A discrete formulation of the wigner distribution function wdf and the wigner transport equation wte is proposed, where the discreteness of the wdf and wte is not just a practical. This limits accuracy of predictions for many topological insulators and materials with complex structure and. Accuracy balancing for the simulation of gateallaround. We are now going to solve this equation by multiplying both sides by e. After a proper fourier transform a wigner function depending on p and w as independent variables is obtained. The quantum version of the boltzmann transport equation wignerboltzmann equation is a quite. Transport equation for thephoton wigner operatorin non. Transport equations for the wigner distribution function in a general linear medium are derived. The fourier transform is one example of an integral transform. Quantum mechanics in phase space, wigner transport equation, quan tum transport. The major novelty compared to the scalar schr\odinger case is the proof of convergence of crossmodes to 0 weakly in. One has a symplectic structure on r2d h d that is naturally induced from the full schr odinger dynamics as well as a hamiltonian function on r2d h d given as the expectation value of the hamiltonian operator h with respect to the gaussian wave packet. Chapter 1 derivation of the navierstokes equations 1. In the latter form, the transport equation becomes very convenient for device simulation.
With its success, however, a certain casualness has been bred concerning its application, without much regard for hypotheses and when they are valid. The wigner eckart theorem is based on an analysis of how operators transform under rotations. Each of these sixteen components is connected with a corresponding physical quantity51, 58. Application of the wigner distribution function in optics. Geometry and dynamics of gaussian wave packets and their wigner transforms 3 r2d h d as opposed to just r2d. This tutorial gives an overview of the use of the wigner function as a tool for modeling optical field propagation. Finally, the most general transport equation in x direction is. For the pure state, the wigner distribution function is defined. Kordyukov, the egorov theorem for transverse diractype operators on foliated manifolds, journal of geometry and physics, 57, 11, 2345, 2007.
Equaltime hierarchies for quantum transport theory abstract. We say that r is a resolving set of d if for every two points x1. Wigner function and quantum transport equation xingyu guo1 ziyue wang2 pengfei zhuang2 1institute of quantum matter south china university 2department of physics tsinghua university particle 2019 meeting guo, wang, zhuang scnu and tsinghua wigner function particle 2019 meeting129. In the absence of scattering and manybody effects, the result reduces to the.
Exact integral operator form of the wigner distribution. Derivatives are turned into multiplication operators. Solutions of differential equations using transforms process. A discrete formulation of the wigner transport equation.
It shows how the liouville equation follows in leading order in an expansion to properties of this expansion and of the resulting equation are discussed with respect to their physical and practical importance. In the first case, the wigner function gives a representation of the field that is similar to a radiance or weight distribution for all. This book gives an overview of the quantum transport approaches for nanodevices and focuses on the wigner formalism. Electronic transport properties of semiconductors with small band gaps are often not well described by semiclassical methods, such as the boltzmann transport equation, because of the missing interaction between carriers whose band energy differences are closer than their linewidth. On the wigner transform of solutions to the schrodinger equation.
Dissipative transport in superlattices within the wigner. It turns out that operators of a certain type, the irreducible tensor operators, are associated with angular momentum quantum numbers and have transformation properties similar to those of kets. The wigner montecarlo method for nanoelectronic devices. Questions tagged wignertransform physics stack exchange. Wigner transport equation thus reduces to the socalled wignerboltzmann equa tion. It details the implementation of a particlebased monte carlo solution of the wigner transport equation and how the technique is applied to typical devices exhibiting quantum phenomena, such as the resonant tunnelling diode, the ultrashort silicon mosfet and the carbon nanotube. A guide to numerical methods for transport equations. It was introduced by eugene wigner in 1932 to study quantum corrections to classical statistical mechanics. The superior scaling of the fast fourier transform 20 is also made use of in the. In the liouville approximation they take the form of a first. The result is given in integral operator form suitable for application to the numerical modeling of quantum tunneling and quantum interference solid state devices. That is, we shall fourier transform with respect to the spatial variable x. The main mathematical tool in the derivation of the kinetic limit is the matrixvalued wigner transform of the vectorvalued dirac solution. The proposed device adaptive inflow boundary condition includes this effect by assigning the wigner distribution to the value obtained from the wigner transform of wave functions inside the device at zero external bias voltage, thus including the dominant.
Physics 221a fall 2019 notes 19 irreducible tensor. An analytical solution for the wignerboltzmann transport equation. Radiative transport limit of dirac equations with random electromagnetic field. A device adaptive inflow boundary condition for wigner. Fourier transform techniques 1 the fourier transform. The wigner transport equation is derived by applying the wigner transform.
A device adaptive inflow boundary condition for wigner equations of quantum transport. Particular emphasis is placed on the spatial propagation of stationary fields, as well as on the propagation of pulses through dispersive media. Conservation of the l2norm implies that for every realization of the random potential v. Use for issues relating to the weyl correspondence the inverse of the wigner transform, the wigner function the wigner transform of the density matrix and, in general, quantum mechanics in phase space issues, such as the product, the wigner transform of the operator multiplication. We will show, however, that the published derivation of the equaltime. Wigner equation by extending the techniques that have already developed for the bte, including the spherical harmonic expansion57. Semiclassical approximations to the collision term are. A wenosolver combined with adaptive momentum discretization. A novel numerical scheme for the wigner transport equation and its. The presented detailed derivation of the wigner equation is based. Solutions of differential equations using transforms. However, later a fully independent derivation was developed based on the moyal.
