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American Journal of Modern Physics

Assessment of Zitterbewegung Interpretation for Free Particle Solution Using the Concept of Relativistic Wave

Emmanuel Ifeanyi Ugwu

Published Jul 29, 2024• 0 citations
Publisher

Abstract

The Assessment of the interpretation of Zitterbewegung for free particle solution using various models has been carried out in this work where we firstly considered by using the free particle solution for which Dirac equation and projection operator were carried out.. ln this case, it was invariably revealed that the solution have two doubly degenerate eigenvalues representing positive and negative state that was further support by the analysis which was carried out using Heisenberg’s equation and the representation in relativistic velocity and semi-classical equation of motion for acceleration., but in this second case, the results obtained were observed to have two terms, first terms standing in for rapidly oscillatory motion and the second term representing average motion of the particle in x-direction. The first term in the expression is the one deemed to signify zitterbewegung which is one considered to be sas a result of the fluctuation resulting from the interference between negative and positive energy state while the other term is the normal state terms signifying the average motion of the particle. This therefore confirms the explicit nature of using Dirac equation in handling problems involving the behavior of free particles in field as compared to the use of semiclassical equation of motion.

DOI
10.11648/j.ajmp.20241303.11

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The Assessment of the interpretation of Zitterbewegung for free particle solution using various models has been carried out in this work where we firstly considered by using the free particle solution for which Dirac equation and projection operator were carried out.. ln this case, it was invariably revealed that the solution have two doubly degenerate eigenvalues representing positive and negative state that was further support by the analysis which was carried out using Heisenberg’s equation and the representation in relativistic velocity and semi-classical equation of motion for acceleration., but in this second case, the results obtained were observed to have two terms, first terms standing in for rapidly oscillatory motion and the second term representing average motion of the particle in x-direction. The first term in the expression is the one deemed to signify zitterbewegung which is one considered to be sas a result of the fluctuation resulting from the interference between negative and positive energy state while the other term is the normal state terms signifying the average motion of the particle. This therefore confirms the explicit nature of using Dirac equation in handling problems involving the behavior of free particles in field as compared to the use of semiclassical equation of motion.
We seek the {\em immediate} description of chiral oscillations in terms of the trembling motion described by the velocity (Dirac) operator {\boldmathααα}. By taking into account the complete set of Dirac equation solutions which results in a free propagating Dirac wave packet composed by positive and negative frequency components, we report about the well-established {\em zitterbewegung} results and indicate how chiral oscillations can be expressed in terms of the well know quantum oscillating variables. We conclude with the interpretation of chiral oscillations as position very rapid oscillation projections onto the longitudinally decomposed direction of the motion.
Around 1930, both Gregory Breit and Erwin Schroedinger showed that the eigenvalues of the velocity of a particle described by wavepacket solutions to the Dirac equation are simply ±\pm±c, the speed of light. This led Schroedinger to coin the term Zitterbewegung, which is German for "trembling motion", where all particles of matter (fermions) zig-zag back-and-forth at only the speed of light. The result is that any finite speed less than ccc, including the state of rest, only makes sense as a long-term average that can be thought of as a drift velocity. In this paper, we seriously consider this idea that the observed velocities of particles are time-averages of motion at the speed of light and demonstrate how the relativistic velocity addition rule in one spatial dimension is readily derived by considering the probabilities that a particle is observed to move either to the left or to the right at the speed of light.
We present a classical optics simulation of the one-dimensional Dirac equation for a free particle. Positive- and negative-energy components are represented by orthogonal polarizations of a free propagating beam, while the spatial profile represents the spatial wave function of the particle. Using a completely tunable time parameter, we observed the oscillation of the average value of the Dirac position operator---known as Zitterbewegung. We are also able to measure the so-called mean-position operator which is a single-particle observable and presents no oscillations. Our work opens the way for optical simulation of interesting phenomenon of relativistic systems, as well as condensed-matter physics, without any requirement for specially engineered medium.
