Dynamics of States with Half-Integer Orbital Momentum in Their Own Field

A nonstationary self-consistent quantum system that intensively interacts with its own field is studied. At a nonzero moment, the psi function cannot be independent of the angles of the spherical coordinate system. Conditions are defined under which, in the case of half-integer values of the moment (l = +/- 1/2), the charge density distribution turns out to be spherically symmetric. In this case, a self-consistent system can be described by a system of ordinary differential equations.