M Loewe, S Mendizabal
An elementary treatment of the Dirac equation in the presence of a three dimensional spherically symmetric delta potential is presented. We show how to calculate the cross section using the relativistic wave expansion method for a one delta potential and two concentric delta potentials. We compare our results with the cross section calculated in the Born approximation.
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The Non-Relativistic Limit of Keldysh Spinors
本文研究Keldysh旋量的非相对论极限，证明其在U(1)电磁耦合下可导出正定Pauli Hamilton量，并发现非相对论Dirac场与Keldysh场在类质量和矢量耦合中的退耦现象。
We have studied a relativistically covariant Stern-Gerlach (SG) experiment for a relativistic spin and a Dirac spin. We have obtained the relativistic spin in an arbitrary frame by using the classical spin dipole tensor, which gives the covariant spin dipole interactions, and the relation between a spin and a spin magnetic dipole moment. The relativistic spin is shown to have problems to become a proper spin operator for a massive relativistic particle because of two reasons. First, the relativistic spin three-vector operators cannot satisfy the spin algebra. Second, the SG experiment for the …
We present a method of simulating the Dirac equation in 3+1 dimensions for a free spin-1/2 particle in a single trapped ion. The Dirac bispinor is represented by four ionic internal states, and position and momentum of the Dirac particle are associated with the respective ionic variables. We show also how to simulate the simplified 1+1 case, requiring the manipulation of only two internal levels and one motional degree of freedom. Moreover, we study relevant quantum-relativistic effects, like the Zitterbewegung and Klein's paradox, the transition from massless to massive fermions, with the rel…
A method to construct non-dissipative non-Dirac-hermitian relativistic quantum system that is isospectral with a Dirac-hermitian Hamiltonian is presented. The general technique involves a realization of the basic canonical (anti-)commutation relations involving the Dirac matrices and the bosonic degrees of freedom in terms of non-Dirac-hermitian operators, which are hermitian in a Hilbert space that is endowed with a pre-determined positive-definite metric. Several examples of exactly solvable non-dissipative non-Dirac-hermitian relativistic quantum systems are presented by establishing/employ…
The formalism of Operational Dynamical Modeling [Phys. Rev. Lett. {\bf 109}, 190403 (2012)] is employed to analyze dynamics of spin half relativistic particles. We arrive at the Dirac equation from specially constructed relativistic Ehrenfest theorems by assuming that the coordinates and momenta do not commute. Forbidding creation of antiparticles and requiring the commutativity of the coordinates and momenta lead to classical Spohn's equation [Ann. Phys. {\bf 282}, 420 (2000)]. Moreover, Spohn's equation turns out to be the classical Koopman-von Neumann theory underlying the Dirac equation.
We show that a relativistic version of Schrodinger cat states, here called Dirac cat states, can be built in relativistic Landau levels when an external magnetic field couples to a relativistic spin 1/2 charged particle. Under suitable initial conditions, the associated Dirac equation produces unitarily Dirac cat states involving the orbital quanta of the particle in a well defined mesoscopic regime. We demonstrate that the proposed Dirac cat states have a purely relativistic origin and cease to exist in the non-relativistic limit. In this manner, we expect to open relativistic quantum mechani…
The derivation becomes possible when we find a new formalism which connects the relativistic mechanics with the quantum mechanics. In this paper, we explore the quantum wave nature from the Newtonian mechanics by using a concept: velocity field. At first, we rewrite the relativistic Newton's second law as a field equation in terms of the velocity field, which directly reveals a new relationship connecting to the quantum mechanics. Next, we show that the Dirac equation can be derived from the field equation in a rigorous and consistent manner.
The theorem known from Pauli equation about operators that anticommute with Dirac's $K$-operator is generalized to the Dirac equation. By means of this theorem the operator is constructed which governs the hidden symmetry in relativistic Coulomb problem (Dirac equation). It is proved that this operator coincides with the familiar Johnson-Lippmann one and is intimately connected to the famous Laplace-Runge-Lenz (LRL) vector. Our derivation is very simple and informative. It does not require a longtime and tedious calculations, as is offten underlined in most papers.
