  Using the quantum covariant Poisson bracket (QCPB) theory, we can accomplish
much more compatible explanations of the quantum mechanics supported by the
G-dynamics. We further study the generalized quantum harmonic oscillator
equipped with the G-dynamics of type I, such as geometric creation and
annihilation operators, and the geometric number operator as an extension of
the number operator is well given for the deep discussions, the geometric
Hamiltonian operator is expressed as another form. Especially, the geometric
quantization rules based on the QCPB theory is then calculated.
