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# The Wigner Function of Ground State and One-Dimensional Numerics

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## Abstract

In this paper, the ground state Wigner function of a many-body system is explored theoretically and numerically. First, an eigenvalue problem for Wigner function is derived based on the energy operator of the system. The validity of finding the ground state through solving this eigenvalue problem is obtained by building a correspondence between its solution and the solution of stationary Schrödinger equation. Then, a numerical method is designed for solving proposed eigenvalue problem in one dimensional case, which can be briefly described by i) a simplified model is derived based on a quantum hydrodynamic model [Z. Cai et al, J. Math. Chem., 2013] to reduce the dimension of the problem, ii) an imaginary time propagation method is designed for solving the model, and numerical techniques such as solution reconstruction are proposed for the feasibility of the method. Results of several numerical experiments verify our method, in which the potential application of the method for large scale system is demonstrated by examples with density functional theory.

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... Indeed, it has already taken over thirty years to develop efficient Wigner solvers, including both deterministic and stochastic algorithms. In contrast to the relatively newer branch of particle-based stochastic methods, [12][13][14] which usually exhibit slower convergence rate, grid-based deterministic solvers allow highly accurate numerical resolutions in the light of their concise principle and solid mathematical foundation, ranging from the finite difference scheme 15 and the spectral collocation method combined with the operator splitting 16,17 to the recent advanced techniques such as the spectral element method, 18-20 the spectral decomposition 21 and the Hermite spectral method, 22,23 as well as those for advection such as the discontinuous Galerkin method, 24 WENO scheme 25 and exponential integrators. 22 Unfortunately, there still remains a huge gap in terms of the applicability of even the state-of-the-art deterministic scheme to full 6-D problems, and the foremost problem is definitely the storage of 6-D grid mesh. ...
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Numerical resolution for 6-D Wigner dynamics under the Coulomb potential faces with the combined challenges of high dimensionality, nonlocality, oscillation and singularity. In particular, the extremely huge memory storage of 6-D grids hinders the usage of all existing deterministic numerical scheme, which is well-known as the curse of dimensionality. To surmount these difficulties, we propose a massively parallel solver, termed the CHAracteristic-Spectral-Mixed (CHASM) scheme, by fully exploiting two distinct features of the Wigner equation: Locality of spatial advection and nonlocality of quantum interaction. Our scheme utilizes the local cubic B-spline basis to interpolate the local spatial advection. The key is to use a perfectly matched boundary condition to give a closure of spline coefficients, so that distributed pieces can recover the global one as accurately as possible owing to the rapid decay of wavelet basis in the dual space, and communication costs are significantly reduced. To resolve the nonlocal pseudodifferential operator with weakly singular symbol, CHASM further adopts the truncated kernel method to attain a highly efficient approximation. Several typical experiments including the quantum harmonic oscillator and Hydrogen 1s state demonstrate the accuracy and efficiency of CHASM. The non-equilibrium electron-proton couplings are also clearly displayed and reveal the uncertainty principle and quantum tunneling in phase space. Finally, the scalability of CHASM up to 16000 cores is presented.
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An accurate description of 2-D quantum transport in a double-gate metal oxide semiconductor filed effect transistor (dgMOSFET) requires a high-resolution solver to a coupled system of the 4-D Wigner equation and 2-D Poisson equation. In this paper, we propose an operator splitting spectral method to evolve such Wigner-Poisson system in 4-D phase space with high accuracy. After an operator splitting of the Wigner equation, the resulting two sub-equations can be solved analytically with spectral approximation in phase space. Meanwhile, we adopt a Chebyshev spectral method to solve the Poisson equation. Spectral convergence in phase space and a fourth-order accuracy in time are both numerically verified. Finally, we apply the proposed solver into simulating dgMOSFET, develop the steady states from long-time simulations and obtain numerically converged current-voltage (I-V) curves.
