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Published: 10 March 2023

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1

Kosik, Robert, Johann Cervenka, and Hans Kosina. "Numerical constraints and non-spatial open boundary conditions for the Wigner equation." Journal of Computational Electronics 20, no. 6 (November 3, 2021): 2052–61. http://dx.doi.org/10.1007/s10825-021-01800-w.

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AbstractWe discuss boundary value problems for the characteristic stationary von Neumann equation (stationary sigma equation) and the stationary Wigner equation in a single spatial dimension. The two equations are related by a Fourier transform in the non-spatial coordinate. In general, a solution to the characteristic equation does not produce a corresponding Wigner solution as the Fourier transform will not exist. Solution of the stationary Wigner equation on a shifted k-grid gives unphysical results. Results showing a negative differential resistance in IV-curves of resonant tunneling diodes using Frensley’s method are a numerical artefact from using upwinding on a coarse grid. We introduce the integro-differential sigma equation which avoids distributional parts at $$k=0$$ k = 0 in the Wigner transform. The Wigner equation for $$k=0$$ k = 0 represents an algebraic constraint needed to avoid poles in the solution at $$k=0$$ k = 0 . We impose the inverse Fourier transform of the integrability constraint in the integro-differential sigma equation. After a cutoff, we find that this gives fully homogeneous boundary conditions in the non-spatial coordinate which is overdetermined. Employing an absorbing potential layer double homogeneous boundary conditions are naturally fulfilled. Simulation results for resonant tunneling diodes from solving the constrained sigma equation in the least squares sense with an absorbing potential reproduce results from the quantum transmitting boundary with high accuracy. We discuss the zero bias case where also good agreement is found. In conclusion, we argue that properly formulated open boundary conditions have to be imposed on non-spatial boundaries in the sigma equation both in the stationary and the transient case. When solving the Wigner equation, an absorbing potential layer has to be employed.

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2

FEDELE, RENATO, SERGIO DE NICOLA, DUSAN JOVANOVIĆ, DAN GRECU, and ANCA VISINESCU. "On the mapping connecting the cylindrical nonlinear von Neumann equation with the standard von Neumann equation." Journal of Plasma Physics 76, no. 3-4 (January 25, 2010): 645–53. http://dx.doi.org/10.1017/s0022377809990870.

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AbstractThe Wigner transformation is used to define the quasidistribution (Wigner function) associated with the wave function of the cylindrical nonlinear Schrödinger equation (CNLSE) in a way similar to that of the standard nonlinear Schrödinger equation (NLSE). The phase-space equation, governing the evolution of such quasidistribution, is a sort of nonlinear von Neumann equation (NLvNE), called here the ‘cylindrical nonlinear von Neumann equation’ (CNLvNE). Furthermore, the phase-space transformations, connecting the Wigner function and the NLvNE with the ‘cylindrical Wigner function’ and the CNLvNE, are found by extending the configuration space transformations that connect the NLSE and the CNLSE. Some examples of phase-space soliton solutions are given analytically and evaluated numerically.

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3

ISAR, A., A. SANDULESCU, H. SCUTARU, E. STEFANESCU, and W. SCHEID. "OPEN QUANTUM SYSTEMS." International Journal of Modern Physics E 03, no. 02 (June 1994): 635–714. http://dx.doi.org/10.1142/s0218301394000164.

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The damping of the harmonic oscillator is studied in the framework of the Lindblad theory for open quantum systems. A generalization of the fundamental constraints on quantum mechanical diffusion coefficients which appear in the master equation for the damped quantum oscillator is presented; the Schrödinger, Heisenberg and Weyl-Wigner-Moyal representations of the Lindblad equation are given explicitly. On the basis of these representations it is shown that various master equations for the damped quantum oscillator used in the literature are particular cases of the Lindblad equation and that not all of these equations are satisfying the constraints on quantum mechanical diffusion coefficients. Analytical expressions for the first two moments of coordinate and momentum are obtained by using the characteristic function of the Lindblad master equation. The master equation is transformed into Fokker-Planck equations for quasiprobability distributions and a comparative study is made for the Glauber P representation, the antinormal ordering Q representation, and the Wigner W representation. The density matrix is represented via a generating function, which is obtained by solving a timedependent linear partial differential equation derived from the master equation. Illustrative examples for specific initial conditions of the density matrix are provided. The solution of the master equation in the Weyl-Wigner-Moyal representation is of Gaussian type if the initial form of the Wigner function is taken to be a Gaussian corresponding (for example) to a coherent wavefunction. The damped harmonic oscillator is applied for the description of the charge equilibration mode observed in deep inelastic reactions. For a system consisting of two harmonic oscillators the time dependence of expectation values, Wigner function and Weyl operator, are obtained and discussed. In addition models for the damping of the angular momentum are studied. Using this theory to the quantum tunneling through the nuclear barrier, besides Gamow’s transitions with energy conservation, additional transitions with energy loss are found. The tunneling spectrum is obtained as a function of the barrier characteristics. When this theory is used to the resonant atom-field interaction, new optical equations describing the coupling through the environment of the atomic observables are obtained. With these equations, some characteristics of the laser radiation absorption spectrum and optical bistability are described.

