26. We are looking for the values of the ratio
E
hm
L
L
n
L
n
L
n
L
nnn
x
x
y
y
z
z
xyz
,,
22
2
2
2
2
2
2
2
222
8
=+
+
F
H
G
I
K
J
+
di
and the corresponding differences.
(a) For
n
x
=
n
y
=
n
z
= 1, the ratio becomes 1 + 1 + 1 = 3.00.
(b) For
n
x
=
n
y
= 2 and
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) For
n
x
=
n
y
= 1 and
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 difference between this and the result of part (b), we obtain
11.0 – 9.00 = 2.00.
(d) For
n
x
=
n
y
= 1 and
n
z
= 2, the ratio becomes 1 + 1 + 4 = 6.00. One can check (by
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 Spring '08
 Any
 Physics

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