A particle is always not pure. It always contains hiding antiparticle ingredient which is the essence of special relativity. Accordingly, the Klein-Gordon (KG) equation and Dirac equation are restudied and compared with the Relativistic Stationary Schrödinger Equation (RSSE). When an electron is bound in a Hydrogenlike atom with pointlike nucleus having charge number $Z$, the critical value of $Z, Z_c$, equals to 137 in Dirac equation whereas $Z_c=\sqrt{M/μ} (137)$ in RSSE with $M$ and $μ$ being the total mass of atom and the reduced mass of the electron.
We give a direct link between description of Dirac particles in the abstract framework of unitary representation of the Poincaré group and description with the help of the Dirac equation. In this context we discuss in detail the spin operator for a relativistic Dirac particle. We show also that the spin operator used in quantum field theory for spin $s=1/2$ corresponds to the Foldy-Woutheysen mean-spin operator.
In this paper we show that the Schrödinger-Newton equation for spherically symmetric gravitational fields can be derived in a WKB-like expansion in 1/c from the Einstein-Klein-Gordon and Einstein-Dirac system.
Computational methods are indispensable to study the quantum dynamics of relativistic light-matter interactions in parameter regimes where analytical methods become inapplicable. We present numerical methods for solving the time-dependent Dirac equation and the time-dependent Klein-Gordon equation and their implementation on high performance graphics cards. These methods allow us to study tunneling from hydrogen-like highly charged ions in strong laser fields and Kapitza-Dirac scattering in the relativistic regime.
We develop the equivalence between the two-dimensional Dirac oscillator and the anti-Jaynes-Cummings model within a q-deformed scenario. We solve the Hamiltonian spectrum and the time evolution for number and coherent initial states, focusing on the appearance of the Zitterbewegung effect. We show the lack of preservation of the q-deformed versions of the total angular momentum that reproduces a collapse-revival structure. We provide suitable relations for the non relativistic limit.
All known solutions of the Dirac equation describing states of electrons endowed with angular momentum are very far from our notion of the electron as a spinning charged bullet because they are not localized in the direction of propagation. We present here analytic exact solutions, eigenstates of the total angular momentum component $M_z$, that come very close to this notion. These new solutions of the Dirac equation have also intricate topological properties similar to the hopfion solutions of the Maxwell equations.
We have shown the covariant relativistic spin operator is equivalent to the spin operator commuting with the free Dirac Hamiltonian. This implies that the covariant relativistic spin operator is a good quantum observable. The covariant relativistic spin operator has the pure quantum contribution which does not exist in the classical covariant spin operator. Based on this equivalence reduced spin states can be claerly defined. We have shown the change in the entropy of a reduced spin density matrix sweeps through the whole range according to the relative motion of an observer.
We investigate the feasibility of performing quantum non-demolition (QND) measurements in relativistic quantum systems, using the one-dimensional Dirac oscillator as a specific example. We derive general expressions for its QND observables and find that they are intricate combinations of the position, momentum, and spin operators, which makes them challenging to realize experimentally in general. However, the situation is considerably simplified in both the weakly and strongly relativistic limits, where their experimental realization will be possible.
By using the general concepts of special relativity and the requirements of quantum mechanics, Dirac equation is derived and studied. Only elementary knowledge of spin and rotations in quantum mechanics and standard handlings of linear algebra are employed for the development of the present work.
We derive the Schrödinger-Newton equation as the non-relativistic limit of the Einstein-Dirac equations. Our analysis relaxes the assumption of spherical symmetry, made in earlier work in the literature, while deriving this limit. Since the spin of the Dirac field couples naturally to torsion, we generalize our analysis to the Einstein Cartan-Dirac (ECD) equations, again recovering the Schrödinger-Newton equation.
Scattering discussion due to Double Dirac Equation in Quaternionic version of relativistic quantum mechanics has been studied in this paper in details. In such a quantum mechanics Dirac equation in presence vector and scalar potential has been considered. Then a Quaternionic double Dirac delta potential comes to our considered system which causes to scatter the particles. Scattering states of the particles have been derived as well as reflected and transmission coefficients are calculated.