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The stationary Wigner inflow boundary value problem (SWIBVP) is modeled as an optimization problem by using the idea of shooting method in this paper. To remove the singularity at $$v=0$$, we consider a regularized SWIBVP, where a regularization constraint is considered along with the original SWIBVP, and a modified optimization problem is established for it. A shooting algorithm is proposed to solve the two optimization problems, involving the limited-memory BFGS (L-BFGS) algorithm as the optimization solver. Numerical results show that solving the optimization problems with respect to the SWIBVP with the shooting algorithm is as effective as solving the SWIBVP with Frensley’s numerical method (Frensley in Phys Rev B 36:1570–1580, 1987). Furthermore, the modified optimization problem gets rid of the singularity at $$v=0$$, and preserves symmetry of the Wigner function, which implies the optimization modeling with respect to the regularized SWIBVP is successful.
Article
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The Wigner function was formulated in 1932 by Eugene Paul Wigner, at a time when quantum mechanics was in its infancy. In doing so, he brought phase space representations into quantum mechanics. However, its unique nature also made it very interesting for classical approaches and for identifying the deviations from classical behavior and the entanglement that can occur in quantum systems. What stands out, though, is the feature to experimentally reconstruct the Wigner function, which provides far more information on the system than can be obtained by any other quantum approach. This feature is particularly important for the field of quantum information processing and quantum physics. However, the Wigner function finds wide-ranging use cases in other dominant and highly active fields as well, such as in quantum electronics—to model the elec- tron transport, in quantum chemistry—to calculate the static and dynamical properties of many- body quantum systems, and in signal processing—to investigate waves passing through certain media. What is peculiar in recent years is a strong increase in applying it: Although originally for- mulated 86years ago, only today the full potential of the Wigner function—both in ability and diversity—begins to surface. This review, as well as a growing, dedicated Wigner community, is a testament to this development and gives a broad and concise overview of recent advancements in different fields.
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Monte Carlo techniques have played an important role in understanding strongly-correlated systems across many areas of physics, covering a wide range of energy and length scales. Among the many Monte Carlo methods applicable to quantum mechanical systems, the path integral Monte Carlo approach with its variants has been employed widely. Since semi-classical or classical approaches will not be discussed in this review, path integral based approaches can for our purposes be divided into two categories: approaches applicable to quantum mechanical systems at zero temperature and approaches applicable to quantum mechanical systems at finite temperature. While these two approaches are related to each other, the underlying formulation and aspects of the algorithm differ. This paper reviews the path integral Monte Carlo ground state (PIGS) approach, which solves the time-independent Schroedinger equation. Specifically, the PIGS approach allows for the determination of expectation values with respect to eigen states of the few- or many-body Schroedinger equation provided the system Hamiltonian is known. The theoretical framework behind the PIGS algorithm, implementation details, and sample applications for sermonic systems are presented.
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Quantum interference and exchange statistical effects can affect the momentum distribution functions making them non-Maxwellian. Such effects may be important in studies of kinetic properties of matter at low temperatures and under extreme conditions. In this work we have generalized the path integral representation for Wigner function to strongly coupled three-dimensional quantum system of particles with Boltzmann and Fermi statistics. In suggested approach the explicit expression for Wigner function was obtained in harmonic approximation and Monte Carlo method allowing numerical calculation of Wigner function, distribution functions and average quantum values has been developed. As alternative more accurate single-momentum approach and related Monte Carlo method have been developed to calculation of the distribution functions of degenerate system of interacting fermions. It allows partially overcoming the well-known sign problem for degenerate Fermi systems.