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4

Gao, Jian-Hua, Zuo-Tang Liang, and Qun Wang. "Quantum kinetic theory for spin-1/2 fermions in Wigner function formalism." International Journal of Modern Physics A 36, no. 01 (January 10, 2021): 2130001. http://dx.doi.org/10.1142/s0217751x21300015.

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We give a brief overview of the kinetic theory for spin-1/2 fermions in Wigner function formalism. The chiral and spin kinetic equations can be derived from equations for Wigner functions. A general Wigner function has 16 components which satisfy 32 coupled equations. For massless fermions, the number of independent equations can be significantly reduced due to the decoupling of left-handed and right-handed particles. It can be proved that out of many components of Wigner functions and their coupled equations, only one kinetic equation for the distribution function is independent. This is called the disentanglement theorem for Wigner functions of chiral fermions. For massive fermions, it turns out that one particle distribution function and three spin distribution functions are independent and satisfy four kinetic equations. Various chiral and spin effects such as chiral magnetic and vortical effects, the chiral separation effect, spin polarization effects can be consistently described in the formalism.

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5

Kapral, R., and A. Sergi. "Quantum-Classical Wigner-Liouville Equation." Ukrainian Mathematical Journal 57, no. 6 (June 2005): 891–99. http://dx.doi.org/10.1007/s11253-005-0237-0.

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6

Chmieliński, J. "On a conditional Wigner equation." aequationes mathematicae 56, no. 1-2 (August 1998): 143–48. http://dx.doi.org/10.1007/s000100050050.

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7

Gasser, I., P. A. Markowich, and B. Perthame. "Dispersion Lemmas Revisited." VLSI Design 9, no. 4 (January 1, 1999): 365–75. http://dx.doi.org/10.1155/1999/81341.

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We investigate regularizing dispersive effects for various classical equations, e.g., the Schrödinger and Dirac equations. After Wigner transform, these dispersive estimates are reduced to moment lemmas for kinetic equations. They yield new regularization results for the Schrödinger equation (valid up to the semiclassical limit) and the Dirac equation.

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8

Li, Ruo, Tiao Lu, Yanli Wang, and Wenqi Yao. "Numerical Validation for High Order Hyperbolic Moment System of Wigner Equation." Communications in Computational Physics 15, no. 3 (March 2014): 569–95. http://dx.doi.org/10.4208/cicp.091012.120813a.

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AbstractA globally hyperbolic moment system upto arbitrary order for the Wigner equation was derived in [6]. For numerically solving the high order hyperbolic moment system therein, we in this paper develop a preliminary numerical method for this system following the NRxx method recently proposed in [8], to validate the moment system of the Wigner equation. The method developed can keep both mass and momentum conserved, and the variation of the total energy under control though it is not strictly conservative. We systematically study the numerical convergence of the solution to the moment system both in the size of spatial mesh and in the order of the moment expansion, and the convergence of the numerical solution of the moment system to the numerical solution of the Wigner equation using the discrete velocity method. The numerical results indicate that the high order moment system in [6] is a valid model for the Wigner equation, and the proposed numerical method for the moment system is quite promising to carry out the simulation of the Wigner equation.

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9

GRUBIN, H. L., and H. L. CUI. "SPIN DEPENDENT TRANSPORT IN QUANTUM AND CLASSICALLY CONFIGURED DEVICES." International Journal of High Speed Electronics and Systems 16, no. 02 (June 2006): 639–58. http://dx.doi.org/10.1142/s0129156406003904.

