HEISENBERG’S uncertainty relation results as condition for convergence of the solution of a difference equation which is an analogue to the SCHROEDINGER equation. From DIRAC’S equations and the relativistic SCHROEDINGER equation (KLEIN-GORDON equation) — written as difference equations — one obtains in the same manner a generalized uncertainty relation, from which HEISENBERG’S is derived as a special case for negligible relativistic mass increase and not too precise position and time measurements.
HEISENBERG’S uncertainty relation results as condition for convergence of the solution of a difference equation which is an analogue to the SCHROEDINGER equation. From DIRAC’S equations and the relativistic SCHROEDINGER equation (KLEIN-GORDON equation) — written as difference equations — one obtains in the same manner a generalized uncertainty relation, from which HEISENBERG’S is derived as a special case for negligible relativistic mass increase and not too precise position and time measurements.
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.
1926, schon zwei Jahre, bevor Dirac die Dirac-Gleichung aufstellte, wurde von O. Klein und W. Gordon eine Gleichung für die relativistische Quantendynamik von Teilchen angegeben.1 Aufgrund von Schwierigkeiten, die wir erörtern werden, wurde sie jedoch vorübergehend aufgegeben. Einige dieser Schwierigkeiten lassen sich auf einfache Weise beseitigen, andere, denen man in ähnlicher Weise auch bei der Dirac-Gleichung begegnet, können erst im Rahmen einer Quantenfeldtheorie, dort aber vollständig, behoben werden. Der letzte Umstand macht die Klein-Gordon-Schrödinger-Gleichung zu einer nützlichen relativistischen “Wellengleichung“, deren Untersuchung sich lohnt.
Since the publication of Heisenberg’s paper1 on the “anschaulichen Inhalt” of quantum mechanics, discussions of the fundamental limitations on the accuracy of physical measurements have been much in the foreground. According to Heisenberg, the quantum mechanics implies that it is impossible to measure simultaneous values of a coordinate and its conjugate momentum with unlimited precision. Instead, if Δp be the estimated error or uncertainty in a momentum and Δq that in the associated coordinate one must have the inequality,
ΔpΔq>2πh
(1)
This inequality has come to be known quite generally as Heisenberg’s uncertainty relation.
Derivation of Klein–Gordon equation Goals and prior knowledge The main wave equations of relativistic quantum mechanics are the Klein–Gordon, Dirac, Weyl , and Rarita–Schwinger equations. The principal aim of this chapter is to derive these, by means of the EPI principles (3.16) and (3.18). The derivations generally follow Frieden (1995), or Frieden and Soffer (1995), with numerous additions. In addition, the Schrödinger wave equation (SWE), which is non -relativistic, is also to be derived. Since rigorous use of EPI gives only relativistic wave equations (Chapter 3), EPI can only derive the SWE in some approximate sense. This is directly carried through in Appendix D, or by taking the well-known non-relativistic limit of the Klein–Gordon equation in Appendix G. Also, in Appendix D a simple EPI derivation of classical mechanics is obtained. In particular, in the limit h → 0 the EPI extremum condition (3.16) is found to give Newton's second law , and the zero-condition (3.18) gives the virial theorem , of classical mechanics. Other derived results are Eq. (4.17) – the equivalence of mass, momentum, and energy; the constancy of the Planck constant h (Sec. 4.1.14); the Compton wavelength as an ultimate resolution length (Sec. 4.1.17); and the Heisenberg uncertainty principle (Sec. 4.3). It is important to state what is assumed a priori . The relativistic properties of space and time coordinates were derived in Sec. 3.5. Hence, these are assumed here. Also, appropriate units for the concepts of mass and energy are assumed to be known, although the basic relation Eq. (4.17) between them is not presumed.
The usual Heisenberg uncertainty relation, ΔX ΔP ≥ ℏ/2, may be replaced by an exact equality for suitably chosen measures of position and momentum uncertainty, which is valid for all wavefunctions. This exactuncertainty relation, δX ΔPnc ≡ ℏ/2, can be generalised to other pairs of conjugate observables such as photon number and phase, and is sufficiently strong to provide the basis for moving from classical mechanics to quantum mechanics. In particular, the assumption of a nonclassical momentum fluctuation, having a strength which scales inversely with uncertainty in position, leads from the classical equations of motion to the Schrödinger equation.