Time-dependent Electromagnetic Potentials Research Articles

A fourth-order compact time-splitting method for the Dirac equation with time-dependent potentials

In this paper, we present an approach to deal with the dynamics of the Dirac equation with time-dependent electromagnetic potentials using the fourth-order compact time-splitting method (S4c). To this purpose, the time-ordering technique for time-dependent Hamiltonians is introduced, so that the influence of the time-dependence could be limited to certain steps which are easy to treat. Actually, in the case of the Dirac equation, it turns out that only those steps involving potentials need to be amended, and the scheme remains efficient, accurate, as well as easy to implement. Numerical examples in 1D and 2D are given to validate the scheme.

Plunging in the Dirac sea using graphene quantum dots

The dynamics of low energy charge carriers in a graphene quantum dot subjected to a time-dependent local field is investigated numerically. In particular, we study a configuration where a Coulomb electric field is provided by an ion traversing the graphene sample. A Galerkin-like numerical scheme is introduced to solve the massless Dirac equation describing charge carriers subjected to space- and time-dependent electromagnetic potentials and is used to evaluate the field induced interband transitions. It is demonstrated that as the ion goes through graphene, electron-hole pairs are generated dynamically via the adiabatic pair creation mechanism around avoided crossings, similar to electron-positron pair generation in low energy heavy ion collisions.

Hölder-stable recovery of time-dependent electromagnetic potentials appearing in a dynamical anisotropic Schrödinger equation

We consider the inverse problem of Holder-stably determining the time- and space-dependent coefficients of the Schrodinger equation on a simple Riemannian manifold with boundary of dimension \begin{document}$ n\geq2 $\end{document} from the knowledge of the Dirichlet-to-Neumann map. Assuming the divergence of the magnetic potential is known, we show that the electric and magnetic potentials can be Holder-stably recovered from these data. Here we also remove the smallness assumption for the solenoidal part of the magnetic potential present in previous results.

Time-evolution of the external field problem in Quantum Electrodynamics

We construct the time-evolution for the second-quantized Dirac equation subject to a smooth, compactly supported, time dependent electromagnetic potential and identify the degrees of freedom involved. Earlier works on this (e.g., Ruijsenaars) observed the Shale–Stinespring condition and showed that the one-particle time-evolution can be lifted to Fock space if and only if the external field had zero magnetic components. We scrutinize the idea, observed earlier by Fierz and Scharf, that the time-evolution can be implemented between time varying Fock spaces. In order to define these Fock spaces we are led to consider classes of reference vacua and polarizations. We show that this implementation is up to a phase independent of the chosen reference vacuum or polarization and that all induced transition probabilities are well-defined and unique.

Inverse problems for the Schrödinger equations with time-dependent electromagnetic potentials and the Aharonov–Bohm effect

We consider the inverse boundary value problem for the Schrödinger operator with time-dependent electromagnetic potentials in domains with obstacles. We extend the resuls of the author’s works [Inverse Probl. 19, 49 (2003); 19, 985 (2003); 20, 1497 (2004)] to the case of time-dependent potentials. We relate our results to the Aharonov–Bohm effect caused by magnetic and electric fluxes.

Rigorous Derivation of the Pauli Equation With Time-dependent Electromagnetic Field

In this work we discuss relativistic corrections for the description of charge carriers in a quantum mechanical framework. The fundamental equation is the Dirac equation which takes into account also the electron's spin. However, this equation intrinsically also incorporates positrons which play no role in applications in solid state physics. We give a rigorous derivation of the Pauli equation describing electrons in a first order approximation of the Dirac equation in the limit of infinite velocity of light. We deal with time-dependent electromagnetic potentials where no rigorous results have been given before. Our approach is based on the use of appropriate projection operators for the electron and the positron component of the spinor which are better suited than the widely used simple splitting into ‘upper (large)’ and ‘lower (small) component’. We also systematically derive corrections at second order in 1/c where we essentially recover the results of the Foldy-Wouthuysen approach. However, due to the non-static problem, differences occur in the term which couples the electric field with the spin.

(Semi)-Nonrelativistic Limits of the Dirac Equation with External Time-Dependent Electromagnetic Field

We perform a mathematical study of the limit of infinite velocity of light for the Dirac equation with given time-dependent electromagnetic potential. Our approach is based on the use of appropriate projection operators for the electron and the positron component of the spinor which are better suited than the widely used simple splitting into “upper (large)” and “lower (small) component”. The “semi-nonrelativistic limit” yields the approximation by the Pauli-equation for the electron component of the 4-spinor where first order corrections are kept. Like in the Foldy–Wouthuysen approach we use a rescaling of time to subtract the rest energy of the electron component and add it for the positron component which is assumed to be “small” initially. We give also rigorous results for the nonrelativistic limit to the Schrodinger equation. In this case we keep the symmetry of electron and positron components in the rescaling, thus avoiding smallness assumptions on the initial data, and obtain a decoupled pair of Schrodinger equations (with negative mass for the positron component). Convergence results for the relativistic current are included.

An Inverse Scattering Problem for Dirac Equations with Time-Dependent Electromagnetic Potentials

We consider an inverse scattering problem, by using a time-dependent method, for the Dirac equation with a time-dependent electromagnetic field. The Fourier transform of the field is reconstructed from the scattering operator on a Lorents invariant set (0.1) D: = {(T,&€=R*R3M<c\ft} in the dual space of the space-time. As corollaries of this result, we can reconstruct the electromagnetic field completely if it is a finite sum of fields each of which is a time-independent one by a suitable Lorentz transform, and we can also determine the field uniquely if the fields satisfies some exponential decay condition. Our assumptions and results are independent of a choice of inertial frames.

Convergence of Nelson diffusions with time-dependent electromagnetic potentials

Some recent results on the convergence of Nelson diffusions are extended to the case of Schrödinger operators with time-dependent electromagnetic potentials. It is proven that the sequence {P n}n≥1 of measures on the canonical space of physical trajectories associated to the solutions of Schrödinger equations in Nelson’s scheme, corresponding to the sequence {(Vn,An)}n≥1 ⊆C1(R;ℛ×L2(R3)), converges in the total variation norm under the assumptions that for every fixed t the scalar potentials Vn(t) converge in ℛ, the space of Rollnik class potentials, and the vector potentials An(t) converge in Lloc∞(R;L2(R3)). In order to prove these results conditions are given under which solutions of Schrödinger equations are continuous in the (time-dependent electromagnetic) potentials in the norm of the Sobolev space H1(R3).