DOES THE WEYL ORDERING PRESCRIPTION LEAD TO THE CORRECT ENERGY LEVELS FOR THE QUANTUM PARTICLE ON THE D-DIMENSIONAL SPHERE?

Abstract

The energy eigenvalues of the quantum particle constrained in a surface of the sphere of D dimensions embedded in a RD+1 space are obtained by using two different procedures: in the first, we derive the Hamiltonian operator by squaring the expression of the momentum, written in Cartesian components, which satisfies the Dirac brackets between the canonical operators of this second-class system. We use the Weyl ordering prescription to construct the Hermitian operators. When D=2 we verify that there is no constant parameter in the expression of the eigenvalues energy, a result that is in agreement with the fact that an extra term would change the level spacings in the hydrogen atom; in the second procedure it is adopted the non-Abelian BFFT formalism to convert the second-class constraints into first-class ones. The non-Abelian first-class Hamiltonian operator is symmetrized by also using the Weyl ordering rule. We observe that their energy eigenvalues differ from a constant parameter when we compare with the second-class system. Thus, a conversion of the D-dimensional sphere second-class system for a first-class one does not reproduce the same values.