Our goal is to add orbital angular momentum with quantum number
to spin
.
We can show in several ways that, for
, that the total angular momentum quantum
number has **two possible values
or
.**For
, only
is allowed.
First lets argue that this makes sense when we are adding two **vectors.** For example
if we add a vector of length 3 to a vector of length 0.5, the resulting
vector could take on a length between 2.5 and 3.5
For quantized angular momentum, we will only have the half integers allowed, rather
than a continuous range.
Also we know that the quantum numbers like
are not exactly the length of the
vector but are close.
So these two values make sense physically.

We can also count states for each eigenvalue of as in the following examples.

* Example:
Counting states for plus spin .*

* Example:
Counting states for any plus spin .*

As in the last section, we could start with the highest
state,
,
and apply the **lowering operator** to find the rest of the
multiplet with
.
This works well for some specific
but is hard to generalize.

We can work the problem in general. We know that each eigenstate of
and
will be a **linear combination of the two product states** with the right
.

The coefficients and must be determined by operating with .

We have made a choice in how to write these equations: must be the same throughout. The negative states are symmetric with the positive ones. These equations will be applied when we calculate the

Jim Branson 2013-04-22