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Journal of Physics A Mathematical and General

The Dirac oscillator

M. Moshińsky, Adam P. Szczepaniak

Published Sep 7, 1989• 734 citations
Publisher

Abstract

Dirac's free particle equation originated in an attempt to express linearly the relativistic quadratic relation between energy and momentum. The authors introduce a Dirac equation which, besides the momentum, is also linear in the coordinates. They call it the Dirac oscillator because in the nonrelativistic limit it becomes a harmonic oscillator with a very strong spin-orbit coupling term. The eigenstates and eigenvalues of the Dirac oscillator can be obtained in an elementary fashion, with the degeneracy of the latter being quite different from that of the ordinary oscillator. They briefly mention the symmetry Lie algebra responsible for this degeneracy and the generalisation of the problem to many-particle systems.

DOI
10.1088/0305-4470/22/17/002

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Dirac's free particle equation originated in an attempt to express linearly the relativistic quadratic relation between energy and momentum. The authors introduce a Dirac equation which, besides the momentum, is also linear in the coordinates. They call it the Dirac oscillator because in the nonrelativistic limit it becomes a harmonic oscillator with a very strong spin-orbit coupling term. The eigenstates and eigenvalues of the Dirac oscillator can be obtained in an elementary fashion, with the degeneracy of the latter being quite different from that of the ordinary oscillator. They briefly mention the symmetry Lie algebra responsible for this degeneracy and the generalisation of the problem to many-particle systems.
Abstract This chapter is seeking relativistic wave equations, which are invariant under Lorentz transformations, in an attempt to obtain a quantum mechanical description of relativistic particles. First, the chapter starts with the Klein–Gordon equation and then it discusses the Dirac equation. It also takes the non-relativistic limit of the Dirac equation to derive the Schrödinger equation with two additional terms, the Zeeman term and the spin–orbit coupling term. These two terms emerge naturally from the Dirac equation, and thus the spin, as an internal quantum number which behaves like angular momentum, is clearly identified. Finally, the existence of antimatter is clearly supported by the nature of the solutions to the Dirac equation.
Dirac sought a relativistic quantum free-particle Hamiltonian that imposes space-time symmetry on the Schroedinger equation in configuration representation; he ignored the Lorentz covariance of energy-momentum. Dirac free-particle velocity therefore is momentum-independent, breaching relativity basics. Dirac also made solutions of his equation satisfy the Klein-Gordon equation via requirements imposed on its operators. Dirac particle speed is thereby fixed to the unphysical value of c times the square root of three, and anticommutation requirements prevent four observables, including the components of velocity, from commuting when Planck's constant vanishes, a correspondence-principle breach responsible for Dirac free-particle spontaneous acceleration (zitterbewegung) that diverges in the classical limit. Nonrelativistic Pauli theory contrariwise is physically sensible, and its particle rest-frame action can be extended to become Lorentz invariant. The consequent Lagrangian yields the corresponding closed-form relativistic Hamiltonian when magnetic field is absent, otherwise a successive-approximation regime applies.
Some years ago, one of the authors~(MM) revived a concept to which he gave the name of single-particle Dirac oscillator, while another~(CQ) showed that it corresponds to a realization of supersymmetric quantum mechanics. The Dirac oscillator in its one- and many-body versions has had a great number of applications. Recently, it included the analytic expression for the eigenstates and eigenvalues of a two-particle system with a new type of Dirac oscillator interaction of frequency~\\omega. By considering the latter together with its partner corresponding to the replacement of~\\omega by~-\\omega we are able to get a supersymmetric formulation of the problem and find the superalgebra that explains its degeneracy.
Some years ago, one of the authors (MM) revived a concept to which he gave the name single-particle Dirac oscillator, while another of the authors (CQ) showed that it corresponds to a realization of supersymmetric quantum mechanics. The Dirac oscillator in its one- and many-body versions has had a great number of applications. Recently, it included the analytic expression for the eigenstates and eigenvalues of a two-particle system with a new type of Dirac oscillator interaction of frequency omega . By considering the latter together with its partner corresponding to the replacement of omega by - omega , we are able to get a supersymmetric formulation of the problem and find the superalgebra that explains its degeneracy.
Abstract Non-inertial physics is seldom considered in quantum mechanics and this contrasts with the ubiquity of non-inertial reference frames. Here, we show an application to the Dirac oscillator which provides a fundamental model of relativistic quantum mechanics. The model emerges from a term linearly dependent on spatial coordinates added to the momentum of the free-particle Dirac Hamiltonian. The definition generates peculiar features (mutating vacuum energy, non-Hermitian momentum, accidental degeneracies of the spectrum, etc). We interpret these anomalies in terms of inertial effects. The demonstration is based on the decoupling of the Dirac equation from the stereographic projection that maps the 3D geometry of the dynamical problem to the complex plane. The decoupling shows that the fundamental mechanical model underpinning the Dirac oscillator reduces to the representation of the oscillator in the rotating reference frame attached to the orbital angular momentum. The resulting Coriolis-like contribution to the Hamiltonian accounts for the peculiarities of the model (mutating vacuum energy, form of the non-minimal correction to the momentum, classical intrinsic spin and gain of its quantum value, accidental degeneracies of the energy spectrum, supersymmetric potential). The suggested interpretation has an interdisciplinary character where stereographic geometry, classical physics of the Coriolis effect and quantum physics of Dirac particles contribute to the definition of one of the few exactly soluble models of relativistic quantum mechanics.
A single-particle Hamiltonian independent of the particle's coordinate ensures the conserves momentum, i.e., is free. Lorentz-covariance of that Hamiltonian's energy-momentum specifies it up to the particle's rest energy; the it describes has speed below c and constant velocity parallel to its conserved momentum. Dirac took his free-particle Hamiltonian to have the same squared value as that relativistic one, but unwittingly blocked Lorentz-covariance of his Hamiltonian's energy-momentum by requiring it to be inhomogeneously linear in momentum. The Dirac free particle badly flouts relativity and even physical cogency; its velocity direction is extremely nonconstant, while its speed is fixed to c times the square root of three even when it interacts electromagnetically. Both its rest energy and total energy can be negative, and its velocity components and rest energy are artificially correlated by being mutually anticommuting; its alleged spin is an artifact of the anticommutation of its velocity components. Unlike the Dirac Hamiltonian, the nonrelativistic Pauli Hamiltonian is apparently physically sensible for speed far below c. Its relativistic extension is worked out via Lorentz-invariant upgrade of its associated action functional at zero velocity, and is obtained in closed form if there is no applied magnetic field; a successive approximation scheme must otherwise be used.