  We introduce the Poisson bracket operator which is an alternative quantum
counterpart of the Poisson bracket. This operator is defined using the operator
derivative formulated in quantum analysis and is equivalent to the Poisson
bracket in the classical limit. Using this operator, we derive the quantum
canonical equation which describes the time evolution of operators. In the
standard applications of quantum mechanics, the quantum canonical equation is
equivalent to the Heisenberg equation. At the same time, the quantum canonical
equation is applicable to c-number canonical variables and then coincides with
the canonical equation in classical mechanics. Therefore the Poisson bracket
operator enables us to describe classical and quantum behaviors in a unified
way. Moreover, the quantum canonical equation is applicable to non-standard
system where the Heisenberg is not applicable. As an example, we consider the
application to the system where a c-number and a q-number particles coexist.
The derived dynamics satisfies the Ehrenfest theorem and the energy and
momentum conservations.
