10.13 (a)
(b)
(c)
(d) The first two electrons will be in the (1,1) level, which is equal
to:
The two lowest-energy states are (n 1 ,n 2 ) equal to (1,1) and then
(1,2). The longest wavelength transition is from the (1,1) state to
the (1,2) and the change in energy is:
= 7.9 × 10
−20 J
10.14 (a) The wavefunction is zero for x ≤ −2a; x ≥ ±3a; y ≤ −b; y ≥ ±2b
(b)
(c)
10.15 −
+
+
Z
2
2
2
2
2
2
2
2m x
x y z
y
x y z
z
∂
∂
ψ
∂
∂
ψ
∂
∂
ψ
(
)
(
)
, ,
, ,
( (
)
(
)
x y z
E x y z
, ,
, ,
⎛
⎝
⎜ ⎜
⎞
⎠
⎟ ⎟ = ψ
ψ
π
( , )
sin
(
)
x y
a
n x
a
a
x
=
+
⎛
⎝
⎜ ⎜
⎞
⎠
⎟ ⎟
⎛
⎝
⎜
⎜
⎞
⎠
⎟
2
5
2
5
⎟ ⎟
+
⎛
⎝
⎜ ⎜
⎞
⎠
⎟ ⎟
⎛
⎝
⎜
⎜
⎞
⎠
⎟
⎟
2
3
3
b
ny y b
b
sin
(
)
π
−
+
⎛
⎝
⎜ ⎜
⎞
⎠
⎟ ⎟ =
Z
2
2
2
2
2
2m x
x y
y
x y
E
∂
∂
ψ
∂
∂
ψ
ψ
( )
( )
,
,
(
( )
x y
,
λ
( .
)(
)
.
=
=
×
×
−
−
hc
E
Δ
6 6 10
3 10
7 9
34
8
1
Js
m s
× ×
=
×
−
−
.
10
2 5 10
20
6
J
m
ΔE
( .
)
( .
) (
=
×
×
−
−
−
6 6 10
8 9 1 10
1
10
34
2
31
9
Js
kg
m m)
.
.
2
2
2
2
1 5
1
1
1 5
⎛
⎝
⎜ ⎜
⎞
⎠
⎟ ⎟ +
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
−
⎛
⎝
⎜ ⎜
⎞ ⎞
⎠
⎟ ⎟ +
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎧
⎨
⎪
⎩
⎪
⎫
⎬
⎪
⎭
⎪
2
2
1
E
h
m
n
L
n
L
g =
⎛
⎝
⎜ ⎜
⎞
⎠
⎟ ⎟ +
⎛
⎝
⎜ ⎜
⎞
⎠
⎟ ⎟
⎡
⎣
⎢
⎢
⎢
2
1
1
2
2
2
2
8
⎤ ⎤
⎦
⎥
⎥
⎥
=
⎛
⎝
⎜ ⎜
⎞
⎠
⎟ ⎟
+
=
(
)
h
m
n
n
E
x
y
2
2
2
2
8
1
20
4
5
Å
0 0
=
×
×
×
−
−
( .
)
( .
)
6 63 10
8 9 11 10
1
10 1
34
2
31
Js
kg
0 0
1
20 10
10
2
10
2
−
−
⎛
⎝
⎜ ⎜
⎞
⎠
⎟ ⎟ +
×
⎛
⎝
⎜ ⎜
⎞
⎠
⎟ ⎟
⎡
⎣
⎢
⎢
m
m
⎢ ⎢
⎤
⎦
⎥
⎥
⎥
E
h
m
n
L
n
L
=
⎛
⎝
⎜ ⎜
⎞
⎠
⎟ ⎟ +
⎛
⎝
⎜ ⎜
⎞
⎠
⎟ ⎟
⎡
⎣
⎢
⎢
⎢
⎤
2
1
1
2
2
2
2
8
⎦ ⎦
⎥
⎥
⎥
ψ
π
( )
( ) ( )
sin
x y
X x Y y
x
,
Å
Å
=
=
⎛
⎝
⎜ ⎜
⎞
⎠
⎟ ⎟
⎛
⎝
⎜ 2
10
10
⎜ ⎜
⎞
⎠
⎟
⎟
⎛
⎝
⎜ ⎜
⎞
⎠
⎟ ⎟
⎛
⎝
⎜
⎜
⎞
⎠
⎟
⎟
2
20
20
Å
Å
sin
π y
−
+
⎛
⎝
⎜ ⎜
⎞
⎠
⎟ ⎟ =
Z
2
2
2
2
2
2m x
x y
y
x y
E
∂
∂
ψ
∂
∂
ψ
ψ
( )
( )
,
,
(
( )
x y
,
ANSWERS TO PROBLEMS
461
9781405124362_5_end.qxd 4/29/08 9:17 Page 461