Non-Relativistic Limit of the Dirac Equation

TL;DR

This paper demonstrates how the non-relativistic limit of the Dirac equation leads to the Schrödinger and Pauli equations, reproducing known quantum results for particles in potential barriers and wells.

Contribution

It establishes that a specific first order form of the Schrödinger equation is derived from the Dirac equation in the non-relativistic limit, clarifying the connection between relativistic and non-relativistic quantum mechanics.

Findings

Derivation of Schrödinger equation from Dirac in non-relativistic limit

Recovery of Pauli Hamiltonian via local gauge invariance

Validation of quantum results for potential barriers and wells

Abstract

We show that the first order form of the Schrodinger equation proposed in [1] can be obtained from the Dirac equation in the non-relativistic limit. We also show that the Pauli Hamiltonian is obtained from this equation by requiring local gauge invariance. In addition, we study the problem of a spin up particle incident on a finite potential barrier and show that the known quantum mechanical results are obtained. Finally, we consider the symmetric potential well and show that the quantum mechanical expression for the quantized energy levels of a particle is obtained with periodic boundary conditions. Based on these conclusions, we propose that the equation introduced in [1] is the non-relativistic limit of the Dirac equation and more appropriately describes spin 1/2 particles in the non-relativistic limit.