Physics 741 – Graduate Quantum Mechanics 1
Solution Set N
1. [10] Let
( )
,,
x
yz
J
JJ
=
J
be three Hermitian operators that commute with the
Hamiltonian
H
, and have angular momentum-like commutation relations
x
J
Ji
J
⎡⎤
=
⎣⎦
=
y
zx
J
J
=
=
[ ]
,.
y
J
J
=
=
Define the operators
2 222
x
JJJ
=++
J
and
x
y
J
J
±
=
±
Show each of the following is true:
(a) [6]
2
,0
=
(this is three identities), and therefore
2
±
=
We simply begin working them out:
[ ] [ ]
()
22
2
2
,0,
,
,
,
,
,
0,
0
,
,
,
x
yx
y yx
yx y
zzx
zxz
zy
y
z
y
x
x
y
x
z
z
y
z
xz
x
J JJ
JJ J JJJ
iJ
J
J
J
J
J
JJ J
i
J
⎡⎤⎡⎤
=+
+
=
+
+
+
⎣⎦⎣⎦
=−
−
+
+
=
⎡
⎤⎡
⎤
⎡
⎤
+=
+
+
+
⎣
⎦⎣
⎦
⎣
⎦
−
−
=
=
[]
222
,,,
0
,
,
0.
z
z
x
x
z
x
z
x
y
y
z
y
xy
J
JJJ J JJ
i
J
J
=
=
+
+
+
−
+
+
=
=
(b) [2]
[ ]
,
z
J
±±
= ±
=
Again, this is most easily done by just working it out:
[ ] [ ] ( )
2
,
.
zz
x
z
y
y
x
x
y
x
y
iJJ
iJ iJ
J iJ
J
±
±
=±
=
=
±
+
=
±
±
=
±
=∓=
=
=
=
=
=
(c) [3]
J
J
±
±
J
∓
=
It is easiest to expand the right side and show that it is equal to the left side.
( )( )
2
2
2
2
2
,
z
z
z
zxx
yy
x
y
z
z
xyz
x
y
zxyz
z
z
J J
J
J
J
iJ
J
iJ
J
J
J
iJ J
iJ J
J
J
J
JJJi
J
J
JJJJi
J
±
+±
=
±
++±