The Hydrogen Atom and the Equivalent Form of Levy-Leblond Equation

TL;DR

This paper derives a non-relativistic form of the Levy-Leblond equation, connects it to the Dirac equation in lower dimensions, and applies it to the hydrogen atom, reproducing known energy levels.

Contribution

It introduces a two-dimensional nilpotent matrix form of the Levy-Leblond equation and extends it to (3+1) dimensions with a Hamiltonian for hydrogen.

Findings

Derived the non-relativistic limit of the (2+1)D Dirac equation

Proposed a Hamiltonian for the (3+1)D Levy-Leblond equation

Obtained hydrogen atom energy levels consistent with quantum mechanics

Abstract

We discuss the equivalent form of Levy-Leblond equation [1, 2] such that the nilpotent matrices are two dimensional. We show that this equation can be obtained in the non-relativistic limit of the (2+1) dimensional Dirac equation. Furthermore, we analyze the case with four dimensional matrices and propose a Hamiltonian for the equation in (3+1) dimensions and solve it for a Coulomb potential. We show that the quantized energy levels for the hydrogen atom are obtained and the result is consistent with non-relativistic quantum mechanics.