###

The Operators

The next step is to figure out how the
operators change the eigenstate
.
What eigenstates of
are generated when we operate with
or
?

Because
commutes with
, we see that we have the same
after operation.
This is also true for operations with
.

The operators
,
and
do not change
.
That is, after we operate, the new state is still an eigenstate of
with the same eigenvalue,
.
The eigenvalue of
is changed when we operate with
or
.

(This should remind you of the raising and lowering operators in the HO solution.)
From the above equation we can see that
is an eigenstate of
.

These operators raise or lower the
component of angular momentum by one unit of
.
Since
, its easy to show that the following is greater than zero.

Writing
in terms of our chosen operators,

we can derive limits on the quantum numbers.

We know that the eigenvalue
is greater than zero.
We can assume that

because negative values just repeat the same eigenvalues of
.
The condition that
then becomes a limit on
.

Now,
raises
by one and
lowers
by one, and does not change
.
Since
is limited to be in the range
, the raising and lowering must stop
for
,

The raising and lowering operators change
in integer steps, so, starting from
,
there will be states in integer steps up to
.

Having the minimum at
and the maximum at
with integer steps only works if
** is an integer or a half-integer**.
There are
states with the same
and different values of
.
We now know what eigenstates are allowed.
The eigenstates of
and
should be normalized

The raising and lowering operators acting on
give

The coefficient
can be computed.

We now have the effect of the raising and lowering operators in terms of the normalized
eigenstates.

Jim Branson
2013-04-22