20. If
j
1
≥
j
2
show that
j
1
+
j
2
X
j
=
j
1
−
j
2
(2
j
+1)=(2
j
1
+1)(2
j
2
+1)
.
21. Consider three particles of spin
1
2
,
and let
~
S
1
,
~
S
2
,
and
~
S
3
denote the corresponding spin operators. In the
combined spin space associatedw
iththetota
lsp
inoperator
~
S
=
~
S
1
+
~
S
2
+
~
S
3
,
what irreducible invariant
subspaces
S
(
τ
,s
) occur, and how many spaces occur for each value of
s
.[Hint:
f
rst determine the invariant
subspaces obtained by combining any two of the spins together, and then combine the third spin with each
of the subspaces of the two previously coupled spins. Make sure the total number of
f
nal basis states

τ
,s,m
i
is the same as the total number of initial basis states

m
1
,m
2
,m
3
i
.
Note: you do not actually have
to produce any of the vectors

τ
,s,m
i
to solve this problem.]
22. Consider three particles of spin
1
2
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 Fall '10
 PaulE.
 mechanics, Angular Momentum, Energy, Work, Photon, Fundamental physics concepts, ground state

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