Chapter 9 The Two-Body Problem

1.13. Generalized Spin Operators
For the construction of a self-adjoint ladder operator for we first need to consider a generalized form of spin operators for dimensions.
Let us define operators by the following set of relations

These generalized spin matrices are related to the matrices defined by Eq. , i.e.

and the relation is

From the aforementioned properties it follows that

Since the spin and orbital angular momentum are independent operators we have

An important result concerning and matrices can be stated by the following result

where the prime on the summation sign means that are all different integers. For the proof of this result see the paper of Joseph [22]. Also as in the case of angular momentum in three dimensions we have

and

when and are all different integers. From these results and Eq. (9.314) we obtain

and

Now we can find an explicit form for the ladder operator.
1.14. The Ladder Operator
The operator defined by

is a self-adjoint ladder operator for the eigenstates of if and only if they are also eigenstates of the self-adjoint operator

This result follows from Eqs. (9.297), (9.299), (9.306) and (9.313).
Next we consider the anti-commutation relation

This result can be proven from the anticommutator

which follows from
Now if we substitute from and (9.308) in (9.322) we find

Using the definitions of and we observe that (9.323) reduces to (9.321).
Next we want to express the total angular momentum in terms of . To this end we square both sides of Eq. (9.319) and use the properties of and s to simplify the result (note that we have set );

We can also obtain another expression for between Eqs. (9.319) and

By eliminating from (9.324) and (9.325) we find

Now for we have

therefore by induction from Eq. (9.326) we get

Since is self-adjoint we can choose a representation in which it is diagonal, and since commutes with we can diagonalize these two operators simultaneously. The common eigenstates for and are doubly degenerate with respect to . As in the case of three-dimensional space we write

and

From these solutions we conclude that

for both and subspaces.
Finally let us write the ladder operators for this problem

These relations show that

and

The matrix elements of can be obtained from (9.324). Apart from a phase factor they are

An interesting result of this rather long derivation is that it shows that except in the three-dimensional space where and may be half-integers, the are integers and they satisfy the following relation

Here we assume, without the loss of generality, that all are positive. In order to justify this assumption we observe that since is positive definite

is compatible with the stepping procedure in only if is a positive integer. Then this guarantees that is an upper bound to the ladder. There is also a lower bound preventing the generation of negative values. Consider the case of , then from (9.335) it follows that for the lowering operator

for

can be interpreted as

When the inequality is satisfied we have a new set of eigenvalues , and which must have as an upper bound. If we combine these with (9.338) we find that and may have integers or half-integer values, a result which is true when .