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

Abstract

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 the 4×44\times 4 Dirac 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.

Keywords

• dirac equation
• maxwell’s equations
• fdtd method
• zitterbewegung
• atomic orbital