎟ ⎟ dx
λ
( )
=
=
=
2
28
3
16
3
L
n
Å
Å
λ
( )
=
=
=
2
28
1
16
L
n
Å
Å
2
2
L
ANSWERS TO PROBLEMS
459
9781405124362_5_end.qxd 4/29/08 9:17 Page 459