Genvalue Equation Research Articles

Deformation quantization of the Pais–Uhlenbeck fourth order oscillator

We analyze the quantization of the Pais–Uhlenbeck fourth order oscillator within the framework of deformation quantization. Our approach exploits the Noether symmetries of the system by proposing integrals of motion as the variables to obtain a solution to the ⋆-genvalue equation, namely the Wigner function. We also obtain, by means of a quantum canonical transformation the wave function associated to the Schrödinger equation of the system. We show that unitary evolution of the system is guaranteed by means of the quantum canonical transformation and via the properties of the constructed Wigner function, even in the so called equal frequency limit of the model, in agreement with recent results.

A deformation quantization theory for noncommutative quantum mechanics

We show that the deformation quantization of noncommutative quantum mechanics previously considered by Dias and Prata [“Weyl–Wigner formulation of noncommutative quantum mechanics,” J. Math. Phys. 49, 072101 (2008)] and Bastos, Dias, and Prata [“Wigner measures in non-commutative quantum mechanics,” e-print arXiv:math-ph/0907.4438v1; Commun. Math. Phys. (to appear)] can be expressed as a Weyl calculus on a double phase space. We study the properties of the star-product thus defined and prove a spectral theorem for the star-genvalue equation using an extension of the methods recently initiated by de Gosson and Luef [“A new approach to the ⋆-genvalue equation,” Lett. Math. Phys. 85, 173–183 (2008)].

A Note on Wigner Functions and *-Genvalue Equation

In deformation quantization, static Wigner functions obey functional *-genvalue equation, which is equivalent to time-independent Schrödinger equation in Hilbert space operator formalism of quantum mechanics. This equivalence is proved mostly for Hamiltonian with form Ĥ = (1/2)p̂2 + V(x̂) [D. Fairlie, Proc. Camb. Phil. Soc. 60 (1964) 581]. In this note we generalize this proof to a very general Hamiltonian Ĥ(x̂, p̂) and give examples to support it.

A New Type of Seiberg–Witten Map and Its Application

Deformation quantization is a powerful tool to deal with systems in noncommutative space to get their energy spectra and corresponding Wigner functions, especially for the case of both coordinates and momenta being noncommutative. In order to simplify solutions of the relevant *-genvalue equation, we introduce a new kind of Seiberg–Witten-like map to change the variables of the noncommutative phase space into ones of a commutative phase space, and demonstrate its role via an example of two-dimensional oscillator with both kinetic and elastic couplings in the noncommutative phase space.

Wigner Functions for Non-Hamiltonian Systems on Noncommutative Space

We propose a modified form of Wigner functions for generic non-Hamiltonian systems on noncommutative space and prove that it satisfies the corresponding *-genvalue equation. In addition, as an example, we derive exact energy spectra and Wigner functions for a non-Hamiltonian toy model on the noncommutative space.

Modified Form of Wigner Functions for Non-Hamiltonian Systems

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

On infinite walls in deformation quantization

We examine the deformation quantization of a single particle moving in one dimension (i) in the presence of an infinite potential wall, (ii) confined by an infinite square well, and (iii) bound by a delta function potential energy. In deformation quantization, considered as an autonomous formulation of quantum mechanics, the Wigner function of stationary states must be found by solving the so-called ⋆-genvalue (“stargenvalue”) equation for the Hamiltonian. For the cases considered here, this pseudo-differential equation is difficult to solve directly, without an ad hoc modification of the potential. Here we treat the infinite wall as the limit of a solvable exponential potential. Before the limit is taken, the corresponding ⋆-genvalue equation involves the Wigner function at momenta translated by imaginary amounts. We show that it can be converted to a partial differential equation, however, with a well-defined limit. We demonstrate that the Wigner functions calculated from the standard Schrödinger wave functions satisfy the resulting new equation. Finally, we show how our results may be adapted to allow for the presence of another, non-singular part in the potential.

A new form of Wigner functions on the noncommutative space

Wigner quasi-probability distribution function in phase space is a special (Weyl) representation of the density matrix. It has been very useful in many areas of quantum mechanics. Starting from fundamental principle of the Weyl correspondence, we derive explicit form of the Wigner function (WF) for noncommutative quantum mechanics (NCQM), and prove that it satisfies a generalized *-genvalue equation. We also give some examples to show that the new form of WF indeed has this property.

Deformation quantization of noncommutative quantum mechanics

The Wigner phase-space distribution function provides the basis for Moyal's deformation quantization alternative to the more conventional Hilbert space and path integral quantization. In this paper, we investigate the basic aspects of deformation quantization for noncommutative quantum mechanics (NCQM). We first prove some general relations of the Weyl correspondence in non-commuting phase-space. Then we derive explicit form of the Wigner Function (WF) for NCQM starting from fundamental principle of the Weyl correspondence, and show that it satisfies a generalized -genvalue equation. We also demonstrate that the new WFs possess orthonormality and completeness, so they can be used as a basis to expand all phase-space functions. Some example is discussed to support our results.

WIGNER FUNCTIONS FOR THE LANDAU PROBLEM IN NONCOMMUTATIVE SPACES

An electron moving on plane in a uniform magnetic field orthogonal to the plane is known as the Landau problem. Wigner functions for the Landau problem when the plane is noncommutative are found employing solutions of the Schrödinger equation as well as solving the ordinary ⋆-genvalue equation in terms of an effective Hamiltonian. Then, we let momenta and coordinates of the phase space be noncommutative and introduce a generalized ⋆-genvalue equation. We solve this equation to find the related Wigner functions and show that under an appropriate choice of noncommutativity relations they are independent of noncommutativity parameter.

Nonlocal, Non-Commutative Picture in Quantum Mechanics and Distinguished Continuous Canonical Maps

It is shown that continuous classical nonlinear canonical (Poisson) maps have a distinguished role in quantum mechanics. They act unitarily on the quantum phase space and generate ℏ-independent quantum nonlinear canonical maps. It is also shown that such maps act in the non-commutative phase space under the classical covariance. A crucial result of the work is that under the action of Poisson maps a local quantum mechanical picture is converted onto a non-local picture which is then represented in a non-local Hilbert space. On the other hand, it is known that a non-local picture is equivalent by the Weyl map to a non-commutative picture which, in the context of this work, corresponds to a phase space formulation of the theory. As a result of this equivalence, a phase space Schrödinger picture can be formulated. In particular, we obtain the ⋆-genvalue equation of Fairlie (1964 Proc. Camb. Phil. Soc. 60 581) and Curtright, Fairlie and Zachos (1998 Phys. Rev. D 58 025002). In a non-local picture entanglement becomes a crucial concept. The connection between the entanglement and non-locality is explored in the context of Poisson maps and specific examples of the generation of entanglement from a local wavefunction are provided by using the concept of generalized Bell states. The results obtained are also relevant for the non-commutative soliton picture in the non-commutative field theories. We elaborate on this in the context of the scalar non-commutative field theory.

Role of the Imaginary Part in the Moyal Quantization

We show that the imaginary part of the $\star$-genvalue equation in the Moyal quantization reveals the symmetries of the Hamiltonian by which we obtain the conserved quantities. Applying to the Toda lattice equation, we derive conserved quantities which are used as the independent variables of Wigner function.