Application of commutation relations to determine the eigenvalues of operators

G. S. Japaridze, Anzor Khelashvili, K. Sh. Turashvili

Eigenfunctions and eigenvalues of the operator of the square of the angular momentum are studied. It is shown that neither from the requirement for the eigenfunctions be normalizable nor from the commutation relations it is possible to prove that the eigenvalues spectrum is a set of only integer numbers (in units $\hbar=1$). We present regular, normalizable eigenfunctions with the non-integer eigenvalues thus demonstrating that a non-integer angular momentum is admissible from the theoretical viewpoint.

Jorge Krause

The canonical commutation relations of quantum mechanics are generalized to the case where appropriate dynamical variables are angular-momentum, rotation-angle, and rotation-axis observables. To this end, SU(2) is “quantized” on the compact group manifold, according to the standard procedure of non-Abelian quantum kinematics. Quantum-kinematic invariant operators are introduced, and their commutation relations with the rotation variables are found in an explicit manner. The quantum-kinematic invariants yield superselection rules in the form of eigenvalue equations of an isotopic structure (which one should solve in the applications, in order to get multiplets that carry the irreducible representations of the underlying quantum kinematic models). A wide range of applicability of SU(2) quantum kinematics is suggested.

A. Lee

David Leo Weaver

The eigenvalues of the matrix operator S⋅Π for a constant magnetic field are derived in a parallel way for spins 1/2, 1, and 3/2 using only the algebra of the spin matrices and the commutation relations of the components of Π.