Fractional Heisenberg Equation

TL;DR

This paper introduces a fractional generalization of the Heisenberg equation by defining a fractional derivative as a fractional power of the commutator, providing exact solutions for free particle and harmonic oscillator Hamiltonians.

Contribution

It proposes a novel fractional Heisenberg equation using fractional derivatives of observables, extending quantum dynamics to include dissipative processes.

Findings

Exact solutions for free particle and harmonic oscillator cases.

Generalizes quantum Hamiltonian systems to include dissipation.

Provides a new mathematical framework for quantum dissipative dynamics.

Abstract

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.