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Physical Review A

Numerical approach to solve the time-dependent Dirac equation

J. W. Braun, Qichang C. Su, R. Grobe

Published Jan 1, 1999• 237 citations
Publisher

Abstract

We discuss how split-operator techniques may be applied to calculate normalized wave-function solutions to the three-dimensional time-dependent Dirac equation on a space-time lattice grid. This approach can be used to investigate various field-free dynamics of electron wave packets as well as the interactions of atomic electrons with space-time varying electromagnetic radiation. We demonstrate this technique by analyzing the time-resolved Klein paradox and the Zitterbewegung.

DOI
10.1103/physreva.59.604

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We discuss how split-operator techniques may be applied to calculate normalized wave-function solutions to the three-dimensional time-dependent Dirac equation on a space-time lattice grid. This approach can be used to investigate various field-free dynamics of electron wave packets as well as the interactions of atomic electrons with space-time varying electromagnetic radiation. We demonstrate this technique by analyzing the time-resolved Klein paradox and the Zitterbewegung.
We have calculated normalized wave function solutions to the time-dependent Dirac equation on a space-time lattice grid. Using these solutions we systematically explore phenomena that are consistent with the classical relativistic Liouville phase space approach and those that do not have a direct classical counterpart. The examples reviewed are the paradoxical Zitterbewegung, the time-resolved Klein-paradox, the dynamics of spin in an inhomogeneous magnetic field, the suppression of the spatial spreading rate and new dynamical features that occur close to relativistic resonances.
A quantum algorithm that solves the time-dependent Dirac equation on a digital quantum computer is developed and analyzed. The time evolution is performed by an operator splitting decomposition technique that allows for a mapping of the Dirac operator to a quantum walk supplemented by unitary rotation steps in spinor space. Every step of the splitting method is decomposed into sets of quantum gates. It is demonstrated that the algorithm has an exponential speedup over the implementation of the same numerical scheme on a classical computer, as long as certain conditions are satisfied. Finally, an explicit decomposition of this algorithm into elementary gates from a universal set is carried out to determine the resource requirements. It is shown that a proof-of-principle calculation may be possible with actual quantum technologies.
A quantum algorithm that solves the time-dependent Dirac equation on a digital quantum computer is developed and analyzed. The time evolution is performed by an operator-splitting decomposition technique that allows for a mapping of the Dirac operator to a quantum walk supplemented by unitary rotation steps in spinor space. Every step of the splitting method is decomposed into sets of quantum gates. It is demonstrated that the algorithm has an exponential speed-up over the implementation of the same numerical scheme on a classical computer, as long as certain conditions are satisfied. Finally, an explicit decomposition of this algorithm into elementary gates from a universal set is carried out to determine the resource requirements. It is shown that a proof-of-principle calculation may be possible with actual quantum technologies.
The time-dependent Dirac equation is solved using the three-dimensional Finite Difference-Time Domain (FDTD) method. The dynamics of the electron wave packet in a vector potential is studied in the arrangements associated with the Aharonov-Bohm effect. The solution of the Dirac equation showed a change in the velocity of the electron wave packet even in a region where no fields of the unperturbed solenoid acted on the electron. The solution of the Dirac equation qualitatively agreed with the prediction of classical dynamics under the assumption that the dynamics was defined by the conservation of generalized or canonical momentum of the electron.
Considers the Dirac equation with external electromagnetic fields and gives a careful formulation of the Stuckelberg-Feynman interpretation. This electron-positron theory may be applied whenever electron and positron states can be separated, which is possible for a wide class of time-independent potentials vanishing at infinity. One then may introduce a system of observables with a position operator X(t) which is free of Zitterbewegung and yields the usual relativistic connection between momentum and velocity. The Stuckelberg-Feynman interpretation is usually associated with a scattering theory (propagator formalism), which differs from the one of a one-particle interpretation (formalism imitating non-relativistic techniques). For the above-mentioned potentials there is a simple relationship between these physically different formalisms. There are, however, more general potentials for which the Stuckelberg-Feynman interpretation may still be applied asymptotically. With the help of Klein's paradox the authors show that the Dirac equation may then have unphysical solutions not conserving the total charge.
Solving the Dirac equation is a formidable task due to the high frequency and the degrees of freedom involved. However, this high frequency allows one to obtain an approximation to the equation. Here, we directly solve the Dirac equation using an envelope method and derive analytical solutions of Dirac wave packets to first order for the small momentum spread. We apply the insight gained from this solution to the Zitterbewegung behavior in a Dirac-like system, where we show that it is crucial to include the first-order term in our solution to correctly describe a Dirac packet.