Unified treatment of complete orthonormal sets for wave functions, and Slater orbitals of particles with arbitrary spin in coordinate, momentum and four-dimensional spaces

The new analytical relations of complete orthonormal sets for the tensor wave functions and the tensor Slater orbitals of particles with arbitrary spin in coordinate, momentum and four-dimensional spaces are derived using the properties of tensor spherical harmonics and complete orthonormal scalar basis sets of ψα-exponential type orbitals, ϕα-momentum space orbitals and zα-hyperspherical harmonics introduced by the author for particles with spin s=0, where the α=1,0,−1,−2,…. All of the tensor wave functions obtained are complete without the inclusion of the continuum and, therefore, their group of transformations is the four-dimensional rotation group O(4). The analytical formulas in coordinate space are also derived for the overlap integrals over tensor Slater orbitals with the same screening constant. We notice that the new idea presented in this work is the combination of tensor spherical harmonics of rank s with complete orthonormal scalar sets for radial parts of ψα-, ϕα- and zα-orbitals, where s=1/2,1,3/2,2,….