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Journal of Statistical Mechanics Theory and Experiment

Wigner function of a quantum system with polynomial potential

E. E. Perepelkin, Б. И. Садовников, Наталья Германовна Иноземцева, Evgenii V. Burlakov

Published May 1, 2020• 15 citations
Publisher

Abstract

Abstract The Moyal equation for the Wigner function was obtained under the assumption that the potential is an analytic function. The polynomial form of the potential is a natural approximation of the analytical potential with any necessary accuracy. The simplest quantum system with a second-order polynomial potential is a quantum harmonic oscillator. In this paper, for a quantum system with a polynomial potential of arbitrary order, explicit expressions are obtained for the matrix elements of the kernel operator in the basis of the eigenfunctions of the harmonic oscillator. Using the explicit representation for the kernel operator matrix elements, we construct the distributions of the Wigner function in the phase space for quantum systems with polynomial potentials. The connection of the modified Vlasov equation with the Moyal equation for the Wigner function is shown. Examples of effective numerical algorithms for finding Wigner functions with high accuracy are given.

DOI
10.1088/1742-5468/ab7bda

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Abstract The Moyal equation for the Wigner function was obtained under the assumption that the potential is an analytic function. The polynomial form of the potential is a natural approximation of the analytical potential with any necessary accuracy. The simplest quantum system with a second-order polynomial potential is a quantum harmonic oscillator. In this paper, for a quantum system with a polynomial potential of arbitrary order, explicit expressions are obtained for the matrix elements of the kernel operator in the basis of the eigenfunctions of the harmonic oscillator. Using the explicit representation for the kernel operator matrix elements, we construct the distributions of the Wigner function in the phase space for quantum systems with polynomial potentials. The connection of the modified Vlasov equation with the Moyal equation for the Wigner function is shown. Examples of effective numerical algorithms for finding Wigner functions with high accuracy are given.
Abstract In this paper, the matrix elements explicit expressions for the kernel operator in the harmonic oscillator eigenfunctions basis are obtained. The matrix elements are expressed in terms of the two complex variables new polynomials that were constructed in this paper. In the particular case, new polynomials degenerate into Laguerre polynomials. The diagonal elements of the kernel operator matrix are the Wigner functions of the harmonic oscillator, which do not introduce dissipation into the quantum system. The off-diagonal elements contain frequency oscillations responsible for dissipations in the quantum systems. Using the explicit representation of the kernel operator matrix elements, we construct the distributions of the Wigner function in the phase space for quantum systems.
The article considers the construction of a parallel algorithm on the GPU computing architecture forfinding the Wigner function of a quantum system with a polynomial potential. A numerical-analyticalmethod for constructing the Wigner function, which is based on calculating the trace of the product ofthe density matrix and the matrix of the Weyl operator, is described. The operators are represented inthe basis of a quantum harmonic oscillator, for which the Moyal equation transforms into the Liouvilleequation. This approach enables to visually analyze the degree of anharmonicity of the system in termsof the off-diagonal elements of the density matrix. The parallel implementation on the GPU massivelyparallel architecture has reduced the calculation time by two orders of magnitude compared to thesingle-threaded version on the x86 architecture.
Abstract When considering quantum systems in phase space, the Wigner function is used as a function of quasi-probability density. Finding the Wigner function is related to the calculation of a Fourier transform of a certain composition of wave functions of the corresponding quantum system. As a rule, knowledge of the Wigner function is not the final goal, and the calculation of mean values of different quantum characteristics of the system is required. An explicit solution of the Schrödinger equation can be obtained only for a narrow class of potentials, therefore, in the majority of cases, numerical methods must be used to find wave functions. As a result, finding the Wigner function is related to the numerical integration of grid wave functions. When considering a one-dimensional system, the calculation of N2 Fourier integrals of the grid wave function is needed. To provide the necessary accuracy for wave functions, corresponding to the higher states of a quantum system, a larger number of grid nodes is needed. The goal of this work was to construct the numerical-analytical method for finding the Wigner function, which allows the number of computational operations to be considerably reduced. The quantum systems with polynomial potentials were considered, for which the Wigner function is represented as a series in some known functions.
Abstract The arbitrary linear quantization is considered for definition of Wigner function. The evolution equation of this function is derived with the use of inversion of linear quantization kernel. In general case the Moyal equation depends on the characteristic function of quantization. It is shown that only for Weyl quantization this equation does not consist the source of quasi-probability. The stationary solutions of Moyal equation are constructed for an arbitrary linear quantization of a harmonic oscillator. The case of anharmonic oscillator is presented as a practical example of quantum system model in the so-called post-newtonian approximation. This model exhibits the intersection effect between quantization dependences of Hamiltonian and Wigner function.
Abstract This paper presents a comprehensive investigation of the problem of a harmonic oscillator with time-depending frequencies in the framework of the Vlasov theory and the Wigner function apparatus for quantum systems in the phase space. A new method is proposed to find an exact solution of this problem using a relation of the Vlasov equation chain with the Schrödinger equation and with the Moyal equation for the Wigner function. A method of averaging the energy function over the Wigner function in the phase space can be used to obtain time-dependent energy spectrum for a quantum system. The Vlasov equation solution can be represented in the form of characteristics satisfying the Hill equation. A particular case of the Hill equation, namely the Mathieu equation with unstable solutions, has been considered in details. An analysis of the dynamics of an unstable quantum system shows that the phase space square bounded with the Wigner function level line conserves in time, but the phase space square bounded with the energy function line increases. In this case the Vlasov equation characteristic is situated on the crosspoint of the Wigner function level line and the energy function line. This crosspoint moves in time with a trajectory that represents the unstable system dynamics. Each such trajectory has its own energy, and the averaging of these energies by the Wigner function results in time-dependent discreet energy spectrum for the whole system. An explicit expression has been obtained for the Wigner function of the 4th rank in the generalized phase space x , p , p ̇ , p ̈ .
Abstract The Moyal equation describes the evolution of the Wigner function of a quantum system in the phase space. The right-hand side of the equation contains an infinite series with coefficients proportional to powers of the Planck constant. There is an interpretation of the Moyal equation as a quantum analogue of the classical Liouville equation. Indeed, if one uses the notion of the classical passage to the limit as the Planck constant tends to zero, then formally the right-hand side of the Moyal equation tends to zero. As a result, the Moyal equation becomes the classical Liouville equation for the distribution function. In this paper, we show that the right side of the Moyal equation does not explicitly depend on the Planck constant, and all terms of the series can make a significant contribution. The transition between the classical and quantum descriptions is related not to the Planck constant, but to the spatial scale. For a model quantum system with a potential in the form of a «quadratic funnel», an exact 3D solution of the Schrödinger equation is found and the corresponding Wigner function is constructed in the paper. Using trajectory analysis in the phase space, based on the representation of the right-hand side of the Moyal equation, it is shown that on the spatial microscale there is an infinite number of «trajectories» of the particle motion (thereby the concept of a trajectory is indefinite), and when passing to the macroscale, all «trajectories» concentrate around the classical trajectory.