Numerical solution of the time-dependent Dirac equation in coordinate space without fermion-doubling

Abstract The validation and parallel implementation of a numerical method for the solution of the time-dependent Dirac equation is presented. This numerical method is based on a split operator scheme where the space–time dependence is computed in coordinate space using the method of characteristics. Thus, most of the steps in the splitting are calculated exactly, making for a very efficient and unconditionally stable method. We show that it is free from spurious solutions related to the fermion-doubling problem and that it can be parallelized very efficiently. We consider a few simple physical systems such as the time evolution of Gaussian wave packets and the Klein paradox. The numerical results obtained are compared to analytical formulas for the validation of the method.

[1]  Bernd Thaller.  Visualizing the kinematics of relativistic wave packets , 2004 .

[2]  Rainer Grobe,et al.  Effects of relativity on the time-resolved tunneling of electron wave packets , 2001 .

[3]  Christoph H. Keitel,et al.  Relativistic high-power laser–matter interactions , 2006 .

[4]  F. Wilczek,et al.  Lattice fermions. , 1987, Physical review letters.

[5]  A. A. Perov,et al.  Space-time evolution of Dirac wave packets , 2010, 1007.1566.

[6]  Bottcher,et al.  Numerical solution of the time-dependent Dirac equation with application to positron production in heavy-ion collisions. , 1985, Physical review letters.

[7]  J. Kogut,et al.  Hamiltonian Formulation of Wilson's Lattice Gauge Theories , 1975 .

[8]  Eva Lindroth,et al.  Solution of the Dirac equation for hydrogenlike systems exposed to intense electromagnetic pulses , 2009 .

[9]  T. Brabec,et al.  Semiclassical Dirac theory of tunnel ionization. , 2002, Physical review letters.

[10]  John B. Kogut,et al.  The lattice gauge theory approach to quantum chromodynamics , 1983 .

[11]  T. Lappi,et al.  Quark-antiquark production from classical fields in heavy-ion collisions: 1+1 dimensions , 2005 .

[12]  G. L. Payne,et al.  Relativistic Quantum Mechanics , 2007 .

[13]  Walter Greiner,et al.  Relativistic Quantum Mechanics. Wave Equations , 1997 .

[14]  Guido R. Mocken,et al.  Quantum dynamics of relativistic electrons , 2004 .

[15]  Chunxiong Zheng,et al.  A time-splitting spectral scheme for the Maxwell-Dirac system , 2005, 1205.0368.

[16]  H. Nielsen,et al.  Absence of neutrinos on a lattice: (I). Proof by homotopy theory , 1981 .

[17]  André D. Bandrauk,et al.  Complex integration steps in decomposition of quantum exponential evolution operators , 2006 .

[18]  R. Stacey,et al.  Eliminating lattice fermion doubling , 1982 .

[19]  Maria J. Esteban,et al.  A short review on computational issues arising in relativistic atomic and molecular physics , 2007 .

[20]  L. Ryder,et al.  Quantum Field Theory , 2001, Foundations of Modern Physics.

[21]  Carleton DeTar,et al.  Lattice methods for quantum chromodynamics , 2006 .

[22]  Joachim Reinhardt,et al.  Quantum electrodynamics of strong fields , 1985 .

[23]  Paul Indelicato,et al.  Computational approaches of relativistic models in quantum chemistry , 2003 .

[24]  Christoph H. Keitel,et al.  A real space split operator method for the Klein-Gordon equation , 2009, J. Comput. Phys..

[25]  S. Salomonson,et al.  Relativistic all-order pair functions from a discretized single-particle Dirac Hamiltonian. , 1989, Physical review. A, General physics.

[26]  Howard R. Reiss.  Quantum Electrodynamics of Strong Fields , 1986 .

[27]  Christoph H. Keitel,et al.  Accelerating the Fourier split operator method via graphics processing units , 2010, Comput. Phys. Commun..

[28]  O. Klein,et al.  Die Reflexion von Elektronen an einem Potentialsprung nach der relativistischen Dynamik von Dirac , 1929 .

[29]  Belkacem,et al.  Numerical treatment of the time-dependent Dirac equation in momentum space for atomic processes in relativistic heavy-ion collisions. , 1996, Physical review. A, Atomic, molecular, and optical physics.

[30]  L. Milne‐Thomson.  A Treatise on the Theory of Bessel Functions , 1945, Nature.

[31]  H. Nielsen,et al.  Absence of neutrinos on a lattice: (II). Intuitive topological proof , 1981 .

[32]  M. I. Katsnelson,et al.  Chiral tunnelling and the Klein paradox in graphene , 2006 .

[33]  Rainer Grobe,et al.  Numerical approach to solve the time-dependent Dirac equation , 1999 .

[34]  Leonard Susskind,et al.  Strong Coupling Calculations of Lattice Gauge Theories: (1+1)-Dimensional Exercises , 1976 .

[35]  Guido R. Mocken,et al.  FFT-split-operator code for solving the Dirac equation in 2+1 dimensions , 2008, Comput. Phys. Commun..

[36]  Q. Su,et al.  Klein paradox in spatial and temporal resolution. , 2004, Physical review letters.

[37]  F. Wilczek.  Quantum Field Theory , 1998, hep-th/9803075.

[38]  Guido R. Mocken,et al.  Relativistic ionization characteristics of laser-driven hydrogenlike ions , 2011 .

[39]  Q. Su,et al.  Relativistic suppression of wave packet spreading. , 1998, Optics express.

[40]  André D. Bandrauk,et al.  A simple and accurate mixed P0–Q1 solver for the Maxwell–Dirac equations , 2011 .

[41]  Werner Scheid,et al.  FINITE ELEMENT FORMULATION OF THE DIRAC EQUATION AND THE PROBLEM OF FERMION DOUBLING , 1998 .

[42]  Fritz Sauter,et al.  Über das Verhalten eines Elektrons im homogenen elektrischen Feld nach der relativistischen Theorie Diracs , 1931 .

[43]  Weizhu Bao,et al.  An efficient and stable numerical method for the Maxwell-Dirac system , 2004 .

[44]  A. Calogeracos,et al.  Seventy years of the Klein paradox , 1999 .

[45]  Werner Scheid,et al.  Solution of the time-dependent Dirac equation by the finite difference method and application for Ca20++U91+ , 1983 .