Spin relevant invariants and the general solution of the Dirac equation for the Coulomb fields

Abstract

The Dirac equation with the Coulomb potential is studied. A new spin relevant invariant is introduced, which does not commute with the Dirac and Johnson–Lippmann integrals of motion. The solutions of the Dirac equation compatible with each of the above three invariants are found and shown to be different. The explicit expressions for the Dirac bispinors are found for each of the three spin related invariants. The generalized invariant is constructed and corresponding to it general solution of the Dirac equation with the Coulomb potential is obtained. The general solution contains free parameters, whose particular values reduce it to any of the three particular solution. It is shown that the distributions of the electron probability density and expectation values of the electron spin direction in the hydrogen-like energy spectrum depend essentially on the spin relevant invariant. This demonstrates physical difference of the states corresponding to different invariants, which can be experimentally manifested.