The technique of differential intertwining operators (or Darboux transformation operators) is systematically applied to the one-dimensional Dirac equation.
The following aspects are investigated: factorization of a polynomial of Dirac Hamiltonians, quadratic supersymmetry, closed extension of transformation operators, chains of transformations, and finally particular cases of pseudoscalar and scalar potentials. The method is widely illustrated by numerous examples.
The behaviour of a Dirac electron in graphene, under magnetic fields which are orthogonal to the layer, is studied. The initial problem is reduced to an equivalent one, where two one-dimensional Schrödinger Hamiltonians $H^{\pm}$ are intertwined by a first order differential operator. Special magnetic field are initially chosen, in order that $V^{\pm}$ will be shape invariant exactly solvable potentials. When looking for more general first order operators, intertwining $H^-$ with a non-necessarily shape invariant Hamiltonian, new magnetic fields associated also to analytic solutions will be ge…
General first- and higher-order intertwining relations between non-stationary one-dimensional Schrödinger operators are introduced. For the first-order case it is shown that the intertwining relations imply some hidden symmetry which in turn results in a $R$-separation of variables. The Fokker-Planck and diffusion equation are briefly considered. Second-order intertwining operators are also discussed within a general approach. However, due to its complicated structure only particular solutions are given in some detail.
The intertwining technique has been widely used to study the Schrödinger equation and to generate new Hamiltonians with known spectra. This technique can be adapted to find the bound states of certain Dirac Hamiltonians. In this paper the system to be solved is a relativistic particle placed in a magnetic field with cylindrical symmetry whose intensity decreases as the distance to the symmetry axis grows and its field lines are parallel to the $x-y$ plane. It will be shown that the Hamiltonian under study turns out to be shape invariant.
In this work, we have extended the factorization method of scalar shape-invariant Schrö\-din\-ger Hamiltonians to a class of Dirac-like matrix Hamiltonians. The intertwining operators of the Schrödinger equations have been implemented in the Dirac-like shape invariant equations. We have considered also another kind of anti-intertwining operators changing the sign of energy. The Dirac-like Hamiltonians can be obtained from reduction of higher dimensional spin systems. Two examples have been worked out, one obtained from the sphere ${\cal S}^2$ and a second one, having a non-Hermitian character,…
The intertwining relations between superpartner Hamiltonians are the main ingredients of well known Supersymmetrical Quantum Mechanics (SUSY QM). In the present paper, the generalized form of intertwining is used for investigation of a massless (zero energy) two-dimensional Dirac equation with scalar external potential. This equation is related to the description of graphene and some other materials in the field of external electrostatic potential. The use of modified intertwining relations allows to find analytically solutions for the wave functions in the field of some external scalar potent…
A matricial Darboux operator intertwining two one-dimensional stationary Dirac Hamiltonians is constructed. This operator is such that the potential of the second Dirac Hamiltonian as well as the corresponding eigenfunctions are determined through the knowledge of only two eigenfunctions of the first Dirac Hamiltonian. Moreover this operator together with its adjoint and the two Hamiltonians generate a quadratic deformation of the superalgebra subtending the usual supersymmetric quantum mechanics. Our developments are illustrated on the free particle case and the generalized Coulomb interactio…
In this work, we construct the exact propagator for Dirac fermions in graphene-like systems immersed in external static magnetic fields with non-trivial spatial dependence. Such field profiles are generated within a first-order supersymmetric framework departing from much simpler (seed) magnetic field examples. The propagator is spanned on the basis of the Ritus eigenfunctions, corresponding to the Dirac fermion asymptotic states in the non-trivial magnetic field background which nevertheless admits a simple diagonal form in momentum space. This strategy enlarges the number of magnetic field p…
We show that, in the relativistic Bohm model of a Dirac-like particle, the electron in the ground state of the hydrogen atom is moving, unlike the prediction for the case of a Schrödinger-like particle, where the electron is stationary. This accounts for the empirically observed dilation of the decay time of the muon in the ground state of muonium.
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 discuss the properties of 1D stationary pulses of light in atomic ensemble with electromagnetically induced transparency in the limit of tight spatial confinement. When the size of the wavepacket becomes comparable or smaller than the absorption length of the medium, it must be described by a two-component vector which obeys the one-dimensional two-component Dirac equation with an effective mass $m^*$ and effective speed of light $c^*$. Then a fundamental lower limit to the spatial width in an external potential arises from Klein tunneling and is given by the effective Compton length $λ_C =…
We propose a mathematical model of \textit{quantum} transistor in which bandgap engineering corresponds to the tuning of Dirac potential in the complex four-vector form. The transistor consists of $n$-relativistic spin qubits moving in \textit{classical} external electromagnetic fields. It is shown that the tuning of the direction of the external electromagnetic fields generates perturbation on the potential temporally and spatially, determining the type of quantum logic gates. The theory underlying of this scheme is on the proposal of the intertwining operator for Darboux transfomations on on…
We show that the Dirac equation in 3+1 dimensions gives rise to supersymmetric patterns when the scalar and vector potentials are (i) Coulombic with arbitrary strengths or (ii) when their sum or difference is a constant, leading to relativistic pseudospin and spin symmetries. The conserved quantities and the common intertwining relation responsible for such patterns are discussed.
