Test II  Physics 463
This is an open book, open note test. You must turn it in, in the Physics department front o
ﬃ
ce, by 4:00PM
one day after you pick it up, or by 4:00PM on Friday at the latest. Work all problems in this test, starting each
problem on a separate piece of paper. Number your pages individually for each problem, organize them in their
original sequence, and staple the results together before turning them in.
1.
The wave function of a particle is given by
ψ
(
~
r
)=
A
h
r
2
+(
x
+
iy
)
2
i
e
−
ar
r
2
,
where
A
and
a
are real constants.
(a) It turns out that
ψ
is not an eigenfunction of
L
2
.
What values can be obtained if
L
2
is measured, and
with what probabilities will they be obtained?
(b) What values can be obtained if
L
z
is measured, and with what probabilities will they be obtained?
[Hint:
ψ
can be expressed in terms of spherical harmonics.]
2.
Let
{
V
x
,V
y
,V
z
}
=
{
V
1
,V
2
,V
3
}
be the Cartesian components of a Hermitian vector operator
~
V
of a quantum
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 Fall '10
 PaulE.
 mechanics, Angular Momentum, Work, wave function, single particle

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