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arXiv:1403.4580

Electron channeling, de Broglie's clock and the relativistic time operator

M. Bauer

Published Mar 18, 2014• 2 citations
arXivHTMLPDF

Abstract

Electron channeling in silicon crystals has brought forward the possibility of having detected a particle's "internal clock", as an intrinsic oscillation with de Broglie's frequency. The transmission probability along a major axial direction is reduced with respect to neighboring angles, except for a sharp peak at the atomic row direction. The pattern observed is a "W" instead of a "U". This central peak is attributed to a process known as "rosette motion", in which the crossing of successive atoms would be related to the de Broglie frequency. A classical multiple scattering calculation found that, to represent the experimental results, the interaction frequency had to be about twice the de Broglie's clock frequency; that is, the "Zitterbewegung" frequency. In the present paper, the observed characteristics of this process are shown to be consistent with a free particle quantum mechanical motion described by Dirac's Hamiltonian, albeit with an effective mass resulting from the interaction with the crystal atoms. The introduction of a self-adjoint dynamic time operator provides the connection with an internal "system time", the de Broglie clock.

DOI
10.48550/arxiv.1403.4580

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Electron channeling in silicon crystals has brought forward the possibility of having detected a particle's "internal clock", as an intrinsic oscillation with de Broglie's frequency. The transmission probability along a major axial direction is reduced with respect to neighboring angles, except for a sharp peak at the atomic row direction. The pattern observed is a "W" instead of a "U". This central peak is attributed to a process known as "rosette motion", in which the crossing of successive atoms would be related to the de Broglie frequency. A classical multiple scattering calculation found that, to represent the experimental results, the interaction frequency had to be about twice the de Broglie's clock frequency; that is, the "Zitterbewegung" frequency. In the present paper, the observed characteristics of this process are shown to be consistent with a free particle quantum mechanical motion described by Dirac's Hamiltonian, albeit with an effective mass resulting from the interaction with the crystal atoms. The introduction of a self-adjoint dynamic time operator provides the connection with an internal "system time", the de Broglie clock.
Electron channeling in silicon crystals has brought forward the possibility of having detected the particle's "de Broglie internal clock", as giving rise to the observed resonance peak at the center of the expected transmission probability dip. A classical multiple scattering calculation fails to represent the experimental results unless, surprisingly, the interaction frequency is twice the de Broglie's clock frequency, that is, the "Zitterbewegung" frequency. In the present paper, the observed characteristics of this process are shown to be consistent with a free particle quantum mechanical motion described by Dirac's Hamiltonian.
De Broglie’s association of a wave to particles is a fundamental concept in the quantum mechanical description of nature. The wave oscillation is referred to alternatively as the “de Broglie clock”, the “Compton clock”, or the “de Broglie periodic phenomenon”. In the present paper it is shown that Dirac’s relativistic quantum mechanics, complemented with the dynamical time operator recently introduced, provides a consistent theoretical description of: (i) the generation of the de Broglie wave through Lorentz boosts; and (ii) the characteristics of the resonance observed in electron channeling through thin crystals as responding to both the periodicity derived from the adjustment of the de Broglie period to the crystal interatomic distance (resonance energy) and the periodicity of the predicted trembling motion (Zitterbewegung). One can conclude that the channeling experiments provide the first direct evidence of the electron Zitterbewegung, and that the de Broglie period is an intrinsic property of matter arising from a self-adjoint dynamical time operator.
The particle internal clock conjectured by L. de Broglie in 1924 was looked for in a channeling experiment with -80 MeV electrons traversing a 1-μm thick silicon crystal aligned along the 〈110〉 direction. Part of them undergo what is called the rosette motion, in which they interact with a single atomic row. When the electron energy is finely varied, the rate of electron transmission at 0° shows an 8% dip within 0.5% of the resonance energy, 80.874 MeV, for which the frequency of atomic collisions matches the internal clock frequency. Our observation is compatible with the de Broglie hypothesis.
An experimental observation compatible with the particle internal clock in a channeling experiment was conducted. The particle internal clock can be an observable quantity only if the wave function is slightly modified. The main momentum resonance may be 162 MeV/c instead of 81 MeV/c, since a dispersion of the electron trajectory is sensitive to the absolute value of the wave function amplitude, which has twice the frequency. In order to design the experimental set-up by means of a Monte-Carlo calculation, a phenomenological model of the electron's motion and internal clock was used on the basis of classical mechanics. The event rate at crystal tilt angle was studied as a function of the electron's momentum. The results show that this rate is constant except in a sharp momentum range of the order of 1% centered at the resonance momentum. The data of the crystal tilt angle analysis shows the usual rosette motion feature.
The existence of an internal frequency associated to any elementary particle conjectured by de Broglie is compared with a classical description of the electron, where this internal structure corresponds to the motion of the centre of charge around the centre of mass of the particle. This internal motion has a frequency twice de Broglie's frequency, which corresponds to the frequency found by Dirac when analysing the electron structure. To get evidence of this internal electron clock a kind of experiment as the one performed by Gouanére et al. \cite{Gouanere} will show a discrete set of momenta at which a resonant scattering effect, appears. The resonant momenta of the electron beam are given by pk=161.748/kp_k=161.748/kpk​=161.748/k MeV/c/c/c, k=1,2,3,...k=1,2,3,...k=1,2,3,..., where only, the corresponding to k=2k=2k=2, was within the range of Gouanére et al. experiment. The extension of the experiment to other values of pkp_kpk​, would show the existence of this phenomenon.
A self-adjoint dynamical time operator is introduced in Dirac's relativistic formulation of quantum mechanics and shown to satisfy a commutation relation with the Hamiltonian analogous to that of the position and momentum operators. The ensuing time-energy uncertainty relation involves the uncertainty in the instant of time when the wave packet passes a particular spatial position and the energy uncertainty associated with the wave packet at the same time, as envisaged originally by Bohr. The instantaneous rate of change of the position expectation value with respect to the simultaneous expectation value of the dynamical time operator is shown to be the phase velocity, in agreement with de Broglie's hypothesis of a particle associated wave whose phase velocity is larger than c. Thus, these two elements of the original basis and interpretation of quantum mechanics are integrated into its formal mathematical structure. Pauli's objection is shown to be resolved or circumvented. Possible relevance to current developments in electron channeling, in interference in time, in Zitterbewegung-like effects in spintronics, graphene and superconducting systems and in cosmology is noted.