V.6 No 1


On the wave-particle duality

To show it, let us go on considering Born’s model. We stopped at the point that after Born’s idea, the atomic field only scatters the wave but does not change its phase. “In this case the wave scattered by the atom in direction OR under an angle gteta.gif (842 bytes)  to the initial direction (Fig. 10) is expressed as


where f (sin gteta.gif (842 bytes) / glumbda.gif (841 bytes))  is called the atomic amplitude, or scattering ability of atoms for electrons” [7, p. 617].


Fig10.gif (3234 bytes)

Fig. 10. The scheme of electron’s interaction with a separate atom (after Born). The exponential term (top, left) factually determines the longitudinal electron wave running on the atom and the stationary appearance of the wave function of atom


In its turn, as we know, the atomic amplitude directly depends on the form-factor F (teta.gif (842 bytes))  that “describes the screening of the nucleus charge by the electron cloud” [2, p. 225].

Or rather, “in case of scattering of quite fast electrons, we yield the following expression for the number of particles deflected by the angle gteta.gif (842 bytes)  in a unit of spatial angle … called differential cross-section:




this is reduced to the Rutherford formula …, if neglecting the term F (gteta.gif (842 bytes))” [2, p. 225].

 “Observation F (gteta.gif (842 bytes)) for all deflection angles gteta.gif (842 bytes) makes possible to determine gslash.gif (825 bytes)gpsi.gif (845 bytes)(r)gslash.gif (825 bytes)2; for it we have to solve … the integral equation


Although mathematically it is simple – we have only to invert a usual Fourier integral, – to do so practically is an extremely complicated task. The matter is, we have to study not separate atoms but assemblages of their giant quantities in gases, liquids or solids. The structure of the medium causes interference patterns superimposed on the patterns related to the form-factor. In gases and liquids whose atoms are chaotic we yield the rings of decreasing intensity around the falling ray. The appearance of rings depends mainly on the average distance between atoms. In solid crystals we yield the well known Laue – Bragg interference patterns. The form-factor modifies the intensity distribution in these effects, but not only it affects. There are mere geometrical factors – such as the width and distance from the used splits, temperature effect related to the thermal motion of atoms, crystal non-ideality etc. So it is not so easy to separate the form-factor effect from the affection of all other causes that produce the observed intensity” [2, p. 225].

This citation from Born shows that after his own opinion, the form-factor, although it takes some part in the electron diffraction, is not definitive, though in the Born’s approximation of interaction of electron with a separate atom the structure of continuum is disregarded, consideration is reduced to the interaction of electron de Broglie wave with the potential of atom. Born [2, p. 226] and his interpreters [6, lecture 26], [8, chapter XVII, section 137], [9, part I, section 14] etc. show the same approach to this approximation. “In collision with atom, we have to account also the wave functions describing the interior structure of atom. In elastic scattering the state of atom does not change. So Up'p   (the matrix element of interaction energy with respect to the wave functions of free particle before and after collision – Authors) has to be determined as the matrix element with respect to wave functions gpsi.gif (845 bytes)p and gpsi.gif (845 bytes)p' of electron being diagonal as to the wave function of atom. In other words, we have to think U… as the potential energy of electron – atom interaction averaged in the wave functions of this last” [8, p. 613–614]. With it, in the beginning of modelling the authors admit the spectrum of free electron non-discrete, which determines the Born interpretation of the de Broglie wave: “The energetic spectrum of free motion is continuous, so we deal with a special case of excitation theory in the continuous spectrum. Conveniently yield the solution here immediately, without general formulas.

The equation for the correction gpsi.gif (845 bytes) (1) of the first approximation to the wave function is


(U is the potential energy). As we know from electrodynamics, the solution of this equation can be written in the form of ‘lagging potentials’, i.e. as


[8, p. 189].

Note that in (12), in full accordance with the statement of problem, the potential function that includes the electron cloud of atom screening the charge of nucleus is stationary. Actually, “Denoting the density of charge distribution in the atom through gro.gif (843 bytes) (r) , we have for the potential gfi.gif (841 bytes)  the Poisson equation


The sought matrix element Up'p is, mainly (! – Authors), the Fourier component of U (i.e. of gfi.gif (841 bytes)) corresponding to the wave vector q = k' - k … The density of charges gro.gif (843 bytes) (r)  is composed of electron charges and charge of nucleus


where en (r)  is the density of electron charge in the atom. Multiplying by eatpsi.gif (878 bytes)  and integrating, yield


Thus, yield for the integral of our interest the expression 


where the value F (q) is determined by the formula


and is called the atomic form-factor. It is the function of the scattering angle and of the speed of falling electron” [8, p. 614].

With it, “Note that same problem can be solved also after the stationary theory of excitations, as the potential energy of interaction does not depend on time” [9, p. 204].

Thus, we clearly see that the conventional quantum-mechanical solution is based on the reason that the electron, being on one hand a particle, has the statistical de Broglie distribution before the interaction with an atom. After the excitation theory, the first and next approximation is known to be well less than the non-excited state – the distribution function of non-excited electron. The authors omit it and do not substantiate the physical causation of interaction of the wave function of electron with the stationary field of atom and the very essence, how the de Broglie wave is formed with the specific vector k = kn, while in words they say of continuous spectrum of this statistical wave function. In the last case, should the initial state contain a continuous spectrum, the difference [8, p. 555]


would be also a continuous, not discrete spectrum, and the whole problem of merely formal substantiation of the discrete function as a consequence of interaction with an atom would be invalid. Thereupon there will appear not the diffraction pattern but some conventional version of Rutherford’s scattering, which caused so many essential reservations of Born.

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