Modified Form of Wigner Functions for Non-Hamiltonian Systems

H. Dessano, R. A. S. Paiva, R. G. G. Amorim +2 more

Tai-Hua Heng, Ping Li, Sicong Jing

Quantization of non-Hamiltonian systems (such as damped systems) often gives rise to complex spectra and corresponding resonant states, therefore a standard form calculating Wigner functions cannot lead to static quasi-probability distribution functions. We show that a modified form of the Wigner functions satisfies a ∗-genvalue equation and can be derived from deformation quantization for such systems.

Wagner Coelho Normando Filho

The present work aimed to discuss quantum mechanics in phase space, propose a modification of the Wigner function through the position-dependent translation operator (PDTO) and apply the proposed theory to systems of physical interest. Thus, using a one-dimensional non-Euclidean metric, we construct a modified Wigner function whose form can be expressed in terms of the position-dependent translation operator for the case of finite translations. Through this metric, we define a point transformation that acts as a parameterization that preserves the Euclidean form of the equations. Thus, it was possible to find a modified expression for the Weyl-Wigner quantization map, whose objective is to map functions in phase space into operators in Hilbert space. We show that, when using this modified map to quantize the classical momentum, we obtain the same result as the modified momentum operator found via the position-dependent translation operator. Furthermore, through the inverse operation of the parameterized Weyl-Wigner map, it was possible to define a modified form of the Wigner function, whose mathematical form from the Euclidean case was preserved. We apply the proposed Wigner function to the particle trapped in the potential well and to a quantum oscillator-type system, with the aim of evaluating the influence of the metric on the system dynamics. We plotted the graph of the modified Wigner function for these two systems, where it was possible to verify the appearance of deformations due to the influence of the metric used. Thus, we concluded that the dynamics of the system undergoes a change when the structure of the space is deformed.

Jens Peder Dahl

Yi Yuan

With great significance in describing the state of quantum system,the Wigner function of the spin half non-commutative Landau problem is studied in this paper.On the basis of the review of the Wigner function in the commutative space,which is subject to the *-eigenvalue equation,Hamiltonian of the spin half Landau problem in the non-commutative phase space is given.Then,energy levels and Wigner functions in the form of a matrix of the spin half Landau problem in the non-commutative phase space are obtained by means of the *-eigenvalue equation(or Moyal equation).