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Wigner’s 1932 quasi-probability Distribution Function in phase-space, his first paper in English, is a special (Weyl) representation of the density matrix. It has been useful in describing quantum flows in semiclassical limits; quantum optics; nuclear and physics; decoherence (eg, quantum computing); quantum chaos; “Welcher Weg” puzzles; molecular Talbot–Lau interferometry; atomic measurements. It is further of great importance in signal processing (time-frequency analysis). Nevertheless, a remarkable aspect of its internal logic, pioneered by H. Groenewold and J. Moyal, has only blossomed in the last quarter-century: It furnishes a third, alternate, formulation of Quantum Mechanics, independent of the conventional Hilbert Space (the gold medal), or Path Integral (the silver medal) formulations, and perhaps more intuitive, since it shares language with classical mechanics: one need not choose sides between coordinate or momentum space variables, since it is formulated simultaneously in terms of position and momentum. This bronze medal formulation is logically complete and self-standing, and accommodates the uncertainty principle in an unexpected manner, so that it offers unique insights into the classical limit of quantum theory. The observables in this formulation are cnumber functions in phase space instead of operators, with the same interpretation as their classical counterparts, only now composed together in novel algebraic ways using star products. One might then envision an imaginary world in which this formulation of quantum mechanics had preceded the conventional Hilbert-space formulation, and its own techniques and methods had arisen independently, perhaps out of generalizations of classical mechanics and statistical mechanics. A sampling of such intriguing techniques and methods has already been published in C. K. Zachos, Int Jou Mod Phys A17 297-316 (2002), and T. L. Curtright, D. B. Fairlie, and C. K. Zachos, A Concise Treatise on Quantum Mechanics in Phase Space, (Imperial Press & World Scientific, 2014).
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The Gibbs canonical state, as a maximum entropy density matrix, represents a quantum system in equilibrium with a thermostat. This state plays an essential role in thermodynamics and serves as the initial condition for non-equilibrium dynamical simulations. We solve a long standing problem for computing the Gibbs state Wigner function with nearly machine accuracy by solving the Bloch equation directly in the phase space. Furthermore, the algorithm is provided yielding high quality Wigner distributions of pure stationary states. The developed numerical methods furnish a long-sought efficient computation framework for non-equilibrium quantum simulations directly in the Wigner representation.
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As a phase space language for quantum mechanics, the Wigner function approach bears a close analogy to classical mechanics and has been drawing growing attention, especially in simulating quantum many-body systems. However, deterministic numerical solutions have been almost exclusively confined to one-dimensional one-body systems and few results are reported even for one-dimensional two-body problems. This paper serves as the first attempt to solve the time-dependent many-body Wigner equation through a grid-based advective-spectral-mixed method. The main feature of the method is to resolve the linear advection in $(\bm{x},t)$-space by an explicit three-step characteristic scheme coupled with the piecewise cubic spline interpolation, while the Chebyshev spectral element method in $\bm k$-space is adopted for accurate calculation of the nonlocal pseudo-differential term. Not only the time step of the resulting method is not restricted by the usual CFL condition and thus a large time step is allowed, but also the mass conservation can be maintained. In particular, for the system consisting of identical particles, the advective-spectral-mixed method can also rigorously preserve physical symmetry relations. The performance is validated through several typical numerical experiments, like the Gaussian barrier scattering, electron-electron interaction and a Helium-like system, where the third-order accuracy against both grid spacing and time stepping is observed.
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A 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 theWigner equation.
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We present a novel numerical scheme for the deterministic solution of the Wigner transport equation, especially suited to deal with situations in which strong quantum effects are present. The unique feature of the algorithm is the expansion of the Wigner function in local basis functions, similar to finite element or finite volume methods. This procedure yields a discretization of the pseudo-differential operator that conserves the particle density on arbitrarily chosen grids. The high flexibility in refining the grid spacing together with the weighted essentially non-oscillatory (WENO) scheme for the advection term allows for an accurate and well-resolved simulation of the phase space dynamics. A resonant tunneling diode is considered as test case and a detailed convergence study is given by comparing the results to a non-equilibrium Green's functions calculation. The impact of the considered domain size and of the grid spacing is analyzed. The obtained convergence of the results towards a quasi exact agreement of the steady state Wigner and Green's functions computations demonstrates the accuracy of the scheme, as well as the high flexibility to adjust to different physical situations.