We analyse a non-relativistic approximation of the Dirac equation for slow fermions, coupled to the chameleon field and torsion in the spacetime with the Schwarzschild metric, taken in the weak gravitational field of the Earth approximation. We follow the analysis of the Dirac equation in the curved spacetime with torsion, proposed by Kostelecky (Phys. Rev. D69, 105009 (2004)), and apply the Foldy--Wouthuysen transformations. We derive the effective low-energy gravitational potentials for slow fermions, coupled to the gravitational field of the Earth, the chameleon field and to torsion with mi…
This paper presents a new Relativistic Symmetrical Interpretation (RSI) of the Dirac equation in (1+1)D which postulates: quantum mechanics is intrinsically time-symmetric, with no arrow of time; the fundamental objects of quantum mechanics are transitions; a transition is fully described by a complex transition amplitude density with specified initial and final boundary conditions; and transition amplitude densities never collapse. This RSI is compared to the Copenhagen Interpretation (CI) for the analysis of Einstein's bubble experiment with a spin-$\frac{1}{2}$ particle. This RSI can predic…
We study the non-relativistic limit of a paradigmatic model in Relativistic Quantum Mechanics, the two-dimensional Dirac oscillator. Remarkably, we find a novel kind of Zitterbewegung which persists in this non-relativistic regime, and leads to an observable deformation of the particle orbit. This effect can be interpreted in terms of a Ramsey Interferometric phenomenon, allowing an insightful connection between Relativistic Quantum Mechanics and Quantum Optics. Furthermore, subsequent corrections to the non-relativistic limit, which account for the usual spin-orbit Zitterbewegung, can be neat…
Relativistic exponential-type spinor orbitals and their use in many-electron Dirac equation solution
本文提出相对论指数型旋量轨道及狄拉克不变量修正形式，结合双向方法与超径向函数解决不完全伽马函数收敛难题，并给出新的相对论辅助函数公式以提升多电子体系库仑能量计算效率。
We show that the first order form of the Schrodinger equation proposed in [1] can be obtained from the Dirac equation in the non-relativistic limit. We also show that the Pauli Hamiltonian is obtained from this equation by requiring local gauge invariance. In addition, we study the problem of a spin up particle incident on a finite potential barrier and show that the known quantum mechanical results are obtained. Finally, we consider the symmetric potential well and show that the quantum mechanical expression for the quantized energy levels of a particle is obtained with periodic boundary cond…
Spin of elementary particles is the only kinematic degree of freedom not having classical corre- spondence. It arises when seeking for the finite-dimensional representations of the Lorentz group, which is the only symmetry group of relativistic quantum field theory acting on multiple-component quantum fields non-unitarily. We study linear transformations, acting on the space of spatial and proper-time velocities rather than on coordinates. While ensuring the relativistic in- variance, they avoid these two exceptions: they describe the spin degree of freedom of a pointlike particle yet at a cla…
We consider the quantum simulation of relativistic quantum mechanics, as described by the Dirac equation and classical potentials, in trapped-ion systems. We concentrate on three problems of growing complexity. First, we study the bidimensional relativistic scattering of single Dirac particles by a linear potential. Furthermore, we explore the case of a Dirac particle in a magnetic field and its topological properties. Finally, we analyze the problem of two Dirac particles that are coupled by a controllable and confining potential. The latter interaction may be useful to study important phenom…
An elegant method to circumvent quantum measurement backaction is the use of quantum mechanics free subsystems (QMFS), with one approach involving the use of two oscillators with effective masses of opposite signs. Since negative energies, and hence masses, are a characteristic of relativistic systems a natural question is to what extent QMFS can be realized in this context. Using the example of a one-dimensional Dirac oscillator we investigate conditions under which this can be achieved, and identify Zitterbewegung or virtual pair creation as the physical mechanism that fundamentally limits t…
We prove that a single-jump quantum stochastic unitary evolution is equivalent to a Dirac boundary value problem on the half line in an extra dimension. This amounts to the equivalence of the quantum measurement boundary-value problem in infinite number particles space to the stochastic calculus in Fock space. It is shown that this exactly solvable model can be obtained from a Schroedinger boundary value problem for a positive relativistic Hamiltonian in the half-line as the inductive ultra relativistic limit, correspondent to the input flow of Dirac particles with asymptotically infinite mome…
Dirac's contributions to the discovery of non-relativistic quantum mechanics and quantum electrodynamics, prior to his discovery of the relativistic wave equation, are described.
