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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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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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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.