Combined theory of two- and four- component complete orthonormal sets of spinor wave functions and Slater type spinor orbitals in position, momentum and four-dimensional spaces

TL;DR

This paper develops complete orthonormal basis sets for spinor wave functions and Slater spinor orbitals in various spaces, facilitating quantum-mechanical calculations for spin-1/2 particles in relativistic contexts.

Contribution

It introduces new basis sets for two- and four-component spinor wave functions expressed through nonrelativistic orbitals, with analytical formulas for overlap integrals in relativistic quantum mechanics.

Findings

Established basis sets for relativistic spinor wave functions.

Derived analytical formulas for overlap integrals.

Applicable in quasirelativistic and relativistic quantum problems.

Abstract

By the use of complete orthonormal sets of nonrelativistic scalar orbitals introduced by the author in previous papers the new complete orthonormal basis sets for two- and four-component spinor wave functions, and Slater spinor orbitals useful in the quantum-mechanical description of the spin- 1/2 particles by the quasirelativistic and Dirac's relativistic equations are established in position, momentum and four-dimensional spaces. These function sets are expressed through the corresponding nonrelativistic orbitals. The analytical formulas for overlap integrals over four-component relativistic Slater spinor orbitals with the same screening constants in position space are also derived. The relations obtained in this study can be useful in the study of different problems arising in the quasirelativistic and relativistic quantum mechanics when the position, momentum and four dimensional spaces are employed.