  Fractional derivative can be defined as a fractional power of derivative. The
commutator (i/h)[H, ], which is used in the Heisenberg equation, is a
derivation on a set of observables. A derivation is a map that satisfies the
Leibnitz rule. In this paper, we consider a fractional derivative on a set of
quantum observables as a fractional power of the commutator (i/h)[H, ]. As a
result, we obtain a fractional generalization of the Heisenberg equation. The
fractional Heisenberg equation is exactly solved for the Hamiltonians of free
particle and harmonic oscillator. The suggested Heisenberg equation generalize
a notion of quantum Hamiltonian systems to describe quantum dissipative
processes.