The importance of the energy spectrum of bound states and their restrictions in quantum mechanics due to the different methods have been used for calculating and determining the limit of them. Comparison of Schrodinger-like equation obtained by Dirac equation with the non-relativistic solvable models is the most efficient methods. By this technique, the exact relativistic solutions of Dirac equation for Hartmann and Ring-Shaped oscillator potentials are accessible, when the scalar potential equals to the vector potential. Using solvable non-relativistic quantum mechanics systems as a basic mod…
Goos-H{ä}nchen Shift for Relativistic Particles Based on Dirac's Equation
本文基于狄拉克方程研究相对论粒子在三维无限势垒上的戈斯-汉肯舍移，发现其可为负值，与非相对论情况不同。
The Dirac Information Carrier for Relativistic Quantum Computation
该研究从相对论量子系统本身推导计算结构，以自旋-1/2的狄拉克信息载体为对象，明确其计算结构，建立单载体可控条件，拓展量子计算的信息载体来源。
The variation problem associated with the solution of Dirac's relativistic electron equation is reviewed here. Derivation of the min-max theorem is discussed. A new observation is that the spurious roots of negative energy satisfy a max-min theorem. The min-max principle (MMP) for solution of Dirac equation, extendable to the Dirac-Fock case, is concisely reviewed. MMP for two-electron Dirac-Coulomb equation is discussed. The min-max theorem is physically interpreted for both Dirac and Dirac-Coulomb problems. Applications of MMP are collated in tables. Associated theoretical and computational …
The dynamics of wave packets in a relativistic Dirac oscillator is compared to that of the Jaynes-Cummings model. The strong spin-orbit coupling of the Dirac oscillator produces the entanglement of the spin with the orbital motion similar to what is observed in the model of quantum optics. The collapses and revivals of the spin which result extend to a relativistic theory our previous findings on nonrelativistic oscillator where they were known under the name of `spin-orbit pendulum'. There are important relativistic effects (lack of periodicity, zitterbewegung, negative energy states). Many o…
It has been observed that a quantum theory need not to be Hermitian to have a real spectrum. We study the non-Hermitian relativistic quantum theories for many complex potentials, and we obtain the real relativistic energy eigenvalues and corresponding eigenfunctions of a Dirac charged particle in complex static and spherically symmetric potentials. Complex Dirac-Eckart, complex Dirac-Rosen-Morse II, complex Dirac-Scarf and complex Dirac-Poschl-Teller potential are investigated.
The relativistic acceleration of an electron in a uniform gravitational field is calculated numerically using the generalization of the Dirac equation to curved spacetime. Equivalent results are also obtained analytically using an iterative approach that is not based on the WKB approximation, which has been used extensively in the past. The acceleration is found to be proportional to a factor of $(1-3v^2/c^2)$ using Schwarzschild coordinates, which is consistent with the classical geodesic of a particle. These techniques may be useful in resolving the differences between commonly-used approxim…
We present a self-consistent mean-field model based on a two-component Pauli-like equation that incorporates quantum and relativistic effects (up to second-order in 1/c) for both external and internal electromagnetic fields. By taking the semi-relativistic limit of the Dirac-Maxwell equations in the presence of an external electromagnetic field we obtain an analytical expression of a coherent light-induced mean-field Hamiltonian. The latter exhibits several mechanisms that involve the internal mean fields created by all the electrons and the external electromagnetic field (laser). The role pla…
Quantum simulation is a powerful tool to study a variety of problems in physics, ranging from high-energy physics to condensed-matter physics. In this article, we review the recent theoretical and experimental progress in quantum simulation of Dirac equation with tunable parameters by using ultracold neutral atoms trapped in optical lattices or subject to light-induced synthetic gauge fields. The effective theories for the quasiparticles become relativistic under certain conditions in these systems, making them ideal platforms for studying the exotic relativistic effects. We focus on the reali…
We construct the relativistic particle model without Grassmann variables which meets the following requirements. A) Canonical quantization of the model implies the Dirac equation. B) The variable which experiences {\it Zitterbewegung}, represents a gauge non-invariant variable in our model. Hence our particle does not experiences the undesirable {\it Zitterbewegung}. C) In the nonrelativistic limit spin is described by three-vector, as it could be expected.
