Spin current evolution in the separated spin-up and spin-down quantum hydrodynamics

We have developed the quantum hydrodynamic model that describes particles with spin-up and with spin-down in separate. We have derived the equation of the spin current evolution as a part of the set of the quantum hydrodynamics (QHD) equations that treat particles with different projection of spin on the preferable direction as two different species. We have studied orthogonal propagation of waves in the external magnetic field and determined the contribution of quantum corrections due to the Bohm potential and to magnetization energy of particles with different projections of spin in the spin current wave dispersion. We have analyzed the limits of weak and strong magnetic fields.