11.19
11.20
ψ 1 (y) =N 1 (2y)e−y
(^2) /2
11.21 (a)
(b)V(x) is continuous at L/2.
a
L
b
L
cL
ab
c
22
2
48 48
+
⎛
⎝
⎜⎜
⎞
⎠
⎟⎟=+=or
−+=≥
Z^22
2
2
2
2
mx
xcL xExxL
d
d
ψψψ() () () /
−++=≤
Z^22
2
23
2 mx
xax bx xExx
d
d
ψψψ() ( ) () () L/2
E
mm
mk k
(^1) m
2
2
2
2
12
3
2
3
2
3
2
/
==
⎛
⎝
⎜⎜
⎞
⎠
⎟⎟ =
ZZ
Z
Z
α
⎛⎛
⎝
⎜⎜
⎞
⎠
⎟⎟ =
12
3
2
/
Zω
−+=−
⎛
⎝
⎜⎜
⎞
⎠
⎟⎟+=
ZZ
Z
2
4
2
222 m^220
k
m
mk k
α
ψ
(^1) αα
2
2
4
2
2 2 1
3
2
()xx
m
k
m
−+ E
⎛
⎝
⎜⎜
⎞
⎠
⎟⎟+−
⎛
⎝
ZZ
⎜⎜⎜
⎞
⎠
⎟⎟
⎡
⎣
⎢
⎢
⎤
⎦
⎥
⎥
= 0
−−
⎛
⎝
⎜⎜
⎞
⎠
⎟⎟
⎛
⎝
⎜
⎜
⎞
⎠
⎟
⎟+
Z^2
1
2
42
2
2
23
m
x
xkx
ψ
αα
()
22
2
xxExψψ 111 ()= ()
=−
⎛
⎝
⎜⎜
⎞
⎠
− / ⎟⎟= ()
223 2 1 2 3
421
Nex 22 xx x
ααα
α ψ
xx^2
42
3
αα
−
⎛
⎝
⎜⎜
⎞
⎠
⎟⎟
=−
⎛
⎝
⎜⎜
⎞
⎠
⎟⎟+−
⎛
⎝
⎜⎜
⎞
⎠
− / ⎟⎟
(^2212)
2
2
2
N 22
e
x xx
ααα
α ⎛−−
⎝
⎜⎜
⎞
⎠
⎟⎟ +−
⎛
⎝
⎜⎜
⎞
⎠
−−⎟⎟
2
2
2
2
x exx 22 x e (^22)
αα
//α αα^2
⎛
⎝
⎜
⎜
⎞
⎠
⎟
⎟
d
d
d
d
2
2 1
(^12)
2
2
(^222)
x
x
N
x
e
x x
ψ
αα
()=−/α ⎛
⎝
⎜⎜
⎞
⎠
− ⎟⎟+e−x/
⎛
⎝
⎜⎜
⎞
⎠
(^222) α⎟⎟
d
d
d
x d
x
N
x
xe
N
ψ xxxe
αα
αα
1
()==^221 −−^22 //^21222 ⎛−−
⎝
⎜⎜
⎞
⎠
⎟⎟+
⎛
⎝
⎜
⎜
⎞
⎠
⎟
⎟
x e−x
α
α
2
(^22) / 2
−+=
Z^22
2
2
22 mx
x
k
xx Ex
d
d
ψψψ() () ()
2 ()Ne 0 x^22 /^22 x
A
−
∞
∫
α d
ANSWERS TO PROBLEMS 465
9781405124362_5_end.qxd 4/29/08 9:17 Page 465