Dirac方程的非相对论极限

In this paper we study the Cauchy problem for the nonlinear Dirac equation in the Sobolev space H^s . We prove the existence and uniqueness of global solutions for small data in H^s with s>1 . The method of proof is based on the Strichartz estimate of L^2_t type for Dirac and Klein-Gordon equations. We also prove that the solutions of the nonlinear Dirac equation after modulation of phase converge to the corresponding solutions of the nonlinear Schröodinger equation as the speed of light tends to infinity.

… Dirac equation converge to the corresponding solutions of a coupled system of nonlinear Schrödinger equations as … for the nonlinear Dirac equation with respect to the speed of light c. …

… the rigorous nonrelativistic limit of the Dirac equation in Sect. 3. … potential than for the nonrelativistic limit and we have to … The semi-nonrelativistic approximation of the Dirac equation …

… non-relativistic limit of this Dirac system, which leads to obtain a deformed Schrödinger–Pauli equation… of the classical and non-relativistic limits of the Dirac equation are clarified. The …

… Since the discovery of the Dirac equations [t] the problem of its non-relativistic limit (nr1.) and … neither the Dirac Hamiltonian, nor the Dirac equation has an obvious nonrelativistic limit. In …

Solving the nonlinear Dirac equation in the nonrelativistic limit regime numerically is difficult, because the solution oscillates in time with frequency of Oε-2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathscr {O}} \! \left( \varepsilon ^{-2}\right) $$\end{document}, where 0<ε≪1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0<\varepsilon \ll 1$$\end{document} is inversely proportional to the speed of light. Yongyong Cai and Yan Wang have shown, however, that such solutions can be approximated up to an error of Oε2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathscr {O}} \! \left( \varepsilon ^2\right) $$\end{document} by solving the semi-nonrelativistic limit system, which is a non-oscillatory problem. For this system, we construct a two-step method, called the explicit exponential midpoint rule, and prove second-order convergence of the semi-discretization in time. Furthermore, we construct a benchmark method based on standard techniques and compare the efficiency of both methods. Numerical experiments show that the new integrator reduces the computational costs per time step to 40% and within a given runtime improves the accuracy significantly.

… in the nonrelativistic limit. The same methods may be applied to a Dirac particle interacting with any type of externalfield (various meson … wave function 4 satisfying the Dirac equation, …

We give from first principles the non-relativistic limit of scalar and Dirac fields in curved spacetime. We aim to find general relativistic corrections to the quantum theory of particles affected by Newtonian gravity, a regime nowadays experimentally accessible. We believe that the ever-improving measurement accuracy and the theoretical interest in finding general relativistic effects in quantum systems require the introduction of corrections to the Schr\"{o}dinger-Newtonian theory. We rigorously determine these corrections by the non-relativistic limit of fully relativistic quantum theories in curved spacetime. For curved static spacetimes, we show how a non-inertial observer (equivalently, an observer in the presence of a gravitational field) can distinguish a scalar field from a Dirac field by particle-gravity interaction. We study the Rindler spacetime and discuss the difference between the resulting non-relativistic Hamiltonians. We find that for sufficiently large acceleration, the gravity-spin coupling dominates over the corrections for scalar fields, promoting Dirac particles as the best candidates for observing non-Newtonian gravity in quantum particle phenomenology.

… , we consider the non-relativistic (scaling) limit of H. We prove that the non-relativistic limit of H … –Fierz Hamiltonian with spin 1/2 in non-relativistic QED. This establishes a mathematically …

… This paper is devoted to the study of the nonrelativistic limit of Amelino-Camelia doubly … and Dirac equations. We show that these equations reduce to the Schrödinger equations for the …

… Limit In this chapter we analyze the behavior of the Dirac equation in the nonrelativistic limit … theory contains the successful nonrelativistic theory as a limiting case. The second reason …

… obtained operators which go over into the expected nonrelativistic form it is fairly clear that the nonrelativistic limit of these theories is readily obtained by a sequence of transformations …

