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Does a Single Electron Have Wave Properties?

  • PHYSICS OF ELEMENTARY PARTICLES AND ATOMIC NUCLEI. THEORY
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Abstract

An attempt is made to substantiate the absence of wave properties for a single electron. What is commonly called the corpuscular-wave duality or the wave properties of matter, which appear in the form of a diffraction pattern on the target of certain kind, is a result of the obligatory involvement of a large number of electrons in the diffraction process. In the context of quantum mechanics, a form of the diffraction pattern can be predicted using the rule of adding probability amplitudes, which, in the simplest case, are solutions to the Schrödinger equation for free particles. Therefore, the answer to the question posed in the title sounds something like this: a single electron has no wave properties; however, they become apparent whenever there are many electrons and a suitable target.

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Notes

  1. We will use the terminology of D.I. Blokhintsev [8], understanding microparticles as objects of the microworld.

  2. Here, \(h = 6.6262 \times {{10}^{{ - 27}}}\) Erg s is the Planck constant.

  3. A. Einstein, The Collection of Scientific Papers, vol. 3, p. 465.

  4. In other words, these are particles without any individual wave properties.

  5. These “exculpatory” statements seem convincing. However, the history of the science and technology development has not yet presented any arguments against the (opposite) point of view that the human mind has always found an opportunity to adequately reflect (understand) new and previously unknown processes and phenomena. It is not obvious that the microcosm will become an insurmountable obstacle here.

  6. Feynman here does not distinguish between diffraction and interference, since he considers the first to be the result of the “work” of the second. He also ignores single-slit diffraction. As is known, the diffraction (deflection) of light takes place even at the macroslit boundary. The impression may be created that wave phenomena do not manifest themselves with the diffraction by one slit, but arise only as a result of the combined action of two slits. In modern work [19], it is said that the results of studying the wave nature of electrons, obtained at an electron microscope, show that the consideration of the electron diffraction by one round hole gives a full proof that an interference experiment on two holes, generally speaking, is not required for demonstrating the superposition of electron waves.

  7. Here it is tacitly assumed that an interaction occurs with probability \({{q}_{{1,2}}} = 1\) at each slit, or, equivalently, both slits are “in action.” If \({{q}_{{1,2}}} \ne 1\), then (3) should be written as \({{P}_{{12}}} = {{\left| {\left\langle {\left. x \right|1} \right\rangle {{q}_{1}}\left\langle {\left. 1 \right|s} \right\rangle + \left\langle {\left. x \right|2} \right\rangle {{q}_{2}}\left\langle {\left. 2 \right|s} \right\rangle } \right|}^{2}}\) and the “cutoff”, e.g., of slit 1, means that \({{q}_{1}} = 0\).

  8. Further in (11), \({\mathbf{r}}\) is the radius vector of an electron; \(C\) is the normalization constant; \({{{\mathbf{k}}}_{i}}\) and \({{{\mathbf{k}}}_{s}}\) are the wavevectors of the initial and scattered electrons; \(\left| {{{{\mathbf{k}}}_{i}}} \right| = \sqrt {{{2mE} \mathord{\left/ {\vphantom {{2mE} {{{\hbar }^{2}}}}} \right. \kern-0em} {{{\hbar }^{2}}}}} \), where \(E,m\) are the energy and mass of an electron, and \(h = 2\pi \hbar \) is the Planck constant. In elastic scattering, \(\left| {{{{\mathbf{k}}}_{i}}} \right| = \left| {{{{\mathbf{k}}}_{s}}} \right|\).

  9. If the combination of words “cannot … simultaneous” is removed, then hope remains that wave phenomena can be associated simply with the presence of a large number of particles.

  10.  The fact that the wave properties of matter were already predicted by de Broglie is secondary in this case.

  11.  With special consideration, it turns out that these wave properties are very unusual, but this is already later.

  12.  Of course, for the sake of respect for the great merits of the founders of quantum mechanics.

  13.  Since today it is already possible to work with a single electron, the proof of the wave–particle duality would be an experiment with one electron, demonstrating the wave properties of this particular electron. It is not at all clear yet how this can be done, though.

  14.  Attempts were made to use the inelastic interaction of electrons in a material of specially prepared nanoslits to determine the place of electron penetration through the slit [34, 35].

  15.  Why is the speed of movement (emission) of a photon is constant? The answer is simple: a photon has zero inertial mass and, because of this, its speed can be changed in no way and it has to be constant! Logically, there are only three options. This constant speed is infinite, but there are no infinities in nature; infinity only exists in the brains of mathematicians. This speed is equal to zero, but then the photon cannot leave its source; i.e., it simply does not exist There is only one constant left. Why is it equal to \(3 \times {{10}^{{10}}}\) cm/s? This is how our world works (anthropic principle); that is the most accurate answer so far.

  16.  Here you can recall the statements of Einstein, e.g., from his book The Evolution of Physics [64].

  17.  He wrote [15] that “in quantum mechanics, a statistical aggregate is analyzed from a certain process of interactions of quantum particles and a macroscopic body. On the basis of a series of such individual processes, the concept of a fictitious average process is established; it is the process that is meant in the uncertainty principle. Quantum mechanics is a theory of the properties of this average fictitious representative of quantum particles and only of the properties which appear during the interaction with macroscopic bodies.”

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  1. Dzhelepov Laboratory for Nuclear Problems, Joint Institute for Nuclear Research, 141980, Dubna, Moscow oblast, Russia

    V. A. Bednyakov

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Bednyakov, V.A. Does a Single Electron Have Wave Properties?. Phys. Part. Nuclei Lett. 18, 413–428 (2021). https://doi.org/10.1134/S1547477121040038

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