Fractional Heisenberg equation

2008/01/30 by Vasily E. Tarasov · 68 citations
Mathematics · Physics and Astronomy · #Advanced Thermodynamics and Statistical Mechanics #Commutator #Dissipative system #Fractional Differential Equations Solutions #Fractional calculus #Hamiltonian (control theory) #Harmonic oscillator #Heisenberg model #Heisenberg picture #Mathematical physics #Mathematics #Observable #Physics #Quantum #Quantum mechanics #Statistical Mechanics and Entropy #quant-ph

paper · pdf · doi:10.1016/j.physleta.2008.01.037

published in Physics Letters A 372(17), 2984-2988 (Elsevier BV) · 11 pahes, LaTeX

openalex publication_date 2008/01/30 · arxiv created 2008/04/03 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

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.

Citations

• Fractional Differential Equations

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