… Foldy–Wouthuysen transformation for relativistic spin-1/2 particles, we investigate the non-relativistic … the non-relativistic limit of the Dirac equation. However, with both Bopp– Shift linear …

… to the Pauli equation in the nonrelativistic limit; it also … transformation that decouples the small and large components of the spinor from one another. In our treatment of the nonrelativistic …

The relationship between Foldy–Wouthuysen and Lorentz transformations has been clarified throughout this paper. We propose a generalized FW transformation connecting two particular realizations of the (m,j) representation of the Poincaré group: the covariant realization and a canonical realization acting on relativistic probability amplitudes. Fermions and bosons must be considered separately because the intrinsic parity of the particle–antiparticle systems is (−1)2j. Thus for fermions we can directly take the 2(2j+1) - dimensional Joos–Weinberg covariant realization, while for bosons we must double it to get a reducible 4(2j+1) - dimensional realization where particles and antiparticles lie in orthogonal subspaces. In short, in momentum space the FW transformation is the matrix representing a Lorentz boost times the factor (m/p0)1/2, while in configuration space the FW transformation does not belong to the Poincaré group. The last part of the paper is devoted to getting quantum-mechanical representations of the Galileo group as a contraction of Poincaré group representations by using mathematical methods earlier developed by Mickelsson and Niederle. The relevance of our generalized FW transformation for getting a smooth, well defined, nonrelativistic limit is a remarkable result.

… the Foldy-Wouthuysen transformations (Foldy and Wouthuysen 1950) … non-relativistic quantum mechanics, cannot be consistent with itself through the Foldy-Wouthuysen transformations…

The Exact Foldy-Wouthuysen transformation (EFWT) method is generalized here. In principle, it is not possible to construct the EFWT to any Hamiltonian. The transformation conditions are the same but the involution operator has a new form. We took a special example and constructed explicitly the new involution operator that allows one to perform the transformation. We treat the case of the Hamiltonian with 160 possible CPT-Lorentz breaking terms, using this new technique. The transformation was performed and physics analysis of the equations of motion is shown.

… discussed in this Section is not a non-relativistic approximation; it is simply an expansion in … The non-relativistic limit of Pauli, Darwin, Foldy and Wouthuysen is simply obtained from our …

… is not trivial, the Foldy-Wouthuysen transformation gives the … be obtained from the Foldy-Wouthuysen transformation. The … particles and we discuss the non-relativistic limit. The proof is …

… Foldy–Wouthuysen transformation for relativistic spin-1/2 particles, we study the nonrelativistic … the efficacy and behavior of the Foldy–Wouthuysen transformation in deriving the NRL in …

… to the Foldy-Wouthuysen transform. There seems to be evidence indicating that only operators in P qualify as observables—ie, can be measured with arbitrary precision—a feature …

It is shown that it is possible to construct, within the framework of a basis set expansion method, the full Foldy–Wouthuysen transformation (i.e., to all orders in the inverse velocity of light) for an arbitrary potential once the Dirac equation has been solved. On this basis an iterative procedure to solve the Dirac equation is suggested that involves only the large component, obviating the time-consuming (at least in molecular calculations) introduction of large basis sets for a proper description of just the small components. The methods are used to compare the expectation value of the radial distance operator in the Dirac picture and in the Schrödinger picture for the orbitals of the Uranium atom.

… spinor" is further developed and becomes the eigenspinor of an elementary particle in the covariant Pauli … to be reduced. There are, as a result, only eight linearly independent products, …

… and spin 5/2 particles and the reduced equations are quantized. It is … By reducing these equations in the nonrelativistic limit it is … to the Dirac spin 1/2 equation and the two spinors A ', B~ …

Properties of observables in the Pauli and Schrödinger theories and first order relativistic approximations to them are derived from the Dirac theory. They are found to be inconsistent with customary interpretations in many respects. For example, failure to identify the ’’Darwin term’’ as the s−state spin−orbit energy in conventional treatments of the hydrogen atom is traced to a failure to distinguish between charge and momentum flow in the theory. Consistency with the Dirac theory is shown to imply that the Schrödinger equation describes not a spinless particle as universally assumed, but a particle in a spin eigenstate. The bearing of spin on the interpretation of the Schrödinger theory is discussed. Conservation laws of the Dirac theory are formulated in terms of relative variables, and used to derive virial theorems and the corresponding conservation laws in the Pauli−Schrödinger theory.