The fact that the Dirac equation predicts Zitterbewegung (ZBW) for a free electron is well known from Schrodinger [E. Schrodinger, Sitzungsber Preuss. Akad. Wiss. Berlin (Math. Phys.) 24, 418 (1930)] from the very early days of Dirac's relativistic quantum theory. Schrodingerdescribed ZBW qualitatively as a persistent interference between positive and negative energy states, and the objective of this and previous papers by this author [Phys. Essays 28, 1 (2015); Phys. Essays 29, 402 (2016)] is to describe ways to measure the presence or absence ofZBW for a free (or nearly free) electron. If ZBW measurements show that it is not present for a free electron, then the Dirac Equation and Schrodinger's analysis [E. Schrodinger, Sitzungsber Preuss. Akad. Wiss. Berlin (Math. Phys.) 24, 418 (1930)] are inaccurate. If ZBW ispresent as Schrodinger's analysis [E. Schrodinger, Sitzungsber Preuss. Akad. Wiss. Berlin (Math. Phys.) 24, 418 (1930)] of the Dirac Equation asserts, then a description of a free electron emerges that is significantly different than the present very small, static point electronwith “intrinsic” properties of spin and magnetic moment accepted by most physicists [Eides et al., Phys. Reports 342, 63 (2001).] today. This paper develops a quantitative theory of the free electron directly from the Dirac Equation and Schrodinger's analyses[E. Schrodinger, Sitzungsber Preuss. Akad. Wiss. Berlin (Math. Phys.) 24, 418 (1930)] without any other assumptions. The fact that ZBW exists for the electron in the hydrogen atom is confirmed by the Darwin term [J. J. Sakurai, Advanced Quantum Mechanics (Addison-Wesley,Reading, MA, 1967)]. It seems unnatural to the author that ZBW would vanish when the electron is removed from the hydrogen atom, and becomes free. The unanswered question is whether ZBW can be confirmed or refuted by measurement for a free electron, and possible ways to measure its presenceor absence are described in this paper. The remainder of the paper describes the free electron, assuming ZBW exists. The Dirac Equation [J. H. Wilson, Phys. Essays 28, 1 (2015); Phys. Essays 29, 402 (2016)] produces a physical description of a free electron's ZBW as rapidly oscillating,quantized Center of Charge (CoC) point rotating about a fictitious Center of Mass (CoM) at a radius of 3/ 2 times the electron's Compton wavelength. The results in the remainder of this paper, as well as Schrodinger's analyses [E. Schrodinger, Sitzungsber Preuss. Akad. Wiss. Berlin (Math. Phys.)24, 418 (1930).], are only true if ZBW can be confirmed by measurement. In case measurements show no ZBW exists for a free electron, the Dirac Equation is not a correct description of a single free electron, and more serious problems arise. The static point like electron deduced fromscattering experiments [Eides et al., Phys. Rep. 342, 63 (2001)] that give estimates of the free electron's internal structure based on far field measurements can be misleading in estimating the free electron's internal structure. In Classical Electrodynamics, when one observesthe electric field from a point electron and from a sphere with the same charge spread uniformly over its surface from the far field, Gauss' Law shows that the electric fields are identical. When one tries to “observe” or “measure” the free electron's internal structurewith the high-energy photons required, the single particle free electron is completely destroyed by the measurement process. In fact, any attempt to observe or measure a free electron's structure ensures that the electron is no longer a totally free single particle, and one must be carefulto use only elastic observation methods [J. H. Wilson, Phys. Essays 28, 1 (2015); Phys. Essays 29, 402 (2016)] for this very difficult measurement. Observing a free Quark, much less its internal structure, has proven very difficult, if not impossible.
The Hamiltonian form of the Klein–Gordon equation is used to study the Zitterbewegung of spin- zero particles. It is shown that the time dependence of the velocity and position operators in the Heisenberg picture is formally the same as that of a spin- 1/2 particle. It is also shown that the Zitterbewegung is the result of the interference between positive and negative energy states. A representation-free expression is found for the mean position operator. This operator is free from Zitterbewegung. The Klein–Gordon equation is solved with an electrostatic potential originally used by Sauter in his investigation of the Klein paradox for spin- 1/2 particles. It is shown that the transmission to states with negative kinetic energy is very small when the electric field is weak.
A linear Hamiltonian in spatial derivative that satisfies Klein-Gordon equation was used starting from energy momentum relation for free particle was solved in agreement with the matrices and bearing in mind their suitability in terms of anticommutation relations in parallel with the definition of algebraic matrices whose hermicity is fulfilled by i += i and += and in turn linked up to explicit representation of the Dirac matrices. The wave packets of plane Dirac wave obtained as a superposition of plane wave yielding a localized wave function was normalized considering only positive energy of plane wave in which the expectation value with respect to the wave packet resulted from 2p/E>+=gr)> was found to agree with the Ehrenfest theorem in relation to Schrodinger theorem as it relates to true velocity of single particle. A comparison was made between the classical concept with Heisenberg representation from where the combined effect of the positive and negative energy components was considered.