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This paper presents of development of quantum transport equations for barrier devices with both electron and hole transport in dilute magnetic semiconductor (DMS) structures. The equations are developed from the time dependent equation of motion of the density matrix equation in the coordinate representation, from which both the spin drift and diffusion and transient Wigner equations are obtained, for a system in which high 'g' factor materials result in significant spin-splitting of the valence and conduction bands. Then for a structure in which the DMS layer is confined to the first barrier solutions to the coupled Poisson's and spin dependent Wigner equations yield the IV and carrier distributions. Negative differential conductance as well as the significant unequal spinup and spin down charge distributions are obtained.

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10

Wang, Zheng-Chuan. "Non-Hermite Spinor Boltzmann Equation and Its Hermitization." Journal of Physics: Conference Series 2370, no. 1 (November 1, 2022): 012008. http://dx.doi.org/10.1088/1742-6596/2370/1/012008.

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It is known that the Wigner distribution function in the quantum Boltzmann equation is not positive elsewhere, which causes trouble for its application to the mesoscopic transport problem. Similarly, if we extend the quantum Boltzmann equation to the spinor Boltzmann equation, which includes the spin freedom in the usual Wigner distribution, and the spinor matrix distribution function therein is not Hermite, we must make this non-Hermite spinor Boltzmann equation Hermitization. So in this paper, we propose a spinor Boltzmann equation with Hermite distribution function and then obtain the equations of continuity satisfied by the charge density, charge current, spin accumulation, and spin current. The numerical results for these physical quantities are illustrated by an example of spin-polarized transport scattered by impurities in a system of spintronics. Their differences compared with the usual spinor Boltzmann equation are shown.

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11

Brunetti, R., A. Bertoni, P. Bordone, and C. Jacoboni. "Dynamical Equation and Monte Carlo Simulation of the Two-time Wigner Function for Electron Quantum Transport." VLSI Design 13, no. 1-4 (January 1, 2001): 375–80. http://dx.doi.org/10.1155/2001/42430.

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Within the Wigner-function formalism for electron quantum transport in semiconductors a two-time Wigner function is defined starting from the Green-function formalism. After a proper Fourier transform a Wigner function depending on p and w as independent variables is obtained. This new Wigner function extends the Wigner formalism to the frequency domain and carries information related to the spectral density of the system. A Monte Carlo approach based on the generation of Wigner paths, already developed for the single-time Wigner function, has been extended to evaluate the momentum and energy-dependent Wigner function. Results will be shown for electrons subject to the action of an external field and in presence of scattering with optical phonons.

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12

Hurst, Jérôme, Paul-Antoine Hervieux, and Giovanni Manfredi. "Phase-space methods for the spin dynamics in condensed matter systems." Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, no. 2092 (March 20, 2017): 20160199. http://dx.doi.org/10.1098/rsta.2016.0199.

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Using the phase-space formulation of quantum mechanics, we derive a four-component Wigner equation for a system composed of spin- fermions (typically, electrons) including the Zeeman effect and the spin–orbit coupling. This Wigner equation is coupled to the appropriate Maxwell equations to form a self-consistent mean-field model. A set of semiclassical Vlasov equations with spin effects is obtained by expanding the full quantum model to first order in the Planck constant. The corresponding hydrodynamic equations are derived by taking velocity moments of the phase-space distribution function. A simple closure relation is proposed to obtain a closed set of hydrodynamic equations. This article is part of the themed issue ‘Theoretical and computational studies of non-equilibrium and non-statistical dynamics in the gas phase, in the condensed phase and at interfaces’.

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13

Pietruczuk, Barbara. "Asymptotic integration of the second order differential equation, resonance effect." Tatra Mountains Mathematical Publications 63, no. 1 (June 1, 2015): 223–35. http://dx.doi.org/10.1515/tmmp-2015-0034.

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Abstract There will be presented asymptotic formulas for solutions of the equation y'' + (1 + φ (x))y = 0, 0 < x0 < x < ∞ , where function is small in a certain sense for large values of the argument. Usage of method of L-diagonal systems allows to obtain various forms of solutions depending on the properties of function φ . The main aim will be discussion about the second order differential equations possesing a resonance effect known for Wigner von Neumann potential. A class of potentials generalizing that of Wigner von Neumann will be presented.

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14

Weber, Hannes, Omar Maj, and Emanuele Poli. "Wigner-function-based solution schemes for electromagnetic wave beams in fluctuating media." Journal of Computational Electronics 20, no. 6 (October 19, 2021): 2199–208. http://dx.doi.org/10.1007/s10825-021-01791-8.

