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+
M
ˆ
u
.
(a) A vector
~v
i
sro
ta
tedinanin
f
nitesimal rotation
A
ˆ
u
(
)in
tothev
ec
to
r
+
(ˆ
u
×
)
,
Thus, the
transformation law can be written
→
0
=
+
M
ˆ
u
=
+
u
×
)
.
Equating the
i
th component
of this vector equation we deduce that
X
k
M
ik
v
k
=
X
j,k
ε
ijk
u
j
v
k
and hence that
M
ik
=
X
j
ε
ijk
u
j
Thus,
M
ii
=0
,
and
M
12
=
ε
132
u
3
=
−
u
z
=
−
M
21
M
13
=
ε
123
u
2
=+
u
y
=
−
M
31
M
23
=
ε
213
u
1
=
−
u
x
=
−
M
32
,
which gives, explicitly
M
=
⎛
⎝
0
−
u
z
u
y
u
z
0
−
u
x
−
u
y
u
x
0
⎞
⎠
=
3
X
i
=1
u
i
M
i
=
u
x
M
x
+
u
y
M
y
+
u
z
M
z
where
M
x
=
⎛
⎝
00 0
00
−
1
01 0
⎞
⎠
M
y
=
⎛
⎝
001
000
−
100
⎞
⎠
M
z
=
⎛
⎝
0
−
10
⎞
⎠
.
(b) De
f
ne the operators/matrices
J
k
=
iM
k
(
k
=1
,
2
,
3)
,
and show that as operators on
R
3
they obey
angular momentum commutation rules.
Introducing the matrices
J
i
=
iM
i
J
x
=
⎛
⎝
−
i
0
i
0
⎞
⎠
J
y
=
⎛
⎝
i
−
i
⎞
⎠
J
z
=
⎛
⎝
0
−
i
0
i
⎞
⎠
.
we
f
nd that [
J
x
,J
y
]isequa
lto
⎛
⎝
−
i
0
i
0
⎞
⎠
⎛
⎝
i
−
i
⎞
⎠
−
⎛
⎝
i
−
i
⎞
⎠
⎛
⎝
−
i
0
i
0
⎞
⎠
=
⎛
⎝
010
−
⎞
⎠
the right hand side of which is
−
M
z
=
i
(
iM
z
)=
iJ
z
.
In a similar fashion we
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 Fall '10
 PaulE.
 mechanics, Angular Momentum, Trigraph, xj, Xj Xk, Vj Wk, Wj Ji

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