dimensions are described by the group SO(3), while rotations of halfintegral spin states
are described by a different group SU(2) which is the “covering group” for SO(3). Don’t
be put off by the jargon—If you have taken linear algebra you know that rotations in three
2
4.
(10 pts) Fun with commutators. We will be working a lot with com
mutators, so you need to get familiar with what you can and cannot
do. Show the following (
ˆ
A,
ˆ
B . . .
are operators;
a, b . . .
are numbers):
(a) [
ˆ
A,
ˆ
B
] =

[
ˆ
B,
ˆ
A
]
(b) [
ˆ
A,
ˆ
B
+
ˆ
C
] = [
ˆ
A,
ˆ
B
] + [
ˆ
A,
ˆ
C
]
(c) [
ˆ
A
+
ˆ
B,
ˆ
C
] = [
ˆ
A,
ˆ
C
] + [
ˆ
B,
ˆ
C
]
(d) [
ˆ
A, b
ˆ
C
] =
b
[
ˆ
A,
ˆ
C
]
(e) [
ˆ
A
ˆ
B,
ˆ
C
] =
ˆ
A
[
ˆ
B,
ˆ
C
] + [
ˆ
A,
ˆ
C
]
ˆ
B
(f) [
ˆ
A,
ˆ
B
ˆ
C
] =
ˆ
B
[
ˆ
A,
ˆ
C
] + [
ˆ
A,
ˆ
B
]
ˆ
C
Can you see the pattern in parts 4e and 4f?
(g) [
ˆ
I
+
a
ˆ
B,
ˆ
I
+
c
ˆ
D
] =
ac
[
ˆ
B,
ˆ
D
]
where
ˆ
I
is the identity.
We used this on rotation operators to get the commutation rela
tions for the generators
ˆ
J
k
.
One way to see this is to use parts
parts 4b and 4c to expand the LHS into four commutators, three
of which vanish because
ˆ
I
commutes with everything.
5.
(10 pts) In some basis three operators are given by the matrices
ˆ
A
↔
"
1
0
0

1
#
,
ˆ
B
↔
"
3
0
0
3
#
,
ˆ
C
↔
"
0
1
1
0
#
.
(a) You can see that these operators are Hermitian, and so can rep
resent observable quantities
A, B, C
.
Find the eigenspinors and
eigenvalues of all three matrices.
What is different about the
eigenvalues of
ˆ
B
compared to the eigenvalues of the other two
operators? If you are careful, you will find that the eigenspinors
for
ˆ
B
are not uniquely defined, which goes along with this fact.
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 Fall '15
 mechanics, Power Series, Work, Photon, Polarization, Townsend Sec., spin1/2 quantum state