15 October 2002
Translated from Polish by Jerzy Bulik, Ontario - Canada.
Quantum mechanics and wave - corpuscular paradigm.
Quantum mechanics - the core of modern theory of physics, commonly recognized as the top achievement of human mind is based on the assumption that diffraction - interference pictures observed in some experiments with light or electrons cannot be explained by the concept of a particle having a determined position and, therefore, the wave equation of continuous media physics has to be taken as the base for a description of microscopic physical phenomena.
Such a statement is at the beginning of the Quantum Mechanics chapter in university textbook "Feynman Physics Lectures"; in the same way L. Landau and L. Lifshitz begin their Quantum Mechanics textbook. Practically, similar reasoning may be found in every text dealing with quantum mechanics. It means that departure from fundamental principles of classical physics has its origin in the puzzle of a wave -corpuscular duality and going further - in Huygen's wave theory of light. As a result, we have a theory according to which neither structure of the atom nor structure of a molecule nor structure of a solid body can be described in terms of deterministic principles of classical physics; the theory - according to which Heisenberg's inequality determines the area in which neither the cause - effect principle nor the conservation of energy principle may be valid or existing. However, the fact that there is no concept of a solution of the wave - corpuscular puzzle does not mean that this puzzle does not have a solution. Using this assumption as a starting point we will attempt to demonstrate falseness of the wave -corpuscular paradigm which is the base of present theory of micro - world. We will also demonstrate how it is possible to obtain interference - diffraction pictures based on principles of Newton dynamics.
Bragg's "wave" rule and its corpuscular alternative. . Among various arguments which are presented in favor of the "wave" nature of the light the Bragg's "wave" rule is situated on the first place. The rule, that has been deduced from scattering of Roentgen rays from a crystal lattice has the form:
n l = 2 d sin q B,            where   n = 1, 2, 3,....
It relates the quantity l which is a measure of hardness of X-rays, with a distance between atomic nuclei situated regularly in a crystal - in the case of simple cubic lattice the parameter d depends upon the lattice constant a in a following way:
d = a / (h 2+k 2+l 2) 1/2,
where h, k, l are integer numbers 0, 1, 2,....
Experiments that have been carried out with electrons and some time later with neutrons have shown that the Bragg's rule interpreted in a spirit of de Broglie "wave" relation l = h / mn, may be applied to light, as well as, to particles. Universality of the simple Bragg's formula implies that it has roots in a very simple physical process. Advocates of the wave theory claim that it is the interference - superposition of light, or electron, or neutron waves reflected from crystal planes. The problem is that speaking about the reflection they do say nothing concrete on the reflection mechanism from abstract planes suspended on dimensionless points distributed uniformly in space and they neglect entirely results of modern experiments, which show that wave pictures arise, point by point, from individual particles striking a screen.

To be not in a contradiction with undoubtedly corpuscular mechanism of creation of diffraction pictures, one must invoke to the old idea of the great Newton for whom the light was a swarm of fast moving vibrators - particles possessing its own internal oscillatory dynamics. But, it was necessary to wait three hundreds years until this oscillatory dynamics has shown its face. For the first time oscillatory nature of the photon has appeared in Planck's formula, while the photon itself was ultimately identified on the grounds of the photo effect by Einstein. If Newton would know the Planck's-Einstein energy relation for photon: E = hn = h / T and would know that the light propagates with speed c, he would write Bragg's formula in a following way:
n c T = 2 d sin q,
and he would search roots of this formula in the interaction of the photon with a pair of the atomic nuclei situated apart of each other on a distance d. Thus, like in the case of Kepler laws, which he derived introducing the gravitational force, he would try to find out the force hidden over the Bragg's rule.

Corpuscular roots of the Bragg's law. Trying to follow Newton, we can simply assume that the interaction between the photon and the nucleus has the following form:
where Q represents the charge of the atomic nucleus and the term in brackets oscillatory field of the photon.
Fig.1. Interaction of the particle accompanied by a periodically varying field with a charged center Q , when the particle moves with a high constant speed c. Vector d p represents the change of momentum of scattered particle, that is a global result of the interaction.
