PHY431 Homework Set 6
Reading:
Lecture Notes
Homework:
See below:
Due date:
Wednesday March 8
Hints and Solutions
Problem VI.1
Using the isospinlowering operator
I

, and starting at the "fully stretched" state
∆
++
= 
uuu
⟩
,
derive the representation of the three other
∆
states in terms of quark states.
Hints:
no hints
Solution:
I

∆
++
=
I


uuu
⟩
=
N
{
duu
⟩
+
udu
⟩
+
uud
⟩
} =
∆
+
, with
N
a normalization constant.
Clearly
N
= 1/
√
3 here.
I

∆
+
=
N
{
ddu
⟩
+
dud
⟩
+
udd
⟩
} =
∆
0
, with again
N
= 1/
√
3 .
I

∆
0
=
N
{
ddd
⟩
} =
∆

, with
N
= 1 .
Problem VI.2
Calculate the Isospin3 (
I
3
) eigenvalues (of the
I
3
operator) of the bottom and the top
quarks.
Hints:
no hints
Solution:
I
3
(
b
) =
Q

1
/
2
(B+S+C+B+T) = 
1
/
3

1
/
2
(
1
/
3
+0+01+0) = 0
I
3
(
t
) =
Q

1
/
2
(B+S+C+B+T) = +
2
/
3

1
/
2
(
1
/
3
+0+0+0+1) = 0, as expected for any quark not a
u
or
d
quark.
Problem VI.3
Show that the states 
π
0
⟩
and 
η
⟩
are othonormal, assuming that the 
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