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.

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Bulletin of the American Physical Society

Time-dependent solutions to the Dirac equation for interacting systems

A. Petridis, Zachary Kertzman, Khinlay Win

Published Oct 25, 2008• 0 citations

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SU(2)

Rashba vector potential

{\bf A}_{R}

. We demonstrate that

{\bf A}_{R}

cannot be eliminated from either the 2D SE or the 2D DE via a gauge transformation. Nevertheless, for a plane wave solution, an

SU(2)

matrix exists that eliminates

{\bf A}_{R}

from the resulting 1D SE. This unitary matrix is an Aharonov-Casher (AC) phase factor, and facilitates the calculation of the AC phase in the Schrödinger scheme. The plane wave solution for the DE contains two components of

{\bf A}_{R}

A_{R, k}

in the direction of the wave vector

{\bf k}

, and

A_{R, n}

normal to

{\bf k}

. The latter generates an effective electron mass that cannot be eliminated from the DE. The former generates an AC phase that can be eliminated by a time-dependent unitary transformation. Thus, the Dirac AC phase is time-dependent, i.e., it is a dynamical phase. This is in contradistinction to the Schrödinger AC phase which is geometrical.