Exact solutions of the Dirac equation, a system of four partial differential equations, are rare. The vast majority of them are for highly symmetric stationary systems. Moreover, only a handful of solutions for time dependent dynamics exists. Given the growing number of applications of high energy electron beams interacting with a variety of quantum systems in laser fields, novel methods for finding exact solutions to the Dirac equation are called for. We present a method for building up solutions to the Dirac equation employing a recently introduced approach for the description of spinorial f…
We consider optical systems where propagation of light can be described by a Dirac-like equation with $PT$-symmetric Hamiltonian. In order to construct exactly solvable configurations, we extend the confluent Crum-Darboux transformation for the one-dimensional Dirac equation. The properties of the associated intertwining operators are discussed and the explicit form for higher-order transformations is presented. We utilize the results to derive a multi-parametric class of exactly solvable systems where the balanced gain and loss represented by the $PT$-symmetric refractive index can imply loca…
By analyzing the Dirac equation with static electric and magnetic fields it is shown that Dirac's theory is nothing but a generalized one-particle quantum theory compatible with the special theory of relativity. This equation describes a {\it quantum} dynamics of a {\it single} relativistic fermion, and its solution is reduced to solution of the generalized Pauli equation for two quasiparticles which move in the Euclidean space with their effective masses holding information about the Lorentzian symmetry of the four-dimensional space-time. We reveal the correspondence between the Dirac bispino…
We investigate the most general form of the one-dimensional Dirac Hamiltonian $H_D$ in the presence of scalar and pseudoscalar potentials. To seek embedding of supersymmetry (SUSY) in it, as an alternative procedure to directly employing the intertwining relations, we construct a quasi-Hamiltonian $\mathcal{K}$, defined as the square of $H_D$, to explore the consequences. We show that the diagonal elements of $\mathcal{K}$ under a suitable approximation reflects the presence of a superpotential thus proving a useful guide in unveiling the role of SUSY. For illustrative purpose we apply our sch…
We prove that for the radial Dirac equation with Coulomb-type potential the generalized dynamical scattering operator coincides with the corresponding generalized stationary scattering operator. This fact is a quantum mechanical analogue of ergodic results in the classical mechanics.
The stationary eigenstates and eigenvalues for the ponderomotive potential of an optical crystal confined in a one-dimensional infinite square well are numerically obtained. The initial states of the incoming particles taken as Gaussian, are expanded in the basis of the stationary eigenstates of the ponderomotive potential, to obtain the subsequent time evolution of the wave function of the particle during the interaction with the optical crystal. From the results of the time evolution of the probability density, it is observed that the particles get localized at equidistant positions in the t…
We derive the universal real time $U(1)$ topological gauge field action for mixed quantum states of weakly correlated fermions in all dimensions, and demonstrate its independence of the underlying equilibrium or non-equilibrium nature of dynamics stabilizing the state. The key prerequisites are charge quantization and charge conservation. The gauge action encodes non-quantized linear responses as expected for mixed states, but also quantized non-linear responses, associated to mixed state topology and accessible in experiment. Our construction furthermore demonstrates how the physical pictures…
In this paper, we introduce an extension of the Dirac equation, very similar to Dirac oscillator, that gives stationary localized wave packets as eigenstates of the equation. The extension to the Dirac equation is achieved through the replacement of the momentum operator by a PT-symmetric generalized momentum operator. In the 1D case, the solutions represent bound particles carrying spin having continuous energy spectrum, where the envelope parameter defines the width of the packet without affecting the dispersion relation of the original Dirac equation. In the 2D case, the solutions are local…
The problem of detection time distribution concerns a quantum particle surrounded by detectors and consists of computing the probability distribution of where and when the particle will be detected. While the correct answer can be obtained in principle by solving the Schrodinger equation of particle and detectors together, a more practical answer should involve a simple rule representing the behavior of idealized detectors. We have argued elsewhere [http://arxiv.org/abs/1601.03715] that the most natural rule for this purpose is the "absorbing boundary rule," based on the 1-particle Schrodinger…
For one dimensional motions, we derive from the Dirac Spinors Equation (DSE) the Quantum Stationary Hamilton-Jacobi Equation for particles with spin 1/2. Then, We give its solution. We demonstrate that the $QSHJES_{1\over2}$ have two explicit forms, which represent the two possible projection of the Spin 1/2.
