VI. Angular momentum
Up to this point, we have been dealing primarily with one dimensional
systems. In practice, of course, most of the systems we deal with live
in three dimensions and 1D quantum mechanics is at best a useful
model
. In this section, we will focus in particular on the quantum
mechanics of 3D systems. Many of the elements we discovered for
one dimensional problems will carry over directly to higher
dimensions; however, we will encounter certain effects that are
qualitatively new, and we will spend most of our time exploring these
new phenomena.
The first change comes in how we associate operators with classical
observables. In one dimension, we had
∂
q
→
q
ˆ
p
→
p
ˆ
−
=
i
∂
q
In three dimensions the position and momentum are vectors and so
we must substitute vector calculus for the single variable results:
r
→
r
ˆ
≡
(
i
x
ˆ
+
j
y
ˆ
+
k
z
ˆ
)
p
→
p
ˆ
≡
≈
Δ
«
’
i
p
j
p
k
ˆ
+
ˆ
+
p
ˆ
÷
◊
x
y
z
Where
r
is the position vector and vector quantities will always be
indicated in
bold face
. Note that the operators that correspond to
different axes (i.e
p
ˆ
x
and
z
ˆ
) commute with one another, while the
position and momentum along a given axis (i.e
p
ˆ
x
and
x
ˆ
) obey the
normal commutation relation. We can summarize this in a few
equations:
r
r
i
p
r
j
]
=
i
Z
δ
[
p
ˆ
i
,
p
ˆ
j
]
=
0
[
ˆ, ˆ
]
=
0
[
ˆ, ˆ
j
i
ij
where
i
and
j
can take the value 1,2 or 3 to indicate the
x
ˆ
,
y
ˆ
and
z
ˆ
components of each vector.
a. Rotations
The first difference between 3D and 1D is the possibility of performing
a rotation of our system about one of the three axes. Let us denote a
rotation of an angle
θ
about a unit vector
n
by
R
(
θ
)
. Clearly,
R
(
)
n
n
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View Full Documentis a matrix (it transforms vectors to vectors). Further, it is clear that
R
(
θ
)
R
(
)
≠
R
(
)
R
(
)
(rotations about different axes do not
n
m
m
n
commute). Note that this has nothing to do with quantum mechanics
and everything to do with geometry!
It is easy to verify that the rotation operators associated with the three
Cartesian axes are:
1
0
0
’
≈
Δ
Δ
Δ
÷
÷
÷
sin
R
x
(
)
0
cos
θ
−
=
0
sin
cos
«
◊
cos
’
0
sin
≈
Δ
Δ
Δ
÷
÷
÷
◊
0
cos
R
(
)
y
0
1
0
=
sin
−
«
cos
−
sin
0
≈
Δ
Δ
Δ
’
÷
÷
÷
◊
R
z
(
)
sin
cos
0
=
0
0
1
«
Note that the rotation matrices for x and y can be obtained from the z
matrix by the
cyclic permutation
x
→
y
,
y
→
z
,
z
→
x
. This
must
always
be the case, because our labeling of the x, y and z axes is
totally arbitrary! The only thing we must be careful of is that the “triple
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 Spring '04
 RobertField
 Angular Momentum, commutation relations

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