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Using the finite difference time domain (FDTD) method based on the eight-element Dirac equation, we found that a stable Dirac field wave packet with low velocity can be created without explicit consideration of Zitterbewegung (the rapid oscillatory motion of elementary particles), which is difficult in one-dimensional simulations. Furthermore, we successfully simulated the formation process of atomic orbitals for the first time without any physical approximations by calculating the eight-element Dirac field propagation in the central electric force potential. Initially, a small unstable orbital appears, which rapidly grows and results in a large stable orbital with a radius equal to the Bohr radius divided by the atomic number, as given by the solution of the Schrödinger equation. The FDTD calculation based on the conventional four-element Dirac equation cannot produce such reasonable orbitals owing to the spatial asymmetry of the4×44\times 4Dirac matrices. This method has the potential to be used for transient analyses of not only atomic or molecular orbitals but also interactions among elementary particles.

Simulation for formation process of atomic orbitals by the finite difference time domain method based on the eight-element Dirac equation