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AbstractElectromagnetic waves are described by Maxwell’s equations together with the constitutive equation of the considered medium. The latter equation in general may introduce complicated operators. As an example, for electron cyclotron (EC) waves in a hot plasma, an integral operator is present. Moreover, the wavelength and computational domain may differ by orders of magnitude making a direct numerical solution unfeasible, with the available numerical techniques. On the other hand, given the scale separation between the free-space wavelength $$\lambda _0$$ λ 0 and the scale L of the medium inhomogeneity, an asymptotic solution for a wave beam can be constructed in the limit $$\kappa = 2\pi L / \lambda _0 \rightarrow \infty$$ κ = 2 π L / λ 0 → ∞ , which is referred to as the semiclassical limit. One example is the paraxial Wentzel-Kramer-Brillouin (pWKB) approximation. However, the semiclassical limit of the wave field may be inaccurate when random short-scale fluctuations of the medium are present. A phase-space description based on the statistically averaged Wigner function may solve this problem. The Wigner function in the semiclassical limit is determined by the wave kinetic equation (WKE), derived from Maxwell’s equations. We present a paraxial expansion of the Wigner function around the central ray and derive a set of ordinary differential equations (phase-space beam-tracing equations) for the Gaussian beam width along the central ray trajectory.

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15

Barletti, Luigi, Giovanni Frosali, and Elisa Giovannini. "Adding Decoherence to the Wigner Equation." Journal of Computational and Theoretical Transport 47, no. 1-3 (April 16, 2018): 209–25. http://dx.doi.org/10.1080/23324309.2018.1520732.

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16

Ilišević, Dijana, and Aleksej Turnšek. "On superstability of the Wigner equation." Linear Algebra and its Applications 542 (April 2018): 391–401. http://dx.doi.org/10.1016/j.laa.2017.05.051.

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17

Arnold, Anton, Horst Lange, and Paul F. Zweifel. "A discrete-velocity, stationary Wigner equation." Journal of Mathematical Physics 41, no. 11 (November 2000): 7167–80. http://dx.doi.org/10.1063/1.1318732.

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18

Lee, Byoungho. "Wigner transport equation for strained heterostructures." Superlattices and Microstructures 14, no. 4 (December 1993): 291. http://dx.doi.org/10.1006/spmi.1993.1142.

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19

Emamirad, Hassan, and Philippe Rogeon. "Scattering theory for the Wigner equation." Mathematical Methods in the Applied Sciences 28, no. 8 (January 6, 2005): 947–60. http://dx.doi.org/10.1002/mma.601.

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20

Bordone, P., A. Bertoni, R. Brunetti, and C. Jacoboni. "Wigner Paths Method in Quantum Transport with Dissipation." VLSI Design 13, no. 1-4 (January 1, 2001): 211–20. http://dx.doi.org/10.1155/2001/80236.

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The concept of Wigner paths in phase space both provides a pictorial representation of the quantum evolution of the system of interest and constitutes a useful tool for numerical solutions of the quantum equation describing the time evolution of the system. A Wigner path is defined as the path followed by a “simulative particle” carrying a σ-contribution of the Wigner function through the Wigner phase-space, and is formed by ballistic free flights separated by scattering processes (both scattering with phonons and with an arbitrary potential profile can be included), as for the case of semiclassical particles. Thus, the integral transport equation can be solved by a Monte Carlo technique by means of simulative particles following classical trajectories, in complete analogy to the “Weighted Monte Carlo” solution of the Boltzmann equation in the integral form.

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21

Davies, Richard W., K. Thomas R. Davies, and Daniel S. Nydick. "Field equations for the massive vector boson from Dirac and Weinberg formalisms." Canadian Journal of Physics 91, no. 7 (July 2013): 506–18. http://dx.doi.org/10.1139/cjp-2012-0433.

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This paper is a follow-up to an earlier paper that discussed the single-particle quantum mechanics of massless bose particles. In the present paper we extend the analysis to the massive vector (j = 1) bosons that occur in electroweak interactions. As in the previous paper we make a connection between a generalization of the Dirac equation and the equations obtained by Weinberg from S-matrix field theory. The starting point is the Bargmann–Wigner generalization of the Dirac equation. This leads to the Proca equations for a vector potential field, then to Maxwell’s equations, which we finally relate to Weinberg’s equations. We spend some time analyzing the quantity Tr(Ψ(x)*Ψ(x)), where Ψ(x) is the Bargmann–Wigner wave function (a symmetric four by four matrix). Using Lagrangian and Hamiltonian density equations, we show that the trace has the interpretation of being the Hamiltonian density for the vector potential field. We also use the Lagrangian analysis to construct a conserved current via Noether’s theorem.

