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Zeitschrift für Naturforschung A

Elementare Herleitung der Dirac-Gleichung / Elementary Derivation of the Dirac Equation

Hans Sallhofer

Published Nov 1, 1978• 8 citations
Publisher

Abstract

As is well known, the Schrödinger equation can be derived by suitably arranging the refraction in the classical equation for light inhomogeneous media. In this paper it is shown that one may derive the Dirac equation in complete analogy by arranging for the refraction in the electrodynamics of inhomogeneous media.
DOI
10.1515/zna-1978-1116

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Der Artikel entwickelt eine vollständige Herleitung der Dirac‑Gleichung aus der nichtrelativistischen Schrödingergleichung und der Energie‑Impuls‑Beziehung der speziellen Relativitätstheorie. Ausgangspunkt ist die Analyse des „duplexity“-Problems der frühen Quantenmechanik, das zur Einführung des Elektronenspins durch Goudsmit, Uhlenbeck und Pauli führte, jedoch ohne zwingende theoretische Begründung blieb. Durch systematische Linearisation der relativistischen Energieformel und Verwendung hermitescher, antikommutierender Matrizen wird gezeigt, dass die Klein–Gordon‑Gleichung strukturell unzureichend ist und erst die Einführung der Dirac‑Matrizen eine lorentzinvariante Gleichung erster Ordnung ermöglicht. Die Konstruktion der 4×4‑Gamma‑Matrizen, die Struktur des Bispinors sowie die vier gekoppelten Gleichungen der Dirac‑Theorie werden ausführlich dargestellt. Die physikalische Interpretation der negativen Energielösungen führt zur Dirac‑See und zur Vorhersage des Positrons. Abschließend wird gezeigt, wie die Dirac‑Gleichung im nichtrelativistischen Grenzfall in die Pauli‑Gleichung übergeht und damit Spin‑Bahn‑Kopplung und Feinstruktur konsistent beschreibt.
This paper re-examines the key aspects of Dirac's derivation of his relativistic equation for the electron in order advance our understanding of the nature of quantum field theory. Dirac's derivation, the paper argues, follows the key principles behind Heisenberg's discovery of quantum mechanics, which, the paper also argues, transformed the nature of both theoretical and experimental physics vis-a-vis classical physics and relativity. However, the limit theory (a crucial consideration for both Dirac and Heisenberg) in the case of Dirac's theory was quantum mechanics, specifically, Schrodinger's equation, while in the case of quantum mechanics, in Heisenberg's version, the limit theory was classical mechanics. Dirac had to find a new equation, Dirac's equation, along with a new type of quantum variables, while Heisenberg, to find new theory, was able to use the equations of classical physics, applied to different, quantum-mechanical variables. In this respect, Dirac's task was more similar to that of Schrodinger in his work on his version of quantum mechanics. Dirac's equation reflects a more complex character of quantum electrodynamics or quantum field theory in general and of the corresponding (high-energy) experimental quantum physics vis-a-vis that of quantum mechanics and the (low-energy) experimental quantum physics. The final section examines this greater complexity and its implications for fundamental physics.
Abstract A new derivation of Schrödinger’s equation is presented, based on Schrödinger’s original discussions on refraction and the optical-mechanical analogy, but adopting a much simpler formalism: Newtonian mechanics and some basic elements of classical wave theory (such as Snell’s law). We compare how particles and waves refract and show that the ‘law of particle refraction’ and the ‘law of wave refraction’ may become consistent if one assumes that a particle can be represented by a wave group. In this case, the differential equation whose solutions represent the waves forming such wave group is the Schrödinger equation. Due to the simplicity of the adopted mathematical formalism, we argue that this derivation can be used in quantum mechanics courses at introductory level to give students an idea of Schrödinger’s original path to his wave equation.