A specific mapping is introduced to reduce the Dirac action to the non-relativistic (Pauli - Schrodinger) action for spinors. Using this mapping, the structures of the vector and axial vector currents in the non-relativistic theory are obtained. The implications of the relativistic Ward identities in the non-relativistic limit are discussed. A new non-abelian type of current in the Pauli - Schrodinger theory is obtained. As we show, this is essential for the closure of the algebra among the usual currents. The role of parity in the non-relativistic theory is also discussed.

The Dirac Hamiltonian is transformed by extracting the operator (σ⋅p)/2mc from the small component of the wave function and applying it to the operators of the original Hamiltonian. The resultant operators contain products of Pauli matrices that can be rearranged to give spin-free and spin-dependent operators. These operators are the ones encountered in the Breit–Pauli Hamiltonian, as well as some of higher order in α2. However, since the transformation of the original Dirac Hamiltonian is exact, the new Hamiltonian can be used in variational calculations, with or without the spin-dependent terms. The new small component functions have the same symmetry properties as the large component. Use of only the spin-free terms of the new Hamiltonian permits the same factorization over spin variables as in nonrelativistic theory, and therefore all the post-self-consistent field (SCF) machinery of nonrelativistic calculations can be applied. However, the single-particle functions are two-component orbitals having a large and small component, and the SCF methods must be modified accordingly. Numerical examples are presented, and comparisons are made with the spin-free second-order Douglas–Kroll transformed Hamiltonian of Hess.

… in-plane Pauli term, whereas we call (A2/8M2)e∇E the Darwin term. Note that the Pauli term can … First, we focus on pure SOI and consequently omit the Darwin term and the potential in …

… the spin-orbit interaction and the Darwin term, it was shown that the electric field exerts a transverse force on an electron spin … equation there appear three terms in addition to the Pauli …

… However, the Dirac equation is often insufficient, especially if the particles have similar masses. For that reason, we will need the energy correction terms of the Breit-Darwin equation …

… values because of the F3 dependence of the spinorbit operator, it seems much more likely that … mass-velocity term -p4/8m3c2 as a correction to the kinetic energy and the Darwin term -(…

… we have used the conventional Pauli form for the mass-velocity and Darwin operators. … spinorbit operator) while the exterior basis functions would be solutions of the free-electron Pauli …

The rotational g factors of the hydrogen halides, HX (X=F,Cl,Br,I), and noble gas hydride cations, XH+ (X=Ne,Ar,Kr,Xe), have been calculated at the level of the random phase approximation (RPA) as relativistic four-component linear response functions as well as nonrelativistic linear response functions. In addition, using perturbation theory with the mass-velocity and Darwin operators as perturbations, the relativistic corrections have been estimated as quadratic response functions. It was found that the four-component relativistic calculations give in general a more negative electronic contribution to the rotational g factor than the nonrelativistic calculations with relativistic corrections ranging from 0.2% for HF and NeH+ to 2.9% for XeH+ and 3.5% for HI. The estimates of the relativistic corrections obtained by perturbation theory with the mass-velocity and Darwin operators are in good agreement with the four-component results for HF, HCl, NeH+, and ArH+, whereas for HI, KrH+, and XeH+ they have the wrong sign.

