  The energy eigenvalues of the quantum particle constrained in a surface of
the sphere of D dimensions embedded in a $R^{D+1}$ 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.
