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: