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Journal of Physics A Mathematical and General

Behaviour of wavepackets of the 'Dirac oscillator': Dirac representation versus Foldy - Wouthuysen representation

F. Masafumi Toyama, Yukihisa Nogami, Francisco Antônio Bezerra Coutinho

Published Apr 7, 1997• 54 citations
Publisher

Abstract

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.

DOI
10.1088/0305-4470/30/7/034

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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.
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 dynamics of wavepackets in a relativistic Dirac oscillator (DO) is compared with that of the Jaynes-Cummings model. The strong spin-orbit coupling of the DO produces the entanglement of the spin with the orbital motion similar to that observed in the model of quantum optics. The resulting collapses and revivals of the spin extend to a relativistic theory our previous findings on a nonrelativistic oscillator where they were known as spin-orbit pendulum . The Foldy-Wouthuysen transformation can be performed exactly for the DO. It produces the well known smoothing effect over the Compton wavelength. Thus, after this transformation, zitterbewegung disappears just as the components of the WP associated to negative energy states.
Zitterbewegung of massless particles with an arbitrary spin is analyzed in various representations. Dynamics of the group velocity of a massless particle as a whole and of the corresponding radius vector is determined. This radius vector defines any fixed point of the envelope of the moving wave packet characterizing the particle and its group velocity differs from the group velocities of any points of the wavefront. We consider free massless scalar and Dirac particles, the photon, and massive and massless particles with an arbitrary spin and describe them in different representations. For particles with an arbitrary spin, the generalized Feshbach-Villars representation and the Foldy-Wouthuysen one are used. Zitterbewegung takes place in any representation except for the Foldy-Wouthuysen one. Formulas describing the motion of a ``trembling'' free particle are the same in any representation. In the Foldy-Wouthuysen representation, the operators of the velocity and momentum of a free particle are proportional and Zitterbewegung does not take place. Since the radius vector (position) and velocity operators are the quantum-mechanical counterparts of the classical position and velocity just in the Foldy-Wouthuysen representation, Zitterbewegung is not observable. The same conclusion has been previously made for free massive particles. For relativistic massive particles with spins 0, 1/2, 1 and massless particles with spins 0, 1/2 in arbitrarily strong electromagnetic fields, independent of the external fields Zitterbewegung does not appear in the Foldy-Wouthuysen representation either. This conclusion is made for leading terms in the Hamiltonian proportional to the zero and first powers of the Planck constant and for such terms proportional to ${\ensuremath{\hbar}}^{2}$ which describe contact interactions.
Abstract Zitterbewegung of a massive Proca (spin-1) boson is analyzed. The equations of motion of a massive Proca particle in the Sakata-Taketani representation are equivalent to the corresponding equations for the Dirac particle. However, Zitterbewegung does not appear in the Foldy-Wouthuysen representation. Zitterbewegung is not observable because the quantum-mechanical counterparts of the classical position and velocity are the position and velocity operators in the Foldy-Wouthuysen representation and their transforms to other representations.