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22

JACOBONI, CARLO, ROSSELLA BRUNETTI, PAOLO BORDONE, and ANDREA BERTONI. "QUANTUM TRANSPORT AND ITS SIMULATION WITH THE WIGNER-FUNCTION APPROACH." International Journal of High Speed Electronics and Systems 11, no. 02 (June 2001): 387–423. http://dx.doi.org/10.1142/s0129156401000897.

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In this paper a review of the research performed in recent years by the group of the authors is presented. The definition and basic properties of the Wigner function are first given. Several forms of its dynamical equation are then derived with the inclusion of potential and phonon scattering. For the case of a potential V(r) the effect of the classical force, for any form of V(r), is separated from quantum effects due to rapidly varying potentials. An elaboration of the dynamical equation is introduced that leads to Wigner paths formed by free flights and scattering events. These are especially suitable for a Monte Carlo solution of the transport equation for the Wigner function very similar to the semiclassical traditional Monte carlo simulation. The Monte Carlo simulation can be extended also to the momentum and frequency dependent Wigner function based on a two-time Green function. Several numerical results are presented throuhout the paper.

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23

Shao, Sihong, Tiao Lu, and Wei Cai. "Adaptive Conservative Cell Average Spectral Element Methods for Transient Wigner Equation in Quantum Transport." Communications in Computational Physics 9, no. 3 (March 2011): 711–39. http://dx.doi.org/10.4208/cicp.080509.310310s.

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AbstractA new adaptive cell average spectral element method (SEM) is proposed to solve the time-dependent Wigner equation for transport in quantum devices. The proposed cell average SEM allows adaptive non-uniform meshes in phase spaces to reduce the high-dimensional computational cost of Wigner functions while preserving exactly the mass conservation for the numerical solutions. The key feature of the proposed method is an analytical relation between the cell averages of the Wigner function in the k-space (local electron density for finite range velocity) and the point values of the distribution, resulting in fast transforms between the local electron density and local fluxes of the discretized Wigner equation via the fast sine and cosine transforms. Numerical results with the proposed method are provided to demonstrate its high accuracy, conservation, convergence and a reduction of the cost using adaptive meshes.

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24

DIAS, NUNO COSTA, and JOÃO NUNO PRATA. "DEFORMATION QUANTIZATION OF CONFINED SYSTEMS." International Journal of Quantum Information 05, no. 01n02 (February 2007): 257–63. http://dx.doi.org/10.1142/s0219749907002712.

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The Weyl–Wigner formulation of quantum confined systems poses several interesting problems. The energy stargenvalue equation as well as the dynamical equation do not display the expected solutions. In this paper, we review some previous results on the subject and add some new contributions. We reformulate the confined energy eigenvalue equation by adding to the Hamiltonian a new (distributional) boundary potential. The new Hamiltonian is proved to be globally defined and self-adjoint. Moreover, it yields the correct Weyl–Wigner formulation of the confined system.

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25

DAYI, ÖMER F., and LARA T. KELLEYANE. "WIGNER FUNCTIONS FOR THE LANDAU PROBLEM IN NONCOMMUTATIVE SPACES." Modern Physics Letters A 17, no. 29 (September 21, 2002): 1937–44. http://dx.doi.org/10.1142/s0217732302008356.

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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.

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26

Chmieliński, Jacek. "Orthogonality preserving property, Wigner equation, and stability." Journal of Inequalities and Applications 2006 (2006): 1–9. http://dx.doi.org/10.1155/jia/2006/76489.

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27

Furtmaier, O., S. Succi, and M. Mendoza. "Semi-spectral method for the Wigner equation." Journal of Computational Physics 305 (January 2016): 1015–36. http://dx.doi.org/10.1016/j.jcp.2015.11.023.

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28

Jacoboni, Carlo, and Paolo Bordone. "Wigner transport equation with finite coherence length." Journal of Computational Electronics 13, no. 1 (October 2, 2013): 257–63. http://dx.doi.org/10.1007/s10825-013-0510-7.

