In-fact, it is well known that the [[gamma].sup.i]-matrices (i = 1,2,3) represent spin (i.e., [mathematical expression not reproducible]) because, together with the

angular momentum operator ([??]), their sum total of the orbital angular momentum and spin [mathematical expression not reproducible] commutes with the Dirac Hamiltonian ([H.sub.D]), i.e.

where [sigma] is the string tension, [[alpha].sub.s] is the strong-interaction fine-structure constant, [f.sub.c] is the color factor which is -4/3 for quark-antiquark and -2/3 for quark-quark, [[sigma].sub.1] and [[sigma].sub.2] are the Pauli matrices, and L is the total orbital

angular momentum operator. Fourier transformation of this potential to momentum space yields

From the definitions of the momentum operator P = -i[??][nabla] and orbital

angular momentum operator L = r x P, we can define the following operators:

For a particle in a spherical (central) field, the total angular momentum [??], and the spin-orbit matrix operator [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] commute with the Dirac Hamiltonian, where L is orbital

angular momentum operator. For a given total angular momentum j, the eigenvalues of [??] are k = -(j +1/2) for aligned spin ([S.sub.1/2], [p.sub.3/2], etc) and k = j + 1/2 for unaligned spin ([p.sub.1/2], [d.sub.3/2], etc).

For ultraspherical expansions, an UP has been proved by Rosler and Voit in [14] making use of the Dunkl operator as the angular momentum operator. Here we give a more general UP for a whole class of "position" operators.

In [10], Narcowich and Ward gave an UP on the sphere using the multiplication with the surface variable [eta] [member of] [S.sup.2] as position operator and the angular momentum operator [OMEGA] -iL* = -I[eta] x [nabla]* as momentum operator where [nabla]* denotes the surface gradient and L* the surface curl gradient.

The orbital

angular momentum operator is [??] = [??]V x [??] and we must transform it into cylindrical coordinates:

For a particle in a spherical field, the total

angular momentum operator J and spin-orbit matrix operator [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] commute with the Dirac Hamiltonian, where a and L are the Pauli matrix and orbital angular momentum, respectively.

This result is just the hyperbolic-space generalization of the standard decomposition of the Laplace operator in spherical coordinates in terms of the radial derivatives plus a term containing the square of the orbital angular momentum operator [L.sup.2]/[r.sup.2].

The generalized Dirac-Konstant equations in Clifford-spaces are obtained after introducing the generalized angular momentum operators [14]

Also appropriate for independent study, this undergraduate textbook introduces the supersymmetric quantum mechanics (SUSYQM) approach to solving the Schr|dinger equation, the concept of shape invariant potentials, and the algebra of

angular momentum operators. Later chapters demonstrate applications to generating orthogonal polynomials by shape invariance, the supersymmetric version of the WKB approximation, isospectral deformations of the conventional shape invariant potentials, supersymmetric quantum Hamilton-Jacobi formalism, and deformation quantization.

In the case of spin, the symbols J and j denote total and single particle

angular momentum operators, respectively.