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The Wigner equation is a promising full quantum model for the simulation of nanodevices. It is also a challenging numerical problem. Two basic Monte Carlo approaches to this model exist exploiting, in the time-dependent case, the so-called particle affinity and, in the stationary case, integer particle signs. In this paper we extend the second approach for time-dependent simulations and present a validation against a well-known benchmark model, the Schrödinger equation. Excellent quantitative agreement is demonstrated by the compared results despite the very different numerical properties of the utilized stochastic and deterministic approaches.
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Wigner's quasi-probability distribution function in phase-space is a special (Weyl) representation of the density matrix. It has been useful in describing quantum transport in quantum optics; nuclear physics; decoherence (e.g. quantum computing); quantum chaos; "Welcher Weg" discussions; semiclassical limits. It is also of importance in signal processing. Nevertheless, a remarkable aspect of its internal logic, pioneered by Moyal, has only emerged in the last quarter-century: It furnishes a third, alternative, formulation of Quantum Mechanics, independent of the conventional Hilbert Space, or Path Integral formulations. In this logically complete and self-standing formulation, one need not choose sides - coordinate or momentum space. It works in full phase-space, accommodating the uncertainty principle. This is an introductory overview of the formulation with simple illustrations.
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For the Wigner equation with discontinuous potentials, a phase space Gaussian beam (PSGB) summation method is proposed in this paper. We first derive the equations satisfied by the parameters for PSGBs and establish the relations for parameters of the Gaussian beams between the physical space (GBs) and the phase space, which motivates an efficient initial data preparation thus a reduced computational cost than previous method in the literature. The method consists of three steps: 1) Decompose the initial value of the wave function into the sum of GBs and use the parameter relations to prepare the initial values of PSGBs; 2) Solve the evolution equations for each PSGB; 3) Sum all the PSGBs to construct the approximate solution of the Wigner equation. Additionally, in order to connect PSGBs at the discontinuous points of the potential, we provide interface conditions for a single phase space Gaussian beam. Numerical examples are given to verify the validity and accuracy of method.
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Wigner quantum mechanics is reformulated in a discrete momentum space and analyzed within a Monte Carlo approach for solving integral equations and thus associated with a particle picture. General quantum phenomena may thereby be modeled in terms of quasi-particles involving attributes such as drift, generation, sign, and annihilation on a phase space grid. The model is examined in an ultimate regime, where classical and quantum dynamics become equivalent. The peculiarities of the transport in this asymptotic regime are analyzed within simulations, benchmarking the behavior of the Wigner function.
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We demonstrate the ability of the phase space formulation of quantum mechanics to provide convenient means and intuitive notions for exploring the process of transition from a quantum to a classical state known as decoherence. The Wigner equation, which is usually relevant for electron transport in nanostructures, augmented by the Boltzmann scattering operator is now applied to the time dependent transport problems which may be considered as benchmark examples for the decoherence role of phonons in semiconductor devices. Simulation results maintained by theoretical analysis show how scattering effectively destroys the interference effects. The initial coherence in the wave vector distribution is pushed towards the equilibrium distribution. In particular scattering by phonons hinders the natural spread of the density with time and advances it towards a classical localization. Furthermore, the decoherence effect due to phonons, is measured by the purity of the Wigner state, which decreases from its initial value of 1, with a rate depending on the lattice temperature, and by a functional comparing diagonal with off-diagonal elements of the density matrix.
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In this paper, we extend the method in Cai et al. (J Math Phys 53:103503, 2012) to derive a class of quantum hydrodynamic models for the density-functional theory (DFT). The most popular implement of DFT is the Kohn–Sham equation, which transforms a many-particle interacting system into a fictitious non-interacting one-particle system. The Kohn–Sham equation is a non-linear Schrödinger equation, and the corresponding Wigner equation can be derived as an alternative implementation of DFT. We derive quantum hydrodynamic models of the Wigner equation by moment closure following Cai et al. (J Math Phys 53:103503, 2012). The derived quantum hydrodynamic models are globally hyperbolic thus locally wellposed. The contribution of the Kohn–Sham potential is turned into a nonlinear source term of the hyperbolic moment system. This work provides a new possible way to solve DFT problems.
