On the theory of the Schroedinger equation with the full set of relativistic corrections

TL;DR

This paper derives a comprehensive set of relativistic corrections to the Schrödinger equation from the Dirac equation, including new terms, and analyzes their impact on electron spectra in quantum well structures.

Contribution

It introduces new relativistic correction terms to the Schrödinger equation and demonstrates their necessity for accurate energy calculations matching Dirac solutions.

Findings

Identified new second-order relativistic correction terms.

Showed that including these corrections aligns Schrödinger energies with Dirac solutions.

Analyzed the physical origin of spin-orbit interactions in quantum wells.

Abstract

All relativistic corrections to the Scr{\"o}dinger equation which determine the interlink between spin and orbit of moving particles, are directly calculated from the Dirac equation using the spin invariant operators. It is shown that among the second order corrections there are not only the well-known Darwin and Thomas terms, but also the new ones. Only with the account of the latter corrections the energies found with the obtained spin-orbit interaction operator, coincide with the energies of the Dirac equation exact solution. The problem of electron spectrum in the quantum well type structures is studied in details and the physical reasons for the appearance of spin-orbit interaction operators in the Dresselhaus or Rashba form, are analyzed.