Abstract
Fractional derivative can be defined as a fractional power of derivative. The commutator ( i / ℏ ) [ 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 Letter, we consider a fractional derivative on a set of quantum observables as a fractional power of the commutator ( i / ℏ ) [ 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.
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