Non-relativistic limit of Dirac Hamiltonians with Aharonov-Bohm fields
该研究通过无穷光速极限将Dirac哈密顿量族映射为具有Aharonov-Bohm场的Schrödinger算子族，保持散射长度并得到物理相关的s波子族。
This paper presents a relativistic symmetrical interpretation of the Dirac equation in 1+1 dimensions which predicts no zitterbewegung for a free spin-1/2 particle. This could resolve the longstanding puzzle of zitterbewegung in relativistic quantum mechanics, and help clarify the physical meaning of the zitterbewegung currently under investigation in many-particle systems. Together with an earlier paper describing a relativistic symmetrical interpretation of the Klein-Gordon equation \cite{Heaney1}, this new interpretation resolves some inconsistencies in the conventional interpretations of b…
Dirac Theory as a Relativistic Flow
本文通过将流体动力学拉格朗日量与 Fisher 信息项结合，并推广到相对论流，提出从经典流推导狄拉克方程的新方法，从而为相对论量子力学提供信息驱动解释。
Quantum walks are not only algorithmic tools for quantum computation but also not trivial models which describe various physical processes. The paper compares one-dimensional version of the free particle Dirac equation with discrete time quantum walk (DTQW). We show that the discretized Dirac equation when compared with DTQW results in interesting relations. It is also shown that two relativistic effects associated with the Dirac equation, namely Zitterbewegung (quivering motion) and Klein's paradox, are present in DTQW, which can be implemented within non-relativistic quantum mechanics.
Closure relations satisfied by the radial Dirac-Coulomb Sturmians are proved analytically. The Sturmian expansion of the Dirac-Coulomb Green function is transformed to the form containing only series with summations running over non-negative indices. The main paper was published in J. Phys. B: At. Mol. Opt. Phys. 30 (1997) 825-61 [Erratum: 30 (1997) 2747], see also J. Phys. A: Math. Gen. 31 (1998) 4963--90 [Erratum: 31 (1998) 7415-6].
The Dirac oscillator is an exactly soluble model recently introduced in the context of many particle models in relativistic quantum mechanics. The model has been also considered as an interaction term for modelling quark confinement in quantum chromodynamics. These considerations should be enough for demonstrating that the Dirac oscillator can be an excellent example in relativistic quantum mechanics. In this paper we offer a solution to the problem and discuss some of its properties. We also discuss a physical picture for the Dirac oscillator's non-standard interaction, showing how it arises …
We uncover a remarkable quantum scattering phenomenon in two-dimensional Dirac material systems where the manifestations of both classically integrable and chaotic dynamics emerge simultaneously and are electrically controllable. The distinct relativistic quantum fingerprints associated with different electron spin states are due to a physical mechanism analogous to chiroptical effect in the presence of degeneracy breaking. The phenomenon mimics a chimera state in classical complex dynamical systems but here in a relativistic quantum setting - henceforth the term "Dirac quantum chimera," assoc…
The most general Dirac Hamiltonians in $(1+1)$ dimensions are revisited under the requirement to exhibit a supersymmetric structure. It is found that supersymmetry allows either for a scalar or a pseudo-scalar potential. Their spectral properties are shown to be represented by those of the associated non-relativistic Witten model. The general discussion is extended to include the corresponding relativistic and non-relativistic resolvents. As example the well-studied relativistic Dirac oscillator is considered and the associated resolved kernel is found in a closed form expression by utilising …
We consider the dynamics of relativistic spin-half particles in quantum graphs with transparent branching points. The system is modeled by combining the quantum graph concept with the one of transparent boundary conditions applied to the Dirac equation on metric graphs. Within such an approach, we derive simple constraints, which turn the usual Kirchhoff-type boundary conditions at the vertex equivalent to the transparent ones. Our method is applied to quantum star graph. An extension to more complicated graph topologies is straightforward.
