New Complete Orthonormal Basis Sets of Relativistic Exponential Type Spinor Orbitals and Slater Spinor Functions of Particles with Arbitrary Half-Integral Spin in Position, Momentum and Four-Dimensional Spaces

TL;DR

This paper introduces new complete orthonormal basis sets of relativistic exponential type spinor orbitals and Slater spinor functions for particles with arbitrary half-integral spin, applicable in various spaces for relativistic quantum mechanics.

Contribution

It presents a unified framework for half-integral spin and scalar particles using complete basis sets in multiple spaces, extending previous methods.

Findings

Derived new formulae for relativistic spinors in position, momentum, and four-dimensional spaces.

Established complete orthonormal sets of spinor functions without continuum inclusion.

Facilitated the linear combination of atomic orbitals in relativistic quantum calculations.

Abstract

Using the complete orthonormal sets of radial parts of nonrelativitistic exponential type orbitals (2,1, 0, 1, 2, ...) and spinor type tensor spherical harmonics of rank s the new formulae for the 2(2s+1)-component relativistic spinors useful in the quantum mechanical description of the arbitrary half-integral spin particles by the generalized Dirac equation introduced by the author are established in position, momentum and four-dimensional spaces, where 1/ 2, 3 / 2, 5 / 2, ... s = . These spinors are complete without the inclusion of the continuum. The 2(2s+1)component spinors obtained are reduced to the independent sets of two-component spinors defined as a product of complete orthonormal sets of radial parts of orbitals and twocomponent spinor type tensor spherical harmonics. We notice that the new idea presented in this work is the unified treatment of half-integral spin and scalar particles in position, momentum and four-dimensional spaces. Relations presented in this study can be useful in the linear combination of atomic orbitals approximation for the solution of different problems arising in the relativistic quantum mechanics when the orthonormal basis sets of relativistic exponential type spinor wave functions and Slater type spinor orbitals in position, momentum and four -dimensional spaces are employed.