PHY 6646 - Quantum Mechanics II - Spring 2011
Homework set # 3, due January 26
1. Show that
√
⎛
Σ(1)
x
0
1⎜√
= ⎝ 2
2
0
0
1⎜ √
= ⎝i 2
2
0
⎛
Σ(1)
z =
1
⎜
⎝0
0
√0 ⎟
2⎠
0
√
−i 2
0
√
i 2
⎛
Σ(1)
y
⎞
2
0
√
2
0
0
0
⎞
0√
⎟
−i 2⎠
0
⎞
0
0 ⎟
⎠
−1
(0.1)
and
⎛
Σ(1)
x
0
⎜
=⎝0
0
⎛
Σ(1)
y
0
⎜
=⎝ 0
−i
0
0
0
0
⎜
= ⎝ +i
0
−i
0
0
⎛
Σ(1)
z
0
0
+i
⎞
0
−i⎟
⎠
0
⎞
+i
0 ⎟
⎠
0
⎞
0
0⎟
⎠
0
(0.2)
are equivalent representations of the rotation group SO(3), i.e. that there exists a unitary
†
(1)
†
(1)
†
matrix U such that Σ(1)
= UΣ(1)
= UΣ(1)
= UΣ(1)
x
x U , Σy
y U and Σz
z U . What is the
matrix U?
2. Problems 15.2.1 (both parts), 15.2.2(2) (second part only), and 15.2.5 in Shankar’s
book. (Note: There is a typo in the statement of problem 15.2.5, which you have to correct
as part of your answer to the question).
1