Ideas and results of the generalized wave operator theory for dynamical and stationary cases are developed further and exact expressions for generalized scattering operators are obtained for wide classes of differential equations. New results on the structure of the generalized scattering operators are derived. Interesting interrelations between dynamical and stationary cases are found for radial Schrödinger and Dirac equations, and for Dirac-type equations as well. For some important examples we explain why the well-known divergences in the higher order approximations of the scattering matric…
The method of intertwining with n-dimensional (nD) linear intertwining operator L is used to construct nD isospectral, stationary potentials. It has been proven that differential part of L is a series in Euclidean algebra generators. Integrability conditions of the consistency equations are investigated and the general form of a class of potentials respecting all these conditions have been specified for each n=2,3,4,5. The most general forms of 2D and 3D isospectral potentials are considered in detail and construction of their hierarchies is exhibited. The followed approach provides coordinate…
Controlling the time evolution of the population of two states in cavity quantum electrodynamics (QED) is necessary by tunning the \textit{modified} Rabi frequency in which the $\textit{extra}$ classical effect of electromagnetic field is taken into account.The theoretical explanation underlying the perturbation of potential on spatial regime of bloch sphere is by the use of (Bagrov, Baldiotti, Gitman, and Shamshutdinova) BBGS-Darboux transformations \cite{bagrov14darboux} on the electromagnetic field potential in one-dimensional stationary Dirac model in which the Pauli matrices are the centr…
Quantum walk (QW) provides a versatile tool to study fundamental physics and also to make a variety of practical applications. We here start with the recent idea of {\it nonlinear} QW and show that introducing {\it nonlinearity} to QW can lead to a wealth of remarkable possibilities, e.g., simulating nonlinear quantum dynamics thus enhancing the applicability of QW above the existing level for a universal quantum simulator. As an illustration, we show that the dynamics of a nonlinear Dirac particle can be simulated on an optical nonlinear QW platform implemented with a measurement-based-feedfo…
Kirchhoff's scalar diffraction theory is applied throughout photon and electron optics. It is based on the stationary electromagnetic or Schrödinger wave equation, and is useful in describing interference phenomena for both light and matter waves. Here, Kirchhoff's diffraction theory is derived from the relativistic Dirac equation, thus reformulated to work on Dirac spinors. The parallels with the "classic" scalar theory are highlighted, and a basic interpretation of the result obtained for the Fraunhofer diffraction limit is given. The goal of this paper is to emphasize the similarity between…
Analytic solutions of the quantum relativistic two-body problem are obtained for an interaction potential modeled as a one-dimensional smooth square well. Both stationary and moving pairs are considered and the limit of the δ-function interaction is studied in depth. Our result can be utilized for understanding excitonic states in narrow-gap carbon nanotubes. We also show the existence of bound states within the gap for a pair of particles of the same charge.
We derive new solutions of the Schrödinger, Klein-Gordon and Dirac equations which describe the motion of particles in a uniform magnetic field. In contrast to the well known stationary solutions, our solutions exhibit the behavior of quantum particles which very closely resembles classical helical trajectories. These solutions also serve as an illustration of the meaning of the Ehrenfest theorem in relativistic quantum mechanics.
We propose an improvement of the basis for the solution of the stationary two-centre Dirac equation in Cassini coordinates using the finite-basis-set method presented in Ref. [1]. For the calculations in Ref. [1], we constructed the basis for approximating the energy eigenfunctions by using smooth piecewise defined polynomials, called B-splines. In the present article, we report that an analysis of the employed representation of the Dirac matrices shows that the above approximation is not efficient using B-splines only. Therefore, we include basis functions which are defined using functions wi…
We consider confinement of Dirac fermions in $AB$-stacked bilayer graphene by inhomogeneous on-site interactions, (pseudo-)magnetic field or inter-layer interaction. Working within the framework of four-band approximation, we focus on the systems where the stationary equation is reducible into two stationary equations with $2\times2$ Dirac-type Hamiltonians and auxiliary interactions. We show that it is possible to find localized states by solving an effective Schrödinger equation with energy-dependent potential. We consider several scenarios where bilayer graphene is subject to inhomogneous (…
We show that the internal stationary state of a black hole for massless Dirac fields can be represented by an entangled state of collapsing matter and infalling Hawking radiation. This implies that the Horowitz-Maldacena conjecture for the black hole final state originally proposed for the massless scalar fields is also applicable to fermionic fields as well. For an initially mixed state we find that the measure of mixedness is expected to decrease under evaporation.
We introduce the new, exactly solvable model of the two-dimensional Dirac fermion in presence of an asymmetric, Pöschl-Teller-like vector potential. Utilizing the translation invariance of the system, the effective one-dimensional stationary equation is brought into the form of the Heun equation and its fundamental solutions are found as an irreducible combination of two Gauss hypergeometric functions. The energy spectrum and the scattering is studied in dependence on the conserved longitudinal momentum as well as on the strength of the coupling.
The classical Lippmann-Schwinger equation (LS equation) plays an important role in the scattering theory for the non-relativistic case (Schrödinger equation). In our previous paper arXiv:1801.05370, we consider the relativistic analogue of the Lippmann-Schwinger equation (RLS equation). We represent the corresponding equation in the integral form. In the present paper, we use the corresponding integral equation and investigate the scattering problems for both stationary and dynamical cases. Our approach allows us to develop an RLS equation theory which is comparable in its completeness with th…
