12. Consider the 3
×
3 rotation operators
A
ˆ
u
(
α
) which rotate vectors in
R
3
.
For in
f
nitesimal rotations these
take the form
A
ˆ
u
(
δα
)=1+
δα
M
ˆ
u
.
(a) Usingthefactthatinanin
f
nitesimal rotation
A
ˆ
u
(
δα
) the vector
~v
is rotated into the vector
~v
+
δα
(ˆ
u
×
~v
)
,
construct the matrix representing
M
ˆ
u
in any standard coordinate system, and show that it can be
written in the form
M
ˆ
u
=
u
x
M
x
+
u
y
M
y
+
u
z
M
z
in terms of matrices
M
x
,M
y
,M
z
.
[Hint: this part is
done in the notes.]
(b) De
f
ne the operators/matrices
J
x
=
iM
x
,J
y
=
iM
y
,
and
J
z
=
iM
z
,
(these are operators on the space
of ordinary vectors in
R
3
). Show that these three matrices obey angular momentum commutation
rules.[It su
ﬃ
ces to compute [
J
x
,J
y
]
,
and to obtain the others by cyclic permutations.
(c) Determine the eigenvalue spectrum of any one of the matrices
J
x
,J
y
,J
z
.
13. Let
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 Fall '10
 PaulE.
 mechanics, Angular Momentum, Work, JZ, total angular momentum

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