SELF

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S.B. Karavashkin and O.N. Karavashkina

2. Diffraction of electrons on a single atom in the quantum-mechanical description

“A simplest problem is to consider the electron scattering by single atoms. A beam of moving electrons is described by a plane monochromatic wave e^ikz propagating along the axis z with unity amplitude and wave vector k, and the module

Atom is considered as a spherically symmetrical region with the (non-zero) potential   whose value consists of the potential of nucleus and potential of electron shell. An essential condition to yield the most simple and widespread solution is the Born approximation according to which the atomic field only scatters the wave e^ikz  but does not change its phase” [7, p. 616–617].

Here we have to make a pause in our consideration of conventional approach to resolving this problem, in order to draw attention that the Born approximation is based on the possibility for the potential field of atom to interact with the de Broglie probability wave. Supposedly, electron, neutron or molecule have such property before the interaction with atom. But it means that such probability wave has revealed itself even in absence of any scattering centre. Taking, for example, two beams of monochromatic electrons and directing them onto a screen, we could expect an interference pattern like shown in Fig. 8 for surface waves.

Fig. 8. Interference of the surface mechanical waves from two pointed sources, http://physics.nad.ru/Physics/Cyrillic/2wa_tmp.htm

But in case of electrons such pattern is basically impossible. For it, the distribution of electrons radiated by cathode or electron gun has to have the appearance shown in Fig. 9, which is in full accordance with the Born representation of physical interpretation of the de Broglie wave.

Fig. 9. Visualisation of the de Broglie wave after Born

Actually, as we could make sure in the introduction, Born has introduced namely the statistic substantiation of the de Broglie wave. Again, “if, for example, at some point of space the probability wave has zero amplitude, this will mean, the probability to reveal an electron at this point is infinitesimal” [2, p. 115]. Consequently, the electrons before meeting the barrier have to be already redistributed according to (2), i.e. to produce the longitudinal wave whose length has to be comparable (in case of diffraction on a crystal) with the inter-atomic distance, within the range of X-rays.

On the other hand, if in accordance with the quantum-wave interpretation electrons are not distributed so, Born’s physical substantiation loses its validity, and we have to think an electron as a wave packet, – which, as we also saw in the introduction, adherents of quantum conception think unsatisfactory. And not in vain. Besides the known discrepancies related to blurring of wave packet in time, such interpretation cannot substantiate the quantum size of electron with little k and the potential; pattern of the electron charge with its wave representation as a chain of waves. If the wave packet has a broad spectrum, after interaction with an atom we factually have to see the electron’s disappearance together with disappearance of the wave packet due to the resonance interaction. As we will see below, this inadmissible duality is permanently present in all problems of quantum theories.

But the main is, even if we do not deepen into the Born’s interpretation of the de Broglie wave, it is basically impossible to yield the diffraction pattern on the atom on some average potential.