We propose a relativistic particle model without Grassmann variables which, being canonically quantized, leads to the Dirac equation. Both $Γ$\,-matrices and the relativistic spin tensor are produced through the canonical quantization of the classical variables which parametrize the properly constructed relativistic spin surface. Although there is no mass-shell constraint in our model, our particle's speed cannot exceed the speed of light. The classical dynamics of the model is in correspondence with the dynamics of mean values of the corresponding operators in the Dirac theory. In particular,…
A theory of relativistic space-charge-limited current (SCLC) is formulated to determine the SCLC scaling, $J\propto V^α/L^β$, for a finite bandgap Dirac material of length $L$ biased under a voltage $V$. In a one-dimensional (1D) bulk geometry, our model allows ($α$, $β$) to vary from (2,3) for the non-relativistic model in traditional solids to (3/2,2) for the ultra-relativistic model of massless Dirac fermions. For a two-dimensional (2D) thin-film geometry, we obtain $α= β$ that varies between 2 and 3/2, respectively, at the non-relativistic and ultra-relativistic limits. We further provide …
Inspired by the reconstituted similarity renormalization group method, the reconstituted Foldy-Wouthuysen (FW) transformation is proposed. Applied to the Dirac equation in the covariant density functional theory, the reconstituted FW transformation shows a fast convergence of the spectrum of the single-particle energy. The single-particle densities and the single-particle scalar densities obtained by this new method are also investigated. In particular, the relativistic corrections to the densities from the picture-change error between the Schrödinger and Dirac pictures are discussed in detail…
By following the Foldy-Wouthuysen (FW) transformation of the Dirac equation, we work out the exact analytic expressions up to the $1/M^4$ order for the general cases in the covariant density functional theory. These results are further compared with the corresponding ones derived from another novel non-relativistic expansion method, the similarity renormalization group (SRG). Based on that, the origin of the difference between the results obtained by the FW transformation and SRG method is explored.
In this paper, three plausible axioms together with two definitions are employed to build an axiomatic framework, and then with the help of the Dirac formalism, it is demonstrated that the time-dependent Schrödinger wave equation is no longer a postulate for the whole theory, but a theorem. Subsequently, a proof for the theorem is presented. The result implies that the other remaining axioms of the picture essentially involve the Schrödinger equation, and this consequence lets the whole theory become stronger.
The formulation of a rigid body in relativistic quantum mechanics is studied. Departing from an alternate approach at the relativistic classical level, the corresponding Klein-Gordon and Dirac operators for the rigid body are obtained in covariant form. The resulting wave equations are shown to be consistent, by construction, with earlier definitions of a relativistic rigid body by Aldinger et al (1983). Wave functions and spectra for both cases are obtained explicitly, including the Dirac gyroscope with asymmetries.
We investigate wavepacket dynamics across supercritical barriers for the Klein-Gordon and Dirac equations. Our treatment is based on a multiple scattering expansion (MSE). For spin-0 particles, the MSE diverges, rendering invalid the use of the usual connection formulas for the scattering basis functions. In a time-dependent formulation, the divergent character of the MSE naturally accounts for charge creation at the barrier boundaries. In the Dirac case, the MSE converges and no charge is created. We show that this time-dependent charge behavior dynamics can adequately explain the Klein parad…
Dirac operators on the half-line: stability of spectrum and non-relativistic limit
本文研究了半直线上带广义无限质量边界条件的 Dirac 算子，推导了保证其谱在非自伴势扰动下稳定的充分条件并分析最优性，最后建立非相对论极限以揭示其与 Robin Laplacian 的关系。
The problem of a particle in a box is probably the simplest problem in quantum mechanics which allows for significant insight into the nature of quantum systems and thus is a cornerstone in the teaching of quantum mechanics. In relativistic quantum mechanics this problem allows also to highlight the implications of special relativity for quantum physics, namely the effect that spin has on the quantized energy spectra. To illustrate this point, we solve the problem of a spin zero relativistic particle in a one- and three-dimensional box using the Klein-Gordon equation in the Feshbach-Villars fo…
Dirac bilinears in condensed matter physics: Relativistic correction for observables and conjugate electromagnetic fields
本文基于狄拉克双线性非相对论极限，重新考虑凝聚态物理中被忽视的微观物理量，明确四分量狄拉克场与二分量薛定谔场的表达关系，并探讨其共轭电磁场，推动物质的电磁调控。
A non-relativistic Dirac equation: An application to photo ionization of highly charged hydrogen-like ions
该研究通过求解 Schrödinger 与 Dirac 方程的半相对论形式，分析强激光场中高电荷类氢离子光电离概率的残余相对论修正来源。
Spin-spin effects from non-relativistic limit of Dirac-Pauli-Maxwell Lagrangian
该研究通过对狄拉克-麦克斯韦拉格朗日量取非相对论极限，推导Breit相互作用并计算其修正项，分析相关自旋效应，给出了实验暂无法获取的理论估计值。
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