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Small semiconductor devices can be separated into regions where the electron transport has classical character, neighboring with regions where the transport requires a quantum description. The classical transport picture is associated with Boltzmann-like particles that evolve in the phase-space defined by the wave vector and real space coordinates. The evolution consists of consecutive processes of drift over Newton trajectories and scattering by phonons. In the quantum regions, a convenient description of the transport is given by the Wigner-function formalism. The latter retains most of the basic classical notions, particularly, the concepts for phase-space and distribution function, which provide the physical averages. In this work we show that the analogy between classical and Wigner transport pictures can be even closer. A particle model is associated with the Wigner-quantum transport. Particles are associated with a sign and thus become positive and negative. The sign is the only property of the particles related to the quantum information. All other aspects of their behavior resemble Boltzmann-like particles. The sign is taken into account in the evaluation of the physical averages. The sign has a physical meaning because positive and negative particles that meet in the phase space annihilate one another. The Wigner and Boltzmann transport pictures are explained in a unified way by the processes drift, scattering, generation, and recombination of positive and negative particles. The model ensures a seamless transition between the classical and quantum regions. A stochastic method is derived and applied to simulation of resonant-tunneling diodes. Our analysis shows that the method is useful if the physical quantities do not vary over several orders of magnitude inside a device.
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We propose a numerical method to solve the Wigner equation in quantum systems of spinless, non-relativistic particles. The method uses a spectral decomposition into $L^2(\mathbb{R}^d)$ basis functions in momentum-space to obtain a system of first-order advection-reaction equations. The resulting equations are solved by splitting the reaction and advection steps so as to allow the combination of numerical techniques from quantum mechanics and computational fluid dynamics by identifying the skew-hermitian reaction matrix as a generator of unitary rotations. The method is validated for the case of particles subject to a one-dimensional (an-)harmonic potential using finite-differences for the advection part. Thereby, we verify the second order of convergence and observe non-classical behavior in the evolution of the Wigner function.
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The theory of the one-dimensional (1D) hydrogen atom was initiated by a 1952 paper but, after more than 60 years, it remains a topic of debate and controversy. The aim here is a critique of the current status of the theory and its relation to relevant experiments. A 1959 solution of the Schrödinger equation by the use of a cut-off at x=a to remove the singularity at the origin in the 1/|x| form of the potential is clarified and a mistaken approximation is identified. The singular atom is not found in the real world but the theory with cut-off has been applied successfully to a range of four practical three-dimensional systems confined towards one dimension, particularly their observed large increases in ground state binding energy. The true 1D atom is in principle restored when the short distance a tends to zero but it is sometimes claimed that the solutions obtained by the limiting procedure differ from those obtained by solution of the basic Schrödinger equation without any cut-off in the potential. The treatment of the singularity by a limiting procedure for applications to practical systems is endorsed. ©2016 The Author(s) Published by the Royal Society. All rights reserved.
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Density Functionals for Non-relativistic Coulomb Systems in the New Century.- Orbital-Dependent Functionals for the Exchange-Correlation Energy: A Third Generation of Density Functionals.- Relativistic Density Functional Theory.- Time-Dependent Density Functional Theory.- Density Functional Theories and Self-energy Approaches.- A Tutorial on Density Functional Theory.
