The next step is to figure out how the
operators change the eigenstate
What eigenstates of
are generated when we operate with
, we see that we have the same
This is also true for operations with
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
(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
, its easy to show that the following is greater than zero.
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
by one and
by one, and does not change
is limited to be in the range
, the raising and lowering must stop
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.
states with the same
and different values of
We now know what eigenstates are allowed.
The eigenstates of
should be normalized
The raising and lowering operators acting on
can be computed.
We now have the effect of the raising and lowering operators in terms of the normalized