The aim of this paper is to illustrate four properties of the non-relativistic limits of relativistic … non-relativistic limit, (b) that a relativistic field may have more than one non-relativistic limit, (c) …

We argue that the process of constructing the quantum mechanical current of the Pauli equation by copying the line of argument used in the spin-0 case, i.e., the Schrodinger equation, is ambiguous. We show that a nonrelativistic reduction of the relativistic Dirac four-vector current is, however, capable of fully resolving the problem. This analysis reveals that the nonrelativistic current of the Pauli equation should include an extra term of the form ∇×(ψ†σψ). We present an initial exploration of the potential consequences of this new “spin term” by solving the Pauli equation for crossed magnetic and electric fields and calculating the corresponding current.

The R-matrix method describing the scattering of low-energy electrons by complex atoms and ions is extended to include terms of the Breit-Pauli Hamiltonian. An application is made to …

… of the full microscopic spin-orbit Hamiltonian (both the spin-orbit and spin-other-orbit terms) and the dipolar spin-spin Hamiltonian. This approach permits the Breit-Pauli interaction to be …

The electron-atom scattering problem is formulated by using the Breit-Pauli hamiltonian, and the Kohn variational principle is derived for this hamiltonian. Two distinct types of relativistic corrections are considered separately: (1) relativistic corrections due to the motion of the colliding electron and its interaction with the target; (2) relativistic corrections due to breakdown of LS-coupling in the target. In both of these cases it is shown that within the Breit-Pauli approximation a collision strength may be written Qvel(i,j) = Qnr(i,j) + a2 C(2)rel(i,J),where Qrelis the collision strength including relativistic corrections and Qnr is the non-relativistic collision strength. The quantities C(2)rel are contributions of orders a2 and a4 respectively, relative to Qnr. In the case of corrections of type (1), consistency problems render it difficult to calculate the term a 4C(4)rel reliably. O n the other hand, strong semi-empirical evidence suggests that in the case of corrections of type (2), the a4 correction can be reliably estimated within the framework of existing theory. By means of Racah algebra it is demonstrated that fine structure interactions between colliding electron and target give no contributions of order a2 provided that that Qrel(i,j) is summed over the fine structure levels of the initial and final target terms.Breakdown of L-S -coupling in the target (due to fine structure interactions among the the target electrons) gives contribution of order a2 to the total collision strength. However, these contributions do not vanish when the collision strengths are summed over the fine structure levels of the initial and final terms. Asymptotic expansions for the dependence of Qrelupon the nuclear charge Z of the target are derived for corrections of types (1) and (2). The present work is discussed in relation to recent work by Carse &amp; Walker (1973) and Walker (1974), who have studied the studied the electron-hydrogen scattering problem in a formulation based upon the Dirac equation. Practical procedures for carrying out calculations in the framework of the present theory are discussed, and one such procedure is formulated in some detail.

Cartesian Gaussian functions are employed to derive general expressions for integrals over all one-electron operators of the Breit–Pauli Hamiltonian. It is shown that in atoms of higher atomic number p6, p8, ⋅⋅⋅ operators can be important in determining relativistic corrections to the kinetic energy. All other operators of this Hamiltonian can be expressed as some derivative of 1/r. Thus, a general expression is derived for the integral over the operator (∂1/∂x1) (∂m/∂ym) (∂n/∂zn) (1/r) by employing its Fourier transform. The operator and charge-distribution-dependent parts can be separately identified in the resulting expression and hence for a given charge distribution, integrals over any number of operators that can be expressed in the above form can be obtained simultaneously. In addition to nuclear attraction, these operators include the spin-orbit and Darwin terms of the Breit-Pauli Hamiltonian, as well as the electric field components and their derivatives, and other interactions over operators required in the study of magnetic shieldings. Furthermore, these expressions serve as a prototype for more difficult and numerous two-electron integrals, as discussed in the second paper in this series.

This work collects the spin-dependent leading-order relativistic and quantum–electrodynamical corrections for the electronic structure of atoms and molecules within the nonrelativistic quantum electrodynamics framework. We report the computation of perturbative corrections using an explicitly correlated Gaussian basis set, which allows high-precision computations for few-electron systems. In addition to numerical tests for triplet Be, triplet H2, and triplet H3 + states and comparison with no-pair Dirac–Coulomb–Breit Hamiltonian energies, numerical results are reported for electronically excited states of the helium dimer, He2, for which the present implementation delivers high-precision magnetic coupling curves necessary for a quantitative understanding of the fine structure of its high-resolution rovibronic spectrum.

