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Free Dirac Current for Superposed States

Anamitra Palit

Published Feb 1, 2017• 0 citations
OpenAlex

Abstract

The article aims to investigate the four current due to the superposition of free Dirac states with a special view towards the terms containing oscillatory factors responsible for “the trembling motion of electrons”----Zitterbewegung. As we shall see that these terms simply disappear. Zitterbewegung is not possible with free Dirac States

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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.

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Zitterbewegung revisited

G C Sherry

Certain inconsistencies in the derivation of the zitterbewegung in Dirac's theory of the free electron are discussed.

European Journal of Physics•Oct 1, 1988•3 citations
The article aims to investigate the four current due to the superposition of free Dirac states with a special view towards the terms containing oscillatory factors responsible for “the trembling motion of electrons”----Zitterbewegung. As we shall see that these terms simply disappear. Zitterbewegung is not possible with free Dirac States
Abstract In this article, we show that the wave equation for a free Dirac electron can be represented in a form that is analogous to Maxwell’s electrodynamics. The electron bispinor wavefunction is explicitly expressed in terms of its real and imaginary components. This leads us to incorporate into it appropriate scalar and pseudo-scalar fields in advance, so that a full symmetry may be accomplished. The Dirac equation then takes on a form similar to that of a set of inhomogeneous Maxwell’s equations involving a particular self-source. We relate plane wave solutions of these equations to waves corresponding to free Dirac electrons, identifying the longitudinal component of the electron motion, together with the corresponding Zitterbewegung (“trembling 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.
The detailed motion of a free Dirac electron is investigated by examining the expectation values of the position r and of r×ṙ in a wave packet. It is shown that the well-known zitterbewegung may be looked upon as a circular motion about the direction of the electron spin, with a radius equal to the Compton wavelength (divided by 2π) of the electron. It is further shown that the intrinsic spin of the electron may be looked upon as the “orbital angular momentum” of this motion. The current produced by the zitterbewegung is seen to give rise to the intrinsic magnetic moment of the electron.
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.
In the Dirac theory for the motion of free relativistic electrons, highly oscillatory components appear in the time evolution of physical observables such as position, velocity, and spin angular momentum. This effect is known as zitterbewegung. We present a theoretical analysis of rather different Hamiltonians with gapped and/or spin-split energy spectrum (including the Rashba, Luttinger, and Kane Hamiltonians) that exhibit analogs of zitterbewegung as a common feature. We find that the amplitude of oscillations of the Heisenberg velocity operator v(t)\mathbf{v}(t)v(t) generally equals the uncertainty for a simultaneous measurement of two linearly independent components of v\mathbf{v}v. It is also shown that many features of zitterbewegung are shared by the simple and well-known Landau Hamiltonian, describing the dynamics of two-dimensional (2D) electron systems in the presence of a magnetic field perpendicular to the plane. Finally, we also discuss the oscillatory dynamics of 2D electrons arising from the interplay of Rashba spin splitting and a perpendicular magnetic field.