Chapter 9 The Two-Body Problem

1.11. Self-Adjoint Ladder Operator
The ladder operators that we have seen so far are not self-adjoint. This is because the adjoint of either of the ladder operators changes the eigenstate in the opposite sense. It is possible to formulate the factorization method in such a way that the ladder operators are self-adjoint [22],[23].
Let be the self-adjoint operator that we want to determine its eigenvalues and and be the non-self-adjoint raising and lowering operators for . We define and as the following operators:

Both of these operators are self-adjoint and satisfy the anti-commutation relation

The operator acts on a pair of states each of which can be represented by a matrix (spinor form)

That is

and

Upon the action of the matrix on and states we get

and

Thus the ladder operator changes the eigenstates from one subspace to the other and at the same time changes the eigenvalues. We observe that in this formulation instead of having one set of eigenstates from one subspace and a pair of mutually adjoint ladder operators, we have one ladder operator and two subspaces.
Let us consider the example of orbital angular momentum where we choose to be the component of the angular momentum, , and and are given by

Substituting for and in (9.287) we have

and

where and are the Pauli spin matrices, Eq. (9.118).
1.12. Self-Adjoint Ladder Operator for Angular Momentum
We now formulate the problem of determination of the eigenvalues of angular momentum in a space of dimensions in terms of the self-adjoint ladder operators. In the following sections, for convenience, we set and we define by

These s are obvious generalization of and operators for the threedimensional space. The operator has the following properties:

and

where and are all different integers, and they all run from 1 to .
The set form the elements of a Lie algebra with a single Casimir invariant

From the properties of we can show that form a commuting set of operators and that the eigenvalues of can be determined from the irreducible representations of the Lie group. The eigenvalues of can also be found from the equation

by separating (9.304) in generalized spherical polar coordinates (hyperspherical coordinates)

However in the present case we are dealing with the representation in which the commuting sets are simultaneously diagonal. Here we can solve the problem by induction. That is we assume that the solution is known in space and the we find in space. For this we can use a ladder operator for which is self-adjoint of the type that we have discussed.