… all operators of the Breit–Pauli Hamiltonian and their tensor … which arise from the Breit–Pauli operators in the Hamiltonian [8– … Besides, our re-derivation of formulae and numerical tests, …

… and differs from the infinite-nuclear-mass Hamiltonian by … by including the term ˆHMP in the Hamiltonian. The energy EF of an … In the derivation of Eq. (13) we assumed that the linear al…

Variational treatment of the Dirac-Coulomb-Gaunt or Dirac-Coulomb-Breit two-electron interaction at the Dirac-Hartree-Fock level is the starting point of high-accuracy four-component calculations of atomic and molecular systems. In this work, we introduce, for the first time, the scalar Hamiltonians derived from the Dirac-Coulomb-Gaunt and Dirac-Coulomb-Breit operators based on spin separation in the Pauli quaternion basis. While the widely used spin-free Dirac-Coulomb Hamiltonian includes only the direct Coulomb and exchange terms that resemble nonrelativistic two-electron interactions, the scalar Gaunt operator adds a scalar spin-spin term. The spin separation of the gauge operator gives rise to an additional scalar orbit-orbit interaction in the scalar Breit Hamiltonian. Benchmark calculations of Aun (n = 2-8) show that the scalar Dirac-Coulomb-Breit Hamiltonian can capture 99.99% of the total energy with only 10% of the computational cost when real-valued arithmetic is used, compared to the full Dirac-Coulomb-Breit Hamiltonian. The scalar relativistic formulation developed in this work lays the theoretical foundation for the development of high-accuracy, low-cost correlated variational relativistic many-body theory.

… the Dirac-Breit Hamiltonian, whose semirelativistic limit (also up to c −2 ) is the so-called Breit-Pauli Hamiltonian. When … The derivation may be also found in modern textbooks [8,16–18]. …

Expressions are derived for the multicenter integrals over all two-electron operators of the relativistic Breit–Pauli Hamiltonian employing Cartesian Gaussian atomic basis functions of any combination of angular momentum quantum numbers. It is shown that by employing the Fourier transform of the (∂1/∂x1) (∂m/∂ym) (∂n/∂zn) (1/r12), a general expression for the integral could be derived which is valid for electron repulsion, three components of electric field, and six components of the space part of the spin–spin interaction. A slight modification of this operator leads to spin-other orbit and orbit–orbit interaction integrals. Replacement of the function fL by gL and modifying the summation limits changes the expression for the integral over spin–spin interaction to that of a spin-orbit operator. It is shown in general that the charge-distribution dependence can be separated from the operator dependence of the integral in question, and thus for a given pair of charge distributions that integrals over all such operators can be derived simultaneously.

… deriving the regularly approximated relativistic Hamiltonians … full four-component relativistic Hamiltonian with respect to the … Hamiltonian recovers all terms of the Breit-Pauli Hamiltonian. …

First-order relativistic corrections to the energy of closed-shell molecular systems are calculated, using all terms in the two-component Breit–Pauli Hamiltonian. In particular, we present the first implementation of the two-electron Breit orbit–orbit integrals, thus completing the first-order relativistic corrections within the two-component Pauli approximation. Calculations of these corrections are presented for a series of small and light molecules, at the Hartree–Fock and coupled-cluster levels of theory. Comparisons with four-component Dirac–Coulomb–Breit calculations demonstrate that the full Breit–Pauli energy corrections represent an accurate approximation to a fully relativistic treatment of such systems. The Breit interaction is dominated by the spin–spin interaction, the orbit–orbit interaction contributing only about 10% to the total two-electron relativistic correction in molecules consisting of light atoms. However, the relative importance of the orbit–orbit interaction increases with increasing nuclear charge, contributing more than 20% in H2S.