Shell model with σ · x potential

The one-nucleon state of a definite j-value is constructed as a linear combination of two states which belong to the azimuthal quantum numbers j−12 and j+12, respectively. The invariance of the state for space inversion is guaranteed by introducing a unit pseudoscalar. The Hartree-Fock equation satisfied by such one-nucleon states and the isospin and spin-angle dependence of the Hartree and exchange fields are discussed on the assumption that the two-nucleon interaction consists of three parts, i.e. the central, the σ · σ′ and the tensor-force type. It is shown that the Hartree-Fock equation reduces to a form of eigenvalue equation of the Hamiltonian with central plus σ · x type potentials. Addition theorems for spin-orbital harmonics are obtained in the ljm scheme.