Fractional Dynamics of Hamiltonian Quantum Systems

Abstract

In the quantum mechanics, the observables are given by self-adjoint operators (Messiah, 1999; Tarasov, 2005). The dynamical description of a quantum system is given by a superoperator (Tarasov, 2008b), which is a rule that assigns to each operator exactly one operator. Dynamics of quantum observable is described by the Heisenberg equation. For Hamiltonian systems, the infinitesimal superoperator of the Heisenberg equation is defined by some form of derivation (Tarasov, 2005, 2008b). The infinitesimal generator (i/ħ)[H, .], which is used in the Heisenberg equation, is a derivation of observables. A derivation is a linear map D, which satisfies the Leibnitz rule D(AB) = (DA)B +A(DB) for all operators A and B. Fractional derivative can be defined as a fractional power of derivative (see Section 5.7 m (Samko et al., 1993)). We consider a fractional derivative on a set of observables as a fractional power of derivative (i/ħ)[H, .]. It allows us to generalize a notion of quantum Hamiltonian systems. In this case, operator equation for quantum observables is a fractional generalization of the Heisenberg equation (Tarasov, 2008a). The suggested fractional Heisenberg equation is exactly solved for the Hamiltonians of free particle and harmonic oscillator. Fractional power of operators (Balakrishnan, 1960; Komatsu, 1966; Berens et al., 1968; Yosida, 1995; Krein, 1971; Martinez and Sanz, 2000) and superoperator (Tarasov, 2008b, 2009) can be used as a possible approach to describe an interaction between the system and an environment. We note that fractional power of the operator, which is defined by Piosson bracket of classical dynamics, was considered in (Tarasov, 2008c).