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Canadian Journal of Physics

Coherent state of the Dirac oscillator

Yukihisa Nogami, F. Masafumi Toyama

Published Mar 1, 1996• 43 citations
Publisher

Abstract

A coherent state of the Dirac oscillator in (1 + 1) dimensions is constructed. This is a relativistic extension of Schrödinger's coherent state of the nonrelativistic harmonic oscillator. There are interesting differences between the Dirac and the Schrödinger cases. In the Dirac case, the coherent state spreads out in time but after a while the original coherent state is nearly restored. Then it spreads again. This process recurs but in an aperiodic manner. In addition, the expectation value of coordinate x for the coherent state exhibits Zitterbewegung. The possibility of an analogue of Ehrenfest's theorem is discussed.

DOI
10.1139/p96-018

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A coherent state of the Dirac oscillator in (1 + 1) dimensions is constructed. This is a relativistic extension of Schrödinger's coherent state of the nonrelativistic harmonic oscillator. There are interesting differences between the Dirac and the Schrödinger cases. In the Dirac case, the coherent state spreads out in time but after a while the original coherent state is nearly restored. Then it spreads again. This process recurs but in an aperiodic manner. In addition, the expectation value of coordinate x for the coherent state exhibits Zitterbewegung. The possibility of an analogue of Ehrenfest's theorem is discussed.
The behaviour of wavepackets of the `Dirac oscillator' in (1 + 1) dimensions is examined. The Dirac oscillator is a relativistic extension of the harmonic oscillator. We construct a wavepacket in the usual Dirac representation and another one in the Foldy - Wouthuysen (FW) representation. In the nonrelativistic limit, these wavepackets are both reduced to Schrödinger's coherent wavepacket for the harmonic oscillator. The wavepacket in the Dirac representation exhibits Zitterbewegung whereas the one in the FW representation does not. We find that the wavepacket in the FW representation behaves much more like a classical particle than that of the Dirac representation. This gives insights into Ehrenfest's theorem for the Dirac equation.
The Dirac oscillator is a relativistic quantum system, characterized by its linearity in both position and momentum. Moreover, considering (1+1)(1{+}1)(1+1) and (2+1)(2{+}1)(2+1) dimensions, the system can be mapped onto the Jaynes-Cummings and anti-Jaynes-Cummings models, as illustrated in an exact manner by Bermudez \textit{et al.} [\href{ https://doi.org/10.1103/PhysRevA.76.041801}{Phys. Rev. A 76, 041801(R) (2007)}]. Using the optical counterparts of the Dirac oscillator, we analyze an extension of the model that incorporates a time-dependent frequency. We focus on the consequences of these time modulations on the angular momentum observables and spin-orbit entanglement. Noticeable changes in the \emph{Zitterbewegung} are found. We show that a specific choice of time dependence yields aperiodic evolution of the observables, whereas an alternative choice allows analytical solutions.
For the Dirac equation in one space dimension, we examine the possibility of harmonic oscillator (HO) potentials. A HO potential is such that it has only bound states and the energy levels are equally spaced. Regarding their Lorentz transformation properties, there are three types of potentials for the Dirac equation: vector, scalar, and pseudoscalar. HO potentials are possible for the scalar and pseudoscalar types, but not for the vector type. We show how HO potentials can be constructed by means of the ``inverse scattering method.'' We also examine the behavior of a wave packet in the HO potentials so constructed. The behavior is, in most of the cases, very similar to that of Schr\"odinger's coherent wave packet of the nonrelativistic harmonic oscillator.
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.
The dynamics of wave packets in a relativistic Dirac oscillator is compared to that of the Jaynes-Cummings model. The strong spin-orbit coupling of the Dirac oscillator produces the entanglement of the spin with the orbital motion similar to what is observed in the model of quantum optics. The collapses and revivals of the spin which result extend to a relativistic theory our previous findings on nonrelativistic oscillator where they were known under the name of `spin-orbit pendulum'. There are important relativistic effects (lack of periodicity, zitterbewegung, negative energy states). Many of them disappear after a Foldy-Wouthuysen transformation.