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  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 , 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 
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