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
=−
⎛
⎝
⎜⎜
⎞
⎠
⎟⎟
⎛
⎝
⎜⎜
⎞
⎠
⎟⎟
⎡
⎣
⎢
⎢
⎤
⎦
(^21) sin ⎥
2
2
L 2
x
x
L
π L
π⎥⎥
=+ =
l
l
/
/
.
4
34
1
2
1
082
π
ψψ
π
*d() () sin
/
/
/
xxx
L
x
l L
l
4 l
34
2
4
3
2
∫ =
⎛
⎝
⎜⎜
⎞
⎠
⎟⎟
ll
x
/4
∫ d
E
mL
h
mL
=
⎛
⎝
⎜⎜
⎞
⎠
⎟⎟ =
Z^2
(^222)
282
ππ
−=−−
⎛
⎝
⎜⎜
⎞
⎠
⎟⎟
⎛
⎝
ZZ⎜
22
2
2 2
22
2
mx
x
mLL
x
L
d
d
ψ
ππ
() ⎜⎜ sin
⎞
⎠
⎟
⎟=
⎛
⎝
⎜⎜
⎞
⎠
⎟⎟ ()= ()
Z^2
2
2 mL
xEx
π
ψψ
d
d
2
2
2
22
xL
x
LLL
x
sin sin
⎛ πππ
⎝
⎜
⎜
⎞
⎠
⎟
⎟
=−
⎛
⎝
⎜⎜
⎞
⎠
⎟⎟
LL
ψ
π
1
2
()x sin
L
x
L
=
⎛
⎝
⎜⎜
⎞
⎠
⎟⎟
−=
Z^22
2 mx^2 xEx
d
d
ψψ() ()
ψψ
π
*d() () sin
/
xx x x
L
x
nx
L
LL
2
0
4
22
0
2
=
⎛
⎝
⎜⎜
⎞
⎠
∫ ⎟⎟
//4
∫ dx
ψψ
π
*d() () sin
//
xxx
L
nx
L
LL
=
⎛
⎝
⎜⎜
⎞
⎠
∫∫⎟⎟
2
0
4
2
0
4
ddx
2
0
2
L
x
Lix
x
L
x
L
sin sin
ππZ d
d
d
⎛
⎝
⎜⎜
⎞
⎠
∫ ⎟⎟
p
L
x
Lix
x
L
L
=
⎛
⎝
⎜⎜
⎞
⎠
⎟⎟
⎛
⎝
⎜⎜
⎞
⎠
⎟⎟
⎛
⎝
∫ ⎜
2
0
sin sin
π∂
∂
Z π
⎜⎜
⎞
⎠
⎟⎟dx
λ
()
== =
228
3
16
3
L
n
Å
Å
λ
()
== =
228
1
16
L
n
Å
Å
2
2
L
ANSWERS TO PROBLEMS 459
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