Taking into account that diffraction phenomena are, as a rule, observed at small scattering angles the analysis of the scattering dynamics can be carried out in the first approximation of the perturbation calculus. To determine the deviation angle for the photon passing by the atomic nucleus at a distance D it is sufficient to calculate the integral:
where d p is a normal component of the momentum changed due to the interaction with the nucleus and F is normal component of the scattering force F. If the scattering force has the form as given by Eq.(3) then integration can be easily carried out and we obtain:
d p = d p max sin f   ,
where f is the phase angle of the oscillatory field at the moment of the closest approach of the photon to the nucles, and the factor d pmax which depends on the impact parameter D is given by:
To relate our considerations to Bragg's rule, which in an open way depends upon the distance between the nuclei, let us consider scattering of the photon from two point charges Q a and Q b situated from each other on a distance d, fig.2.
Fig.2. Periodical localization of electrical charges in space (on the left) and the moving particle with periodically varying electric field of (in the middle). d sin q effective distance between charges of uniform chain as seen by the moving particle (on the right).
At small scattering angles, that is when the photon moves along almost straight line with a constant speed, the scattering angle is, in the first approximation, determined by a sum of changes of moments d p a i d p b arising, respectively, from the interaction with the charge Q a and with charge Q b. Let us assume now that the unperturbed trajectory of our pseudo-photon crosses a central point between the two charges. In such a case impact parameters with respect to the one charge, as well as, with respect to the other have identical value and we have:
d pab = d pmax [sin fa - sin fb]
where the phase angle f a corresponds to the moment of time when the photon passes by the first charge and the phase angle f b when it passes by the second charge. Let us denote now the phase angle at the moment of crossing the central point by f and the phase shift on a distance ½ d sin q by Df. The value of the phase shift Df depends on the oscillation period T and the interval of time needed for traversing the distance ½ d sin q. At a constant speed c we have:
Dt = ½ d sin q (1/c).
Df = 2 p Dt / T.
Taking into account that:
f a = f - Df     oraz     f b = f + Df
equation (7) may be written in a following way:
It follows from the above that the photon will pass between centers being not scattered, if the argument of the trigonometric function sin is a total multiple of p, then d p ab = 0. Thus, the photon will not see the scattering system when the following time relation will be satisfied:
Dt = n ½ T.
Taking into account relation (8) and denoting by L a distance traversed by the photon during one period of time (L = c T we obtain an equivalent of Bragg's formula:
d sin q = n L.
The obtained relation is a little bit different from Bragg's formula - on the left hand part of the above relation should stay the numerical factor 2 . Looking for the origin of this difference one must remember that a in a solid body there are positive as well as negative charges, the system as a whole is neutral. In a crystal lattice we have, in fact, to do with a chain of positively and negatively charged centers - static in the case of ionic bond, like in the case of NaCl, and dynamic in the case of a covalent bond, due to the binding electron moving from the one nucleus to the other ( about this we will talk in our of our lectures at the end of the course). Thus. electrical lattice constant is two times greater then a lattice constant of a solid body theory.
Thus, concluding our considerations we may say that the Bragg's formula represents time-spatial resonance between time periodic field of the photon and periodic distribution of electric charges in space as seen by the moving photon.
Diffraction. Angular modulation of scattered photons. Unearthing of the mechanism which is hidden behind Bragg's rule is the key for corpuscular description of diffraction and for reconstruction of periodically modulated structure of a picture of a wide beam of scattered photons. So, using equation (8) and equation (11) and taking into account that scattering angle, in the case of small angles, is approximately given by:
tgJ ~ d p / p0   ,
we obtain the key formula of the corpuscular diffraction theory:
J (f, q) = Jmax • cosf • sin (p d sinq / L )   ,
Jmax = 2 d pmax / p0   .
Using formula (14) we are able to determine not only locations of diffraction minima and maxima but also to calculate intensity of scattered photons resulting from statistical spread of the phase angle f. In order to calculate the intensity distribution for the case shown in fig. 3 one has to calculate the integral:
where d is the Dirac function whose argument is defined by equation (14).
Fig.3. Corpuscular mechanism of creation of diffraction pictures.
