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PHYS852 Quantum Mechanics II, Spring 2010
HOMEWORK ASSIGNMENT 9: Solutions
Topics covered: hydrogen hyperﬁne structure, WignerEkert theorem, Zeeman eﬀect
1.
Relations between
~
V
and
~
J
: For a rotation by
φ
about the zaxis, we have
U
†
V
z
U
=
V
z
,
U
†
V
x
U
=
cos
φV
x

sin
φV
y
, and
U
†
V
y
U
= sin
φV
x
+ cos
φV
y
, where
U
=
e

(
i/
~
)
φJ
z
.
(a) Consider an inﬁnitesimal rotation by
δφ
, and use these expressions to show:
[
J
z
,V
z
] = 0
,
(1)
[
J
z
,V
x
] =
i
~
V
y
,
(2)
[
J
z
,V
y
] =

i
~
V
x
.
(3)
Write out the six additional commutators generated by cyclic permutation of the indices.
For an inﬁnitesimal rotation, we can expand
U
as
U
≈
1

i
~
φJ
z
, so that keeping terms up
to ﬁrstorder in
φ
gives
V
x
+
i
~
φ
[
J
z
,V
x
] =
V
x

φV
y
(4)
V
y
+
i
~
[
J
z
,V
y
] =
φV
x
+
V
y
(5)
V
z
+
i
~
[
J
,
V
z
] =
V
z
(6)
from which we can read oﬀ:
[
J
z
,V
x
] =
i
~
V
y
(7)
[
J
z
,V
y
] =

i
~
V
x
(8)
[
J
z
,V
z
] = 0
(9)
Cyclic permutation of indices then gives:
[
J
x
,V
y
] =
i
~
V
z
[
J
y
,V
z
] =
i
~
V
x
(10)
[
J
x
,V
z
] =

i
~
V
y
[
J
y
,V
x
] =

i
~
V
z
(11)
[
J
x
,V
x
] = 0
[
J
y
,V
y
] = 0
(12)
b.) Use the results from (a) to show:
[
J
z
,V
±
] =
±
~
V
±
(13)
[
J
±
,V
±
] = 0
(14)
[
J
±
,V
∓
] =
±
2
~
V
z
(15)
1
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View Full Documentwhere
V
±
=
V
x
±
iV
y
.
[
J
z
,V
±
] = [
J
z
,V
x
]
±
i
[
J
z
,V
y
]
=
i
~
V
y
±
~
V
x
=
±
~
(
V
x
±
iV
y
)
=
±
~
V
±
(16)
[
J
±
,V
±
] = [
J
x
,V
x
]
±
i
[
J
x
,V
y
]
±
i
[
J
y
,V
x
]

[
J
y
,V
y
]
=
∓
~
V
z
±
~
V
z
= 0
(17)
[
J
±
,V
∓
] = [
J
x
,V
x
]
∓
i
[
J
x
,V
y
]
±
i
[
J
y
,V
x
] + [
J
y
,V
y
]
=
±
~
V
z
±
~
V
z
=
±
2
~
V
z
(18)
2.
Derivation of WignerEkert theorem
: Verify Eqs. (108)(127) in the Atomic Physics lecture
notes.
Eq. (108):
[
J
z
,V
z
] = 0
[
J
z
,V
z
]

kjm
i
= 0
J
z
(
V
z

kjm
i
) =
V
z
J
z

kjm
i
J
z
(
V
z

kjm
i
) =
~
mV
z

kjm
i
(19)
Eq. (109):
[
J
z
,V
±
] =
±
~
V
±
[
J
z
,V
±
]

kjm
i
=
±
~
V
±

kjm
i
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 Spring '11
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