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The aim of ab-initio approaches is the simulation of many-body quantum systems from the first principles of quantum mechanics. These methods are traditionally based on the many-body Schrödinger equation which represents an incredible mathematical challenge. In this paper, we introduce the many-body Wigner Monte Carlo method in the context of distinguishable particles and in the absence of spin-dependent effects. Despite these restrictions, the method has several advantages. First of all, the Wigner formalism is intuitive, as it is based on the concept of a quasi-distribution function. Secondly, the Monte Carlo numerical approach allows scalability on parallel machines that is practically unachievable by means of other techniques based on finite difference or finite element methods. Finally, this method allows time-dependent ab-initio simulations of strongly correlated quantum systems. In order to validate our many-body Wigner Monte Carlo method, as a case study we simulate a relatively simple system consisting of two particles in several different situations. We first start from two non-interacting free Gaussian wave packets. We, then, proceed with the inclusion of an external potential barrier, and we conclude by simulating two entangled (i.e. correlated) particles. The results show how, in the case of negligible spin-dependent effects, the many-body Wigner Monte Carlo method provides an efficient and reliable tool to study the time-dependent evolution of quantum systems composed of distinguishable particles.
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In order to simulate quantum N-body systems, stationary and time-dependent density functional theories rely on the capacity of calculating the single-electron wave-functions of a system from which one obtains the total electron density (Kohn–Sham systems). In this paper, we introduce the use of the Wigner Monte Carlo method in ab-initio calculations. This approach allows time-dependent simulations of chemical systems in the presence of reflective and absorbing boundary conditions. It also enables an intuitive comprehension of chemical systems in terms of the Wigner formalism based on the concept of phase-space. Finally, being based on a Monte Carlo method, it scales very well on parallel machines paving the way towards the time-dependent simulation of very complex molecules. A validation is performed by studying the electron distribution of three different systems, a Lithium atom, a Boron atom and a hydrogenic molecule. For the sake of simplicity, we start from initial conditions not too far from equilibrium and show that the systems reach a stationary regime, as expected (despite no restriction is imposed in the choice of the initial conditions). We also show a good agreement with the standard density functional theory for the hydrogenic molecule. These results demonstrate that the combination of the Wigner Monte Carlo method and Kohn–Sham systems provides a reliable computational tool which could, eventually, be applied to more sophisticated problems.
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The Schrödinger eigenvalue problem is solved with the imaginary time propagation technique. The separability of the Hamiltonian makes the problem suitable for the application of splitting methods. High order fractional time steps of order greater than two necessarily have negative steps and cannot be used for this class of diffusive problems. However, there exist methods which use fractional complex time steps with positive real parts which can be used with only a moderate increase in the computational cost. We analyze the performance of this class of schemes and propose new methods which outperform the existing ones in most cases. On the other hand, if the gradient of the potential is available, methods up to fourth order with real and positive coefficients exist. We also explore this case and propose new methods as well as sixth-order methods with complex coefficients. In particular, highly optimized sixth-order schemes for near integrable systems using positive real part complex coefficients with and without modified potentials are presented. A time-stepping variable order algorithm is proposed and numerical results show the enhanced efficiency of the new methods.
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In this paper, a framework of using h-adaptive finite element method for the Kohn–Sham equation on the tetrahedron mesh is presented. The Kohn–Sham equation is discretized by the finite element method, and the h-adaptive technique is adopted to optimize the accuracy and the efficiency of the algorithm. The locally optimal block preconditioned conjugate gradient method is employed for solving the generalized eigenvalue problem, and an algebraic multigrid preconditioner is used to accelerate the solver. A variety of numerical experiments demonstrate the effectiveness of our algorithm for both the all-electron and the pseudo-potential calculations.
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This is a study of simple kinetic models of open systems, in the sense of systems that can exchange conserved particles with their environment. The system is assumed to be one dimensional and situated between two particle reservoirs. Such a system is readily driven far from equilibrium if the chemical potentials of the reservoirs differ appreciably. The openness of the system modifies the spatial boundary conditions on the single-particle Liouville-von Neumann equation, leading to a non-Hermitian Liouville operator. If the open-system boundary conditions are time reversible, exponentially growing (unphysical) solutions are introduced into the time dependence of the density matrix. This problem is avoided by applying time-irreversible boundary conditions to the Wigner distribution function. These boundary conditions model the external environment as ideal particle reservoirs with properties analogous to those of a blackbody. This time-irreversible model may be numerically evaluated in a discrete approximation and has been applied to the study of a resonant-tunneling semiconductor diode. The physical and mathematical properties of the irreversible kinetic model, in both its discrete and its continuum formulations, are examined in detail. The model demonstrates the distinction in kinetic theory between commutator superoperators, which may become non-Hermitian to describe irreversible behavior, and anticommutator superoperators, which remain Hermitian and are used to evaluate physical observables.
