The Dirac equation: historical context, comparisons with the Schrödinger and Klein-Gordon equations, and elementary consequences

TL;DR

The paper reconstructs the Dirac equation within its historical development, contrasting it with the Schrödinger and Klein–Gordon equations to illustrate Lorentz covariance and spin. It derives and analyzes free-particle Dirac solutions, explains negative-energy states through the Dirac sea and antiparticles, and shows how spin and helicity emerge from a four-component bispinor. In the nonrelativistic limit, it derives the Pauli equation and highlights spin–orbit and Darwin corrections via Foldy–Wouthuysen transformations, connecting to hydrogen fine structure. It also presents relativistic covariant notation to underscore invariance and provides a pedagogical bridge from historical context to modern relativistic quantum mechanics and quantum field theory.

Abstract

This paper offers educational insight into the Dirac equation, examining its historical context and contrasting it with the earlier Schrödinger and Klein-Gordon (KG) equations. The comparison highlights their Lorentz transformation symmetry and potential probabilistic interpretations. We explicitly solve the free-particle dynamics in Dirac's model, revealing the emergence of negative-energy solutions. This discussion examines the Dirac Sea Hypothesis and explores the solutions' inherent helicity. Additionally, we demonstrate how the Dirac equation accounts for spin and derive the Pauli equation in the non-relativistic limit. The Foldy-Wouthuysen transformation reveals how the equation incorporates spin-orbit interaction and other relativistic effects, ultimately leading to the fine structure of hydrogen. A section on relativistic covariant notation is included to emphasize the invariance of the Dirac equation, along with more refined formulations of both the KG and Dirac equations. Designed for undergraduate students interested in the Dirac equation, this resource provides a historical perspective without being purely theoretical. Our approach underscores the significance of a pedagogical method that combines historical and comparative elements to profoundly understand the role of the Dirac equation in modern physics.

Figures (7)

• Figure 1: Not to scale timeline concerning the most relevant events for this work.
• Figure 2: Schematic representation of $\psi(x)$ and $|\psi(x)|^2$ for the scenario of a particle in a box of length L, a simple solution to the Schrödinger equation.
• Figure 3: The inertial frames $\mathcal{O}$ and $\mathcal{O}'$.
• Figure 4: SG experiment illustrated schematically. The beam of silver atoms is expelled from the furnace, passes through a collimator, and is subjected to a heterogeneous magnetic field $\mathbf{B}$. In the detector, two preferential regions are perceived.
• Figure 5: Dirac equation's energy diagram.