Electron vortices in spin-orbit coupling system

There exists an electron vortex solution with orbital angular momentum quantum in a non-spin-orbit coupling system which has nonconservative momentum. We discuss the spin-orbit and momentum, we can find that electrons total numbers also have beam solutions. And is expressed as entangled wave function of spin function. Taking central force field for example, this paper constructed spinor structure caused by propagation carrying fixed number along <i>z</i>-axis. The under condition non-conserved but conserved. corresponding beams spin-vortex entanglement are solved perturbation method, Foldy-Wouthuysen transformation utilized to show four-component does exist case relativity, when propagates <i>z</i>-axis field. provides theoretical support existence where not conserved due coupling.