Article
This paper deals with the ground state of an interacting electron gas in an external potential v(r). It is proved that there exists a universal functional of the density, Fn(r), independent of v(r), such that the expression Ev(r)n(r)dr+Fn(r) has as its minimum value the correct ground-state energy associated with v(r). The functional Fn(r) is then discussed for two situations: (1) n(r)=n0+n(r), n/n01, and (2) n(r)= (r/r0) with arbitrary and r0. In both cases F can be expressed entirely in terms of the correlation energy and linear and higher order electronic polarizabilities of a uniform electron gas. This approach also sheds some light on generalized Thomas-Fermi methods and their limitations. Some new extensions of these methods are presented.
Article
We present a method for the direct computation of the Wigner function by solving a coupled system of linear partial differential equations. Our procedure is applicable to arbitrary binding potentials. We introduce a modified spectral tau method that uses Chebyshev polynomials as shape functions to approximate the solution. Since two differential equations are solved simultaneously, the resulting linear equation system is overdetermined. We approximate its solution by a least-squares method. We prove the stability and convergence of our scheme. As an application, we compute numerically the Wigner function for the harmonic oscillator. Our calculations show excellent agreement with known analytic results.
Article
. We show how the quantum analog of the Fokker-Planck equation for describing Brownian motion can be obtained as the diffusive limit of the quantum linear Boltzmann equation. The latter describes the quantum dynamics of a tracer particle in a dilute, ideal gas by means of a translation-covariant master equation. We discuss the type of approximations required to obtain the generalized form of the Caldeira-Leggett master equation, along with their physical justification. Microscopic expressions for the diffusion and relaxation coefficients are obtained by analyzing the limiting form of the equation in both the Schrödinger and the Heisenberg picture.
Article
We have constructed the Wigner function for the ground state of the hydrogen atom and analysed its variation over phase space. By means of the Weyl correspondence between operators and phase space functions we have then studied the description of angular momentum and resolved a dilemma in the comparison with early quantum mechanics. Finally we have discussed the introduction of local energy densities in coordinate space and demonstrated the validity of a local virial theorem.
Article
The problem of two particles in a common harmonic oscillator potential interacting through harmonic oscillator forces is discussed in the Weyl–Wigner phase-space representation. The Wigner function of the system is an ordinary function of the phase-space constants of the motion. The density functional description of the system is touched upon. The functional that determines the confining harmonic potential from the particle density is also an ordinary function.
Article
We generalize our modified spectral method for the solution of the coupled real partial differential equations in phase space for the stationary Wigner function of an energy eigenstate. This generalization allows us to apply our algorithm to arbitrary high-order partial derivatives without increasing the numerical costs. This is possible since we can derive a sum factorization formula converting a multiple sum into a simple product. We apply our method to evaluate the Wigner function of the Morse oscillator and an asymmetric double-well potential, and compare our results with the exact solution when it is known.
Article
We present a unified description of the position‐space wave functions, the momentum‐space wave functions, and the phase‐space Wigner functions for the bound states of a Morse oscillator. By comparing with the functions for the harmonic oscillator the effects of anharmonicity are visualized. Analytical expressions for the wave functions and the phase space functions are given, and it is demonstrated how a numerical problem arising from the summation of an alternating series in evaluating Laguerre functions can be circumvented. The method is applicable also for other problems where Laguerre functions are to be calculated. The wave and phase space functions are displayed in a series of curves and contour diagrams. An Appendix discusses the calculation of the modified Bessel functions of real, positive argument and complex order, which is required for calculating the phase space functions for the Morse oscillator.
