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Homework 3
1
11/1/2005
Homework III and Solutions
Problems:
III.1
Show that the rotation (in spin or isospin space)
U
(
θ
) = exp(
i
θ
·
I
) with
I
= (
I
1
,
I
2
,
I
3
) =
σ
i
/2 can be
written as:
()
cos
(/2
)
s
in
)
Ui
θ
⋅
=+
θσ
θ
1
Solution:
First, note that:
(
)
33
3
22
11
2 2
,1
1
1
3
2
1
2
32
1
1
21
3
3
13
2
1
1
jk j k
j
j
jk
j
j k
jk j
k
l l
ii
θθσσ
θ σ
θθ ε σ
θθσ
θθσ θθσ
==
≠
=
≠=
⋅=
+
+
=
=
+
−
+
−
+
−
=
∑∑
∑
∑
θ
1
θ
1
θ
1
Taylor expansion of the exponential gives:
1
00
0
2
2
1
!2
2 !2
2 1!2
1
1
2!2
2
2 1
1
1
2
1
kk
k
k
k
i
U
k
i
i
k
+
∞∞
∞
=
⎛⎞
=⋅
=
⋅
+
⋅
⎜⎟
+
⎝⎠
−⋅−
+
⋅
+
−⋅
−
+
−
=
⎜
⎝
∑
θθ
σ
θ
σ
θ
σ
1
θ
1
±
±
1
1
cos
sin
!
2
2
2
k
k
+
−
+⋅
=
⎟⎜
⎟
+
⎠⎝
⎠
θ
1
III.2
Show that the matrix
U
(
θ
) = exp(
i
θ
·
σ
/2) is unitary and that its determinant is unity. In your proof,
be wary of the fact that matrices do not necessarily commute!
Solution:
First, note that det(
U
(
θ
)) = exp{Tr(
i
θ
·
σ
/2)} = exp{0} = 1 (the proof of Pietch’s Theorem can be
found in many books on linear algebra). Unitarity of
U
(
θ
) means
UU
†
=
1
:
††
†
exp
exp
i
U
U
e
e
⋅− ⋅
⎛
⎞
=−
⋅
⋅
⇒
=
=
⎜
⎟
⎝
⎠
1
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View Full Document Homework 3
2
11/1/2005
III.3
Take the deuteron as an analog of a quarkquark system. For the strong interactions, the deuteron
is simply a nucleonnucleon system. The nucleon comes in two flavors: the proton
p
= ½,+½
⟩
,
and the neutron
n
= ½,½
⟩
.
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This note was uploaded on 02/22/2009 for the course PHYSICS 557 taught by Professor Michaelrijssenbeek during the Fall '05 term at SUNY Stony Brook.
 Fall '05
 MichaelRijssenbeek
 Physics, Work

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