Electrical structure of an edge. Young experiment. Electrical structure of an edge. Young experiment. In wave theory, an edge is a line on paper without any physical meaning, a line determining only a formal location of an edge, a line which is required for mathematical operations. However, as it results from presented above analysis, the phenomenon of light refraction (bending) at an edge has to have its origin in electric charges present at the boundary between two media. So,
the starting point of any analysis regarding diffraction of light has to be the physical definition of a concept of an edge and of a surface.
Nowadays, when we know that solid state is a collection of positive and negative electric charges it seems to be obvious. Surface of a solid state and especially its edge is the area, where homogeneity of positive and negative electric charges is damaged. Taking the distribution of space charges in double layer as a base, known from plasma experiments, we may create supposed electrical picture of a slot. Thus, at the border of the slot there is a narrow band of negative charges and at a certain distance from the border there is wider band of positive charges - see fig. 4.
Fig.4. Qualitative picture of electric charge distribution at a boundary of solid state surface. Quantitative picture will depend obviously on the material used in the experiment, on diaphragm thickness and on slot width. Obviously when the distance between the edges is decreasing then layering will be diminishing and in the limit it will approach zero - in homogeneous material there will be obviously homogeneous distribution of electric charges.
Taking into consideration that photon's electric field decreases when distance increases, we see that photons moving towards a slot will be affected first by the electric field originating from the narrow band of charges located near the edge. If the slot is sufficiently narrow, then the scattering force may achieve high values; then only the photons that arrived in the proximity of the slot with electric field being close to zero - with phase angle close to zero, will have a chance to cross the slot. Thus the narrow slot will function as a phase selector.
Fig.5. Slot as phase identifier. Only the photons whose electric field was close to zero during their passage through the slot will cross the slot without deflection.
These photons which penetrated through the strong scattering field of charges located at the border of the edge and are moving on its other side will be affected by a further located layer of positive charges and will be deflected, some of them more and some of them less by them.
Fig.6. Geometry of Young's experiment. Photon which is scattered in electric field of negative charges located at the border of one slot changes its direction of movement first under the influence of a band of positive charges located closer to this slot and then under the influence of electric charges of the next slot.
This influence may be easily calculated using formula (5). For a narrow stream of photons moving along the axis of a slot there will be:
where the two terms in the curly bracket describe interaction of two bands of positive charges located at the distance D+, where Dt is the time interval required to travel between the point on the photon path which is a minimum distance away from the band of charges and the axis of the slot:
Dt = D+ sinq / c   .
Taking into account that equation (16) maybe converted to a form:
after integration one obtains an expression determining angle modulation of a stream of photons leaving the slot:
In this case, the scale of angle modulation is determined by the ratio D+ / L. The obtained picture is in accordance with the picture which is observed in the experiment, see fig.7.
Fig.7. Wide dark bands which are seen in this picture originate from wide bands of positive charges located in the proximity of each slot. Fine modulation originates from charges associated with an adjacent slot.
In the case of two slots, as it has taken place in famous Young's experiment, modulation caused by electric charges located significantly further from the second slot will be imposed on the modulated picture of a single slot. Then the scale of angle modulation is determined by the ratio d/ L, where d determines distance between slots.

And yet: Newton was right. Light is particles. The above presented discussion obviously cannot provide answers for many questions appearing from a multitude of diffraction results observed in various experiments. Similarly, they do not say anything about phenomena which appear under the name of interference, as well as they do not say anything about dynamics of reflection of photons from a surface. However, they show a solution for a troubling situation that was created by the puzzle of wave - corpuscular duality. Periodic structure of diffraction pictures has its origin in time - space resonance of oscillating photon's field and periodic structure of scattering object. Interested readers may find several other arguments confirming particle theory of light in my book, published in the last year "Sprawa atomu". Readers who want to get some knowledge about the electron wave field which de Broglie worked on and about quantum effects associated with it and which will be the topic of the next lecture are advised to read my paper "Spin dynamical theory of the wave corpuscular duality" which was published in 1987, in the International Journal of Theoretical Physics, Vol. 26, page 11.