Conjugate Spinor Solution of the Dirac Equation for the Hydrogen Atom

TL;DR

This paper introduces a simplified method for solving the Dirac equation for the hydrogen atom using conjugate spinors, reducing the problem to a single differential equation and facilitating wave function construction.

Contribution

It presents a novel approach employing real conjugate spinors to simplify the Dirac equation for hydrogen, enabling easier solution and wave function construction.

Findings

Reduction of Dirac equation to a single second-order differential equation

Simplification of wave function construction using conjugate spinors

Potential for more efficient solutions of atomic Dirac equations

Abstract

It is shown the central field Dirac equation can be simplified through the use of real conjugate spinors to substitute for the upper and lower components of the bi-spinor eigensolutions. This substitution reduces the Dirac equation for the hydrogen atom to the problem of solving a single second order differential equation similar to a Klein-Gordon equation but containing additional terms to take account of the spin on the electron. The bi-spinor wave functions are readily constructed once the solution is known in terms of the conjugate spinors.