This preview shows pages 1–3. Sign up to view the full content.
VIII. Addition of Angular Momenta
a. Coupled and Uncoupled Bases
When dealing with two different sources of angular momentum,
J
ˆ
1
and
J
ˆ
2
, there are two obvious bases that one might choose to work
in. The first is called the
uncoupled basis.
Here the basis kets are
eigenstates of
both
operators:
2
j
,
m
;
j
2
,
m
2
=
j
(
j
+
1
29
j
,
m
1
;
j
2
,
m
2
J
ˆ
1
1
1
1
1
1
j
,
m
1
;
j
2
,
m
2
=
m
1
j
,
m
1
;
j
2
,
m
2
J
ˆ
1
z
1
1
2
j
,
m
;
j
2
,
m
2
=
j
(
j
+
1
29
j
,
m
;
j
2
,
m
2
J
ˆ
2
1
1
2
2
1
1
j
,
m
;
j
2
,
m
2
=
m
2
j
,
m
;
j
2
,
m
2
J
ˆ
2
z
1
1
1
1
In the case of spin orbit coupling, this would mean that our basis
states would be simultaneous eigenfunctions of orbital
and
spin
angular momentum, and each state would have a particular value for
,
{
m
l
}
and
m
. Note that this is only possible if (as we assume):
s
ˆ
ˆ
[
J
,
J
2
]
=
0
1
otherwise, the two operators would not have simultaneous
eigenfunctions. This is clearly true for
L
ˆ
and
S
ˆ
since the two
operators act on different spaces. This really defines what we mean
by
different
angular momenta, since operators that do not commute
will, in some sense, define overlapping – and thus not completely
distinct – forms of angular momentum.
This basis is appropriate if
J
ˆ
1
and
J
ˆ
2
do not interact. However, when
the Hamiltonian contains an interaction between these two angular
momenta (such as
J
ˆ
1
⋅
J
ˆ
2
), the eigenstates will be mixtures of the
uncoupled basis functions and this basis becomes somewhat
awkward. In these cases, it is easiest to work in the
coupled
basis,
which we now develop.
First, note that the total angular momentum is given by:
This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentJ
ˆ
=
J
ˆ
1
+
J
ˆ
2
It is easy to show that this is, in fact, an angular momentum (i.e.
ˆ
ˆ
ˆ
[
J
,
J
y
]
=
J
i
). We can therefore associate two quantum numbers,
j
x
z
and
m
, with the eigenstates of total angular momentum indicating its
magnitude and projection onto the z axis. The coupled basis states
are eigenfunctions of the
total
angular momentum operator. This
specifies two quantum numbers for our basis states (
j
and
m
).
However, as we saw above, the uncoupled basis states were
specified by four quantum numbers (
j
,
j
2
,
m
1
and
m
2
) and we
1
therefore need to specify two more quantum numbers to fully specify
the coupled states. To specify these last two quantum numbers, we
note that
2
ˆ
ˆ
ˆ
2
ˆ
ˆ
2
ˆ
ˆ
ˆ
ˆ
[
J
,
J
1
z
]
=
[
(
J
+
2
J
ˆ
1
⋅
J
+
J
2
29
,
J
]
=
[
2
J
ˆ
1
⋅
J
2
,
J
ˆ
]
=
2
[
J
ˆ
1
,
J
1
z
]
⋅
J
≠
0
1
2
1
z
1
z
2
and similarly for
J
ˆ
2
z
. Thus
J
ˆ
1
z
and
J
ˆ
2
z
do not
share common
eigenfunctions with
J
ˆ
2
. To put it another way, to obtain a definite
state of the total angular momentum, one must generally mix states
with different
m
1
and
m
2
. All thus leads to the conclusion that neither
m
1
nor
m
2
can be one of the other quantum numbers that specify the
coupled basis.
What about
This is the end of the preview. Sign up
to
access the rest of the document.
 Spring '04
 RobertField
 Bases

Click to edit the document details