Physics 463  Test I:
Solve 4 of the following 6 problems
1.
Consider a quantum system in an eigenstate
j
¿; j; m
i
.of
J
2
and
J
z
.
(a) Compute for this state the mean values
h
J
x
i
;
h
J
y
i
;
h
J
z
i
;
h
J
2
x
i
;
h
J
2
y
i
;
and
h
J
2
z
i
:
(b) Use the results to compute the statistical uncertainties
¢
J
x
and
¢
J
y
associated with a measurement
of a component of angular momentum perpendicular to the
z
axis.[Recall
¢
A
=
p
h
A
2
i ¡ h
A
i
2
].
(c) If the system is in such an eigenstate of
J
2
and
J
z
;
is “the component of
~
J
perpendicular to the
z
axis”
a wellde…ned quanity?
2.
The wave function of a particle is given by
Ã
(
~
r
) =
A
h
a
2
+ (
x
+
iy
)
2
i
e
¡
ar
r
2
:
where
A
and
a
are real constants.
(a) Is
Ã
an eigenfunction of
L
2
?
If so, what is the eigenvalue? If not, what values can be obtained if
L
2
is
measured?
(b) Is
Ã
an eigenfunction of
L
z
?
If so what is the eigenvalue? If not, what values can be obtained if
L
z
is
measured?
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 Fall '10
 PaulE.
 Physics, mechanics, Angular Momentum, M0

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