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29

Dimov, Ivan, Mihail Nedjalkov, Jean-Michel Sellier, and Siegfried Selberherr. "Boundary conditions and the Wigner equation solution." Journal of Computational Electronics 14, no. 4 (July 19, 2015): 859–63. http://dx.doi.org/10.1007/s10825-015-0720-2.

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30

Lange, Horst, and P. F. Zweifel. "Periodic solutions to the Wigner-Poisson equation." Nonlinear Analysis: Theory, Methods & Applications 26, no. 3 (February 1996): 551–63. http://dx.doi.org/10.1016/0362-546x(94)00298-v.

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31

Singer, K., and W. Smith. "Quantum dynamics and the Wigner-Liouville equation." Chemical Physics Letters 167, no. 4 (March 1990): 298–304. http://dx.doi.org/10.1016/0009-2614(90)87171-m.

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Nedjalkov, M., S. Selberherr, D. K. Ferry, D. Vasileska, P. Dollfus, D. Querlioz, I. Dimov, and P. Schwaha. "Physical scales in the Wigner–Boltzmann equation." Annals of Physics 328 (January 2013): 220–37. http://dx.doi.org/10.1016/j.aop.2012.10.001.

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33

KOSINA, HANS. "NANOELECTRONIC DEVICE SIMULATION BASED ON THE WIGNER FUNCTION FORMALISM." International Journal of High Speed Electronics and Systems 17, no. 03 (September 2007): 475–84. http://dx.doi.org/10.1142/s0129156407004667.

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Coherent transport in mesoscopic devices is well described by the Schrödinger equation supplemented by open boundary conditions. When electronic devices are operated at room temperature, however, a realistic transport model needs to include carrier scattering. In this work the kinetic equation for the Wigner function is employed as a model for dissipative quantum transport. Carrier scattering is treated in an approximate manner through a Boltzmann collision operator. A Monte Carlo technique for the solution of this kinetic equation has been developed, based on an interpretation of the Wigner potential operator as a generation term for numerical particles. Including a multi-valley semiconductor model and a self-consistent iteration scheme, the described Monte Carlo simulator can be used for routine device simulations. Applications to single barrier and double barrier structures are presented. The limitations of the numerical Wigner function approach are discussed.

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BAL, GUILLAUME, GEORGE PAPANICOLAOU, and LEONID RYZHIK. "SELF-AVERAGING IN TIME REVERSAL FOR THE PARABOLIC WAVE EQUATION." Stochastics and Dynamics 02, no. 04 (December 2002): 507–31. http://dx.doi.org/10.1142/s0219493702000522.

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We analyze the self-averaging properties of time-reversed solutions of the paraxial wave equation with random coefficients, which we take to be Markovian in the direction of propagation. This allows us to construct an approximate martingale for the phase space Wigner transform of two wave fields. Using a prioriL2-bounds available in the time-reversal setting, we prove that the Wigner transform in the high frequency limit converges in probability to its deterministic limit, which is the solution of a transport equation.

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35

Grubin, H. L., and R. C. Buggeln. "Wigner Function Methods in Modeling of Switching in Resonant Tunneling Devices." VLSI Design 13, no. 1-4 (January 1, 2001): 221–27. http://dx.doi.org/10.1155/2001/37412.

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Issues associated with modeling of quantum devices within the framework of the transient Wigner equation are addressed. Of particular importance is the structure of the Wigner function, hysteresis, and device switching time, whose value is determined by the large signal device properties.

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GRUBIN, H. L., and R. C. BUGGELN. "Wigner Function Simulations of Quantum Device-Circuit Interactions." International Journal of High Speed Electronics and Systems 13, no. 04 (December 2003): 1255–86. http://dx.doi.org/10.1142/s0129156403002162.

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Issues associated with modeling of quantum devices within the framework of the transient Wigner equation are addressed. Of particular importance is the structure of the Wigner function, hysteresis, and device switching time, whose value is determined by the large signal device properties.

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37

Hiesmayr, Beatrix C. "The GKLS Master Equation in High Energy Physics." Open Systems & Information Dynamics 24, no. 03 (September 2017): 1740008. http://dx.doi.org/10.1142/s123016121740008x.

