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Multiple choice
Applying the lowering operator to
∣
1
,
1
⟩
|1,1\rangle
∣1
,
1
⟩
gives
S
−
∣
1
,
1
⟩
=
j
(
j
+
1
)
−
m
(
m
−
1
)
∣
1
,
0
⟩
S_- |1,1\rangle = \sqrt{j(j+1) - m(m-1)} |1,0\rangle
S
−
∣1
,
1
⟩
=
j
(
j
+
1
)
−
m
(
m
−
1
)
∣1
,
0
⟩
with
j
=
1
j=1
j
=
1
,
m
=
1
m=1
m
=
1
, so the normalization is
N
=
2
N = \sqrt{2}
N
=
2
. The state
∣
1
,
0
⟩
=
(
1
/
N
)
(
∣
d
o
w
n
,
u
p
⟩
+
∣
u
p
,
d
o
w
n
⟩
)
|1,0\rangle = (1/N)(|down,up\rangle + |up,down\rangle)
∣1
,
0
⟩
=
(
1/
N
)
(
∣
d
o
w
n
,
u
p
⟩
+
∣
u
p
,
d
o
w
n
⟩)
. What is the Clebsch-Gordan coefficient of
∣
u
p
,
d
o
w
n
⟩
|up,down\rangle
∣
u
p
,
d
o
w
n
⟩
in
∣
1
,
0
⟩
|1,0\rangle
∣1
,
0
⟩
?
1
/
3
≈
0.577
1/\sqrt{3} \approx 0.577
1/
3
≈
0.577
1/2
1
/
2
≈
0.707
1/\sqrt{2} \approx 0.707
1/
2
≈
0.707
2
≈
1.414
\sqrt{2} \approx 1.414
2
≈
1.414
Check answer