Article
The probability of a configuration is given in classical theory by the Boltzmann formula exp[−VhT] where V is the potential energy of this configuration. For high temperatures this of course also holds in quantum theory. For lower temperatures, however, a correction term has to be introduced, which can be developed into a power series of h. The formula is developed for this correction by means of a probability function and the result discussed.
Article
In previous publications (especially by Hohenberg, Kohn, and Sham), a theory of the ground state of an inhomogeneous interacting electron gas was developed, in which the electronic density $n(\mathrm{r})$ played a dominant role. The present paper extends this approach to the one-particle Green's function and physical properties related to it, such as single-particle-like excitations and, in the case of metals, the Fermi surface. The Dyson mass operator $$\Sigma{} is studied as a function of its spatial arguments and as a functional of n(\mathrm{r}), and, in both senses, it is found to have important short-range properties. An approximation for$$\Sigma${}$, which is exact for systems of slowly varying density, is proposed. This leads to simple, explicit, Schr\"odinger-like equations for the single-particle-like excitations, whose solution determines their energies and lifetimes. In particular, we show how to apply this procedure to metals.
Article
Correlation energy makes a small but very important contribution to the total energy of an electronic system. Among the traditional methods used to study electronic correlation are coupled clusters (CC), configuration interaction (CI) and manybody perturbation theory (MBPT) in quantum chemistry, and density functional theory (DFT) in solid state physics. An alternative method, which has been applied successfully to systems ranging from the homogeneous electron gas, to atoms, molecules, solids and clusters is quantum Monte Carlo (QMC). In this method the Schrödinger equation is transformed to a diffusion equation which is solved using stochastic methods. In this work we review some of the basic aspects of QMC in two of its variants, variational (VMC) and diffusion Monte Carlo (DMC). We also review some of its applications, such as the homogeneous electron gas, atoms and the inhomogeneous electron gas (jellium surface). The correlation energy obtained by Ceperley and Alder (D.M. Ceperley and B.J. Alder, Physical Review, 45 (1980) 566), as parameterized by Perdew and Zunger (J.P. Perdew and A. Zunger, Phys. Rev. B23 (1980) 5469), is one of the most used in DFT calculations in the local density approximation (LDA). Unfortunately, the use of the LDA in inhomogeneous systems is questionable, and better approximations are desired or even necessary. We present results of the calculations performed on metallic surfaces in the jellium model which can be useful to obtain better approximations for the exchange and correlation functionals. We have computed the electronic density, work function, surface energy and pair correlation functions for a jellium slab at the average density of magnesium (rs = 2.66). Since there is an exact expression for the exchange and correlation functional in terms of the pair correlation functions, the knowledge of such functions near the edge of the surface may be useful to obtain exchange and correlation functionals valid for inhomogeneous systems. From the exchange and correlation functional we can conclude that the exchange-correlation hole is nearly spherical in the bulk region but elongated in the direction perpendicular to the surface as the electron approaches the edge of the surface, showing the anisotropic character of the electronic correlation near the surface.
Article
Our problems are aboutα the correspondence a ↔ a between physicial quantities a and quantum operators a (quantization) andβ the possibility of understanding the statistical character of quantum mechanics by averaging over uniquely determined processes as in classical statistical mechanics (interpretation).α and β are closely connected. Their meaning depends on the notion of observability.We have tried to put these problems in a form which is fit for discussion. We could not bring them to an issue. (We are inclined to restrict the meaning of α to the trivial correspondence a → a (for lim ħ → 0) and to deny the possibility suggested in β).Meanwhile special attention has been paid to the measuring process (coupling, entanglement; ignoration, infringement; selection, measurement).For the sake of simplicity the discussion has been confined to elementary non-relativistic quantum mechanics of scalar (spinless) systems with one linear degree of freedom without exchange. Exact mathematical rigour has not been aimed at.