On the Eigenvalue Problem in Terms of a Complete Set of the Casimir Operators

Abstract

The existence of the group of contact transformations, regarded as the so-called continuous Schrödinger group whose elements commute with Hamiltonian,is intimately related not only to the problem of degeneracy, as shown by Jauch and Hill, but also to the problem of quantization. In the present paper, it is shown that on the basis of representations of the Lie ring generated by the operators which have a certain definite correspondence with a set of independent integrals of the equations of motion in Newtonian mechanics, an energy eigenvalue problem can be constructed in terms of a complete set of the Casimir operators. It is further pointed out that this quantization procedure is, in spite of various restrictions, applicable to quantum-mechanical systems having the continuous Schrödinger group, and then the degree of degeneracy in regard to the angular momentum can be obtained simultaneously.