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Utilizing the GKLS master equation we show that the decay property of a particle can be straightforwardly incorporated. In standard particle physics the decay is often described by an efficient non-hermitian Hamiltonian, in accord with the seminal Wigner-Weisskopf approximation. We show that by enlarging the Hilbert space and defining specific GKLS operators we have attained a formalism with a hermitian Hamiltonian and probability conserving states. This proves that the Wigner-Weisskopf approximation is Markovian and completely positive. In addition, this formalism allows a straightforward generalization to many-particle decays. Last, but not least, some impacts of the GKLS master equation onto systems at high energies are reported, such as for neutral meson, neutrino and hyperon systems.

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MATERDEY, TOMAS B., and CHARLES E. SEYLER. "THE QUANTUM WIGNER FUNCTION IN A MAGNETIC FIELD." International Journal of Modern Physics B 17, no. 25 (October 10, 2003): 4555–92. http://dx.doi.org/10.1142/s0217979203022957.

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The Wigner function is shown related to the quantum dielectric function derived from the quantum Vlasov equation (QVE), with and without a magnetic field, using a standard method in plasma physics with linear perturbations and a self-consistent mean field interaction via Poisson's equation. A finite-limit-of-integration Wigner function, with oscillatory behavior and negative values for free particles, is proposed. In the classical regimes, where the problem size is huge compared to the particle wavelength, these limits go to infinity, and for free particles, the Wigner function becomes a positive delta function as expected. For the harmonic oscillator potential, there is no distinction between finite and infinite limits of integration when these are larger than the eigenfunction localization length.

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Frommlet, Florian, Peter A. Markowich, and Christian Ringhofer. "A Wignerfunction Approach to Phonon Scattering." VLSI Design 9, no. 4 (January 1, 1999): 339–50. http://dx.doi.org/10.1155/1999/30381.

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We consider the motion of a single electron under phonon scattering caused by a crystal lattice. Starting from the Fröhlich Hamiltonian in the second quantization formalism we derive a kinetic transport model by using the Wigner transformation. Under the assumption of small electron-phonon interaction we derive asymptotically the operator representing electron-phonon scattering in the Wigner equation. We then consider some scaling limits and finally we give the connection of our result to the well known Barker-Ferry equation.

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Tsekov, Roumen. "On the Stochastic Origin of Quantum Mechanics." Reports in Advances of Physical Sciences 01, no. 03 (September 2017): 1750008. http://dx.doi.org/10.1142/s2424942417500086.

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The quantum Liouville equation, which describes the phase space dynamics of a quantum system of fermions, is analyzed from stochastic point of view as a particular example of the Kramers–Moyal expansion. Quantum mechanics is extended to relativistic domain by generalizing the Wigner–Moyal equation. Thus, an expression is derived for the relativistic mass in the Wigner quantum phase space presentation. The diffusion with an imaginary diffusion coefficient is discussed. An imaginary stochastic process is proposed as the origin of quantum mechanics.

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Weber, Hannes, Omar Maj, and Emanuele Poli. "Paraxial beams in fluctuating fusion plasmas: Diffusive limit and beyond." EPJ Web of Conferences 277 (2023): 01003. http://dx.doi.org/10.1051/epjconf/202327701003.

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A paraxial expansion of the (ensemble-averaged) Wigner function in the relevant wave kinetic equation for electron cyclotron waves in fluctuating plasmas allows the derivation of phase-space equations similar to the equations for the Gaussian beam parameters in the paraxial WKB method [G.V. Pereverzev, Phys. Plasmas 5, 3529 (1998)]. This is relatively straightforward when the scattering of the wave field by density fluctuations can be described by a diffusion operator in refractive-index space. The general case is rather more complicated, yet we could find a heuristic construction of a paraxial Wigner function. Here we use a simple model, which has an analytical solution, to test both the theoretical validity of the diffusion approximation and the heuristic paraxial approach beyond the diffusion approximation.

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Zhou, Jing-Rong, and David K. Ferry. "2-D Simulation of Quantum Effects in Small Semiconductor Devices Using Quantum Hydrodynamic Equations." VLSI Design 3, no. 2 (January 1, 1995): 159–77. http://dx.doi.org/10.1155/1995/93452.

