Fractional Heisenberg equation
Abstract
Fractional derivative can be defined as a fractional power of derivative. The commutator (i/ℏ)[H,ṡ], 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,ṡ]. 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.
- Publication:
-
Physics Letters A
- Pub Date:
- April 2008
- DOI:
- 10.1016/j.physleta.2008.01.037
- arXiv:
- arXiv:0804.0586
- Bibcode:
- 2008PhLA..372.2984T
- Keywords:
-
- 03.65.-w;
- 03.65.Ca;
- 45.10.Hj;
- 03.65.Db;
- Quantum mechanics;
- Formalism;
- Perturbation and fractional calculus methods;
- Functional analytical methods;
- Quantum Physics
- E-Print:
- 11 pahes, LaTeX