The inverse transform of fk is given by the formula 2. He received the nobel prize in physics in 1963 for his contributions to the theory of the atomic nucleus and the elementary particles, particularly through the discovery and. The derivation of the wigner transport equation properties of the wigner fucntion another form of the wigner function derivation of the wigere equation q,k reference the derivation of the wigner transport equation i taking the derivative of pwq,p,t w. Wigner distribution function in the case of a canonical ensemble can be derived in a completely. Vasak, transport equations for the qcd gluon wigner operator, phys. Note that gaussian random matrix ensembles can be also noninvariant. Wigner equation for general electromagnetic fields. We show that unphysical simulation results are obtained unless we pay attention to the accuracy balancing problem. An important peculiarity is that the differential part of the derived wigner equation resembles the liouville operator corresponding to the case of the magnetic. An analytical solution for the wignerboltzmann transport. Instead of capital letters, we often use the notation fk for the fourier transform, and f x for the inverse transform.
Extending the spherical har monic methodology to the wigner equation allows for reduction of dimensionality, as well as for a rigorous treat ment of collisions. Wigner transport equation thus reduces to the socalled wigner boltzmann equation. Wigner function approach institute for microelectronics. The relativistically covariant wigner operator is the fourier transform eq. Complete and consistent chiral transport from wigner. A monte carlo solution of the wigner transport equation. We report the application of the discrete wigner transport equation to the simulation of gateallaround junctionless nanowire transistors jlnwts. Alternative computational methods for boltzmann and wigner models in charged transport systems. We are living in a 3 dimensional space, where the same rules for the general mass balance and transport are valid in all dimensions. Regardless, the weyl wigner transform is a welldefined integral transform between the phasespace and operator representations, and yields insight into the workings of quantum mechanics.
Alternative computational methods for boltzmann and wigner. Denote the fourier transform with respect to x, for each. We discuss the details of such a derivation in appendix a. The wigner transform is the bridge between hilbert space operators to phasespace quantities cnumbers. Politis, exact equations for smoothed wigner transforms and homogenization of wave propagation, pamm, 7, 1, 20400192040020, 2008. Zweifel center for transport theory and mathematical physics virginia polytechnic institute and state university blacksburg, virginia 240610435, usa received january 20, 1993 in 1932, eugene wigner 1 invented a phasespace representation of. The generic noninvariant gaussian rme is determined by the pdf of the form. The approach is based on an mc solution of the wigner transport equation wm the equation describing the time evolution of the wigner function w on the lines described in 3. Moments and regularity for a boltzmann equation via wigner transform. Take transform of equation and boundaryinitial conditions in one variable. The transport equation obeyed by the relativistic wigner function is derived in terms of discrete derivatives and degenerates in the continuum limit into the transport equation obeyed by the wigner function of 2d dirac fermions. Here we propose a generalized mc method for the analysis of hotcarriertransportphenomena in quantum devices.
Transport equations for the onebody density oxford. Second, the quantumkinetic framework makes it easier to formulate boundary conditions at the device contacts, which may be inspired from classical kinetic considerations 1. Radiative transport limit of dirac equations with random. Dissipative transport in superlattices within the wigner function.
Most importantly, the wigner quasiprobability distribution is the wigner transform of the quantum density matrix, and, conversely, the density matrix is. Wigner transport equation with the model collision inte gral, we. Schiff the laplace transform is a wonderful tool for solving ordinary and partial differential equations and has enjoyed much success in this realm. A formal derivation of a generalized equation of a wigner distribution function including all manybody effects and all scattering mechanisms is given. Complete and consistent chiral transport from wigner function formalism. A relativistic wigner function for free discrete time quantum walks dtqws on the square 2d spacetime lattice is defined. Moments and regularity for a boltzmann equation via wigner. Problem 3 show that for any smooth scalar function, u x,t, and any constant vector v, div u x,t v v grad u x,t it follows from the result of the problem that since b is an arbitrary ball in u, and t1,t2 is similarly arbitrary, then if u and its derivatives of order one are all continuous in u, t u x,t v grad u x,t 0, in u for all t.
Relativistic quantum transport theory for electrodynamics. The equation governing the temperature dependence of the. November 17, 1902 january 1, 1995 was a hungarianamerican theoretical physicist and mathematician. The wigner boltzmann equation the time evolution for the density matrix. The generalized strichartz estimate see st, gv, cz gives a similar estimate for but it involves a norm on the initial data which. The wigner transform of this equality gives rise to f wp,r f wp, 4 together with the evolution equation for f wp. This is the celebrated gaussian random matrix ensemble of wigner and dyson wd.
123 193 1222 1331 673 900 1519 665 671 268 139 1128 275 115 1258 319 402 1555 1472 926 948 214 730 397 683 841 1426 1644 988 1222 1366 1320 435 1457 986 446 289 545 1379 6 303 5 798 117