### 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