# Closed-form spin-relativistic corrections from the Dirac equation enabling a modified Schrödinger solver

## TL;DR

This paper introduces a new method to include relativistic effects in quantum calculations using a modified Schrödinger equation, reducing computational costs.

## Contribution

A novel closed-form framework for spin-relativistic corrections derived from the Dirac equation with an open-source solver.

## Key findings

- A Schrödinger-like equation with spin–relativistic corrections was derived for general central potentials.
- An open-source solver was developed to handle the quadratic eigenvalue problem from finite-difference discretization.
- First-order corrections for energy and wavefunctions were obtained for harmonic oscillator and Coulomb potentials.

## Abstract

We revisit the non-relativistic limit of the Dirac equation in finite scalar and vector potentials and derive a Schrödinger-like equation that retains leading spin–relativistic corrections in closed form. For general central potentials, we cast the radial equation into a quadratic eigenvalue problem (QEP) using a finite-difference discretization method and develop an open-source solver to address it. We study Coulomb, harmonic oscillator, Woods–Saxon, and Yukawa potentials. We further obtain first-order energy and wavefunction corrections for the three-dimensional isotropic harmonic oscillator and Coulomb potentials via perturbation theory. This framework provides a practical bridge between non-relativistic and fully relativistic treatments, enabling accurate quantification of relativistic effects without the computational cost of full four-component calculations.