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28. We are looking for the values of the ratio
E
n
x
,n
y
,n
z
h
2
/
8
mL
2
=
L
2
Ã
n
2
x
L
2
x
+
n
2
y
L
2
y
+
n
2
z
L
2
z
!
=
(
n
2
x
+
n
2
y
+
n
2
z
)
and the corresponding diFerences.
(a) ±or
n
x
=
n
y
=
n
z
=1
,
the ratio becomes 1 + 1 + 1 = 3
.
00.
(b) ±or
n
x
=
n
y
=2and
n
z
=1
,
the ratio becomes 4 + 4 + 1 = 9
.
00. One can check (by computing
other (
n
x
,n
y
,n
z
) values) that this is the third lowest energy in the system. One can also check
that this same ratio is obtained for (
n
x
,n
y
,n
z
)=(2
,
1
,
2) and (1
,
2
,
2).
(c) ±or
n
x
=
n
y
=1and
n
z
=3
,
the ratio becomes 1 + 1 + 9 = 11
.
00. One can check (by computing
other (
n
x
,n
y
,n
z
) values) that this is three “steps” up from the lowest energy in the system. One
can also check that this same ratio is obtained for (
n
x
,n
y
,n
z
)=(1
,
3
,
1) and (3
,
1
,
1). If we take
the diFerence between this and the result of part (b), we obtain 11
.
00
−
9
.
00 = 2
.
00.
(d) ±or
n
x
=
n
y
=1and
n
z
=2
,
the ratio becomes 1 + 1 + 4 = 6
.
00. One can check (by computing
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 Fall '08
 SPRUNGER
 Physics

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