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We discuss the basis of a set of quantum hydrodynamic equations and the use of this set of equations in the two-dimensional simulation of quantum effects in deep submicron semiconductor devices. The equations are obtained from the Wigner function equation-of-motion. Explicit quantum correction is built into these equations by using the quantum mechanical expression of the moments of the Wigner function, and its physical implication is clearly explained. These equations are then applied to numerical simulation of various small semiconductor devices, which demonstrate expected quantum effects, such as barrier penetration and repulsion. These effects modify the electron density distribution and current density distribution, and consequently cause a change of the total current flow by 10-15 per cent for the simulated HEMT devices. Our work suggests that the inclusion of quantum effects into the simulation of deep submicron and ultra-submicron semiconductor devices is necessary.

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Martins, A. X., R. A. S. Paiva, G. Petronilo, R. R. Luz, R. G. G. Amorim, S. C. Ulhoa, and T. M. R. Filho. "Analytical Solution for the Gross-Pitaevskii Equation in Phase Space and Wigner Function." Advances in High Energy Physics 2020 (April 30, 2020): 1–6. http://dx.doi.org/10.1155/2020/7010957.

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In this work, we study symplectic unitary representations for the Galilei group. As a consequence a nonlinear Schrödinger equation is derived in phase space. The formalism is based on the noncommutative structure of the star product, and using the group theory approach as a guide a physically consistent theory is constructed in phase space. The state is described by a quasi-probability amplitude that is in association with the Wigner function. With these results, we solve the Gross-Pitaevskii equation in phase space and obtained the Wigner function for the system considered.

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LI, BIN, and HAN YANG. "THE MODIFIED QUANTUM WIGNER SYSTEM IN WEIGHTED -SPACE." Bulletin of the Australian Mathematical Society 95, no. 1 (October 13, 2016): 73–83. http://dx.doi.org/10.1017/s0004972716000666.

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This paper is concerned with the modified Wigner (respectively, Wigner–Fokker–Planck) Poisson equation. The quantum mechanical model describes the transport of charged particles under the influence of the modified Poisson potential field without (respectively, with) the collision operator. Existence and uniqueness of a global mild solution to the initial boundary value problem in one dimension are established on a weighted $L^{2}$-space. The main difficulties are to derive a priori estimates on the modified Poisson equation and prove the Lipschitz properties of the appropriate potential term.

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BONILLA, L. L., and R. ESCOBEDO. "WIGNER–POISSON AND NONLOCAL DRIFT-DIFFUSION MODEL EQUATIONS FOR SEMICONDUCTOR SUPERLATTICES." Mathematical Models and Methods in Applied Sciences 15, no. 08 (August 2005): 1253–72. http://dx.doi.org/10.1142/s0218202505000728.

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A Wigner–Poisson kinetic equation describing charge transport in doped semiconductor superlattices is proposed. Electrons are assumed to occupy the lowest miniband, exchange of lateral momentum is ignored and the electron–electron interaction is treated in the Hartree approximation. There are elastic collisions with impurities and inelastic collisions with phonons, imperfections, etc. The latter are described by a modified BGK (Bhatnagar–Gross–Krook) collision model that allows for energy dissipation while yielding charge continuity. In the hyperbolic limit, nonlocal drift-diffusion equations are derived systematically from the kinetic Wigner–Poisson–BGK system by means of the Chapman–Enskog method. The nonlocality of the original quantum kinetic model equations implies that the derived drift-diffusion equations contain spatial averages over one or more superlattice periods. Numerical solutions of the latter equations show self-sustained oscillations of the current through a voltage biased superlattice, in agreement with known experiments.

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Wagner, Wolfgang. "A random cloud model for the Wigner equation." Kinetic and Related Models 9, no. 1 (October 2015): 217–35. http://dx.doi.org/10.3934/krm.2016.9.217.

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Truong, T. T. "Moyal equation—Wigner distribution functions for anharmonic oscillators." Journal of Mathematical Physics 62, no. 10 (October 1, 2021): 102103. http://dx.doi.org/10.1063/5.0021132.

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Truong, T. T. "Moyal equation—Wigner distribution functions for anharmonic oscillators." Journal of Mathematical Physics 62, no. 10 (October 1, 2021): 102103. http://dx.doi.org/10.1063/5.0021132.

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Orlov, Yurii Nikolaevich. "Evolution equation for Wigner function for linear quantization." Keldysh Institute Preprints, no. 40 (2020): 1–22. http://dx.doi.org/10.20948/prepr-2020-40.

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GALLEANI, LORENZO, and LEON COHEN. "Wigner equation of motion for time-dependent potentials." Journal of Modern Optics 49, no. 3-4 (March 2002): 561–69. http://dx.doi.org/10.1080/09500340110088515.

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