On the problem of the physical sense of the wave equation’S icon

On the problem of the physical sense of the wave equation’S

НазваниеOn the problem of the physical sense of the wave equation’S
Дата конвертации29.07.2012
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N. Chavarga

Uzhgorod National University, 46, Pidhirna Str., 88 000, Uzhgorod, Ukraine

E-mail: chavarga@mail.uzhgorod.ua

The energetic interpretation of the -function, which is based upon the assumption that all elementary particles are soliton formations of an electromagnetic field, is proposed. The norming condition for the energetic interpretation of -function is proposed.

With only “probably” a theory cannot be made up. Even if it is correct empirically and logically, in its own depth it is false. A.Einstein

1. Introduction

Nowadays the problem of the physical sense of the wave equation’s -function is officially considered as solved in favor of the probabilistic interpretation proposed by M.Born and approved by Bohr, Heisenberg, Pauli, Dirac. It is known, however, that such scientists as Einstein, Lorentz, Planck, de Broglie, Feynman, and even the author of the wave equation Schrödinger, did not accept this viewpoint. In literature, many very effective expressions of outstanding scientists against the probabilistic interpretation can be found. Schrödinger’s expression concerning the probabilistic interpretation we consider the most brilliant (it took place in his old age – in 1957): “I want to say clearly that from now onwards I pledge myself to all the responsibility for my self-will. I drift upstream, but the direction of the flow will be changed”, [1]. Einstein has considerably more similar expressions. We give one of them made in his letter to Schrödinger. “Heisenberg-Born’s soothing philosophy – or religion? – is made up so subtly that provides a soft pillow for a believer till some time; and it is not so easy to frighten him from this pillow. Let him sleep”, [2]. We hope that these words of one of the creators of quantum theory will at least to some extent prevent the reader from prejudiced attitude towards the problem discussed here.

The adherents of quantum theory explain Einstein’s stubborn non-accepting of the probabilistic interpretation by the insufficient level of his abilities. M.Born, for example, frankly vented in his book: “Even today (1965) I consider Einstein’s meditations to rest on insufficient understanding of quantum mechanics”, [3]. The simplest conclusion that can be made from this information is the following – either the probabilistic interpretation is incorrect, or even a number of classics were unable to understand it. Nowadays ordinary tutors of physics and even students are obliged to have these abilities.

The interpretation of the sense of -function, proposed for reader’s attention, is based upon the notion of soliton.
It is known that solitons have not only wave properties, but also properties of corpuscles – in collisions they behave as elastic balls, do not lose their motion energy (inertia law is carried out), do not give the energy, concentrated in them, to their environment (they do not stop existing without contacting with anti-solitons), they do not flow apart (in contrast with wave packets), etc., [4]. The image of elementary particles as electromagnetic solitons allowed us to suggest a simple and non-contradictory relativity theory, [5]. Evidently, this model is not agreed with the image of point particles; nowadays, practically all elementary particles are considered point (or close to point ones). It is also evident that, for soliton formations, a wave function must describe real oscillation processes, which take place in these objects. It means that in the given case, for a wave function, it is necessary to search for a new interpretation, not connected with the probability of finding out a point particle at one or another place, in one or another situation.

^ 2. Energetic Interpretation of the Physical Sense of -Function

For solving the problem of making up physical theories explaining some experimental facts, usually a physical idea, which explains this phenomenon qualitatively, in a simplified way, is first advanced. Then the idea should be tested concerning quantitative agreement with the experiment, if one succeeds in correct representing it in the notions of mathematics. In this procedure, all quantities belonging to the equations obtain their explanation at the stage of making up the equations, therefore, problems of the interpretation of their physical sense do not appear.

In some cases, however, due to the absence of another possibility, one does in the opposite way. First, after a number of mathematical operations, one obtains the result, then the question of the physical sense of the quantities, belonging to the equations, is solved. So it was in the case with Maxwell’s equations, Lorentz’s transformations, uncertainty relation, so it was with Schrödinger’s equation.

It is known that Schrödinger’s equation is not strictly derived, but it can be obtained from the so-called classical wave equation, which describes any wave motion – from string or pendulum oscillations up to a sound or electromagnetic wave.


The physical sense of the quantity depends on the process, which we are going to describe with the help of this equation. It can be the value of linear deviation from the middle position (an oscillator, a string), angular deviation (a pendulum), the charge value on a capacitor plate (in electric circuits), the degree of medium deformation, the intensity of electric or magnetic field (if it is an electromagnetic wave), etc.

If we want to investigate, how this equation describes the behavior of the electron, we can make use of that fact, that the electron has wave properties, which are described by de Broglie’s relations. Assume that the quantity in equation (2.1) represents de Broglie’s harmonic wave in the space with a constant potential and depends on time in the following way [6]:


where  is the frequency of oscillations connected, according to de Broglie, with the complete energy of the particle by the following expression:


Through differentiating from (2.2) one can obtain:



For an electromagnetic wave, V=C , therefore, one can write down as follows, instead of (2.1):


where 2 is Laplace’s operator.

Then: (2.7)

where Т is the oscillation period,  is the wavelength. Now we can rewrite (2.6) in the following way:


According to conservation law, the complete energy of the electron is equal to the sum of the kinetic energy T and potential energy U, which is determined by the potential of the field. Since T = p2/2m , where p is the momentum of the particle, we have:


thus: (2.10)

In accordance with de Broglie’s hypothesis, the electron is characterized not only by its momentum, but also by its wavelength, and the photon is characterized not only by its wavelength, but also by its momentum:


As a matter of fact, just (2.11) contains the information that the investigated objects have properties of solitons. Taking (2.10) into account, we have:


Now, instead of (2.8), one can write down:


thus: (2.14)

Equation (2.14) is that form of writing down Schrödinger’s equation (not containing time), which occurs in literature most frequently.

Taking into consideration that , one can represent the quantity Е in the following way:


Taking (2.15) into account, one can represent equation (2.14) as follows:


Equation (2.16) is the second form of writing wave Schrödinger’s equation (taking into consideration the dependence of the -function on time). It would be all right, if only when using expression (2.2), we did not forget to make it more exact, what physical sense the quantity has.

One of the most significant steps, which is made when investigating this equation, consists in the assumption that equations (2.14) or (2.16) are also true in case, when potential energy U depends on the distance, i.e. the potential of the force field has spatial distribution. If instead of U(x,y,z) one substitutes the expression for the potential energy of the electron in the proton field into (2.14), then the solution of the equation can be obtained only for discrete values of energy En . It is known that these values of energies coincide with the positions of the experimentally observed energetic levels of a hydrogen atom with high exactness. Just the experimental confirmation of the obtained results solves the questions concerning the validity of the made assumptions.

As for the physical sense of the -function, the officially accepted today’s viewpoint does not need any special commentaries, however, let us analyze this question. In literature, it is very often represented with the help of the following example. If one assumes that the curve of the potential of the force field acting upon the electron has the form = kx2, then the curve of the potential energy of a point electron has the form U = ekx2 . If one substitutes this expression into (2.14), then, as well as in case of a hydrogen atom, the solution can be obtained only for a discrete series of energy values – the so-called own values. It is one of the few cases, when Schrödinger’s equation can be solved analytically:


where n= 0,1,2,…

For each value of n , there is the appropriate form of -function, the so-called own functions. In Fig. 1, the graph of the dependence of the potential energy U(x) for a point electron in a force field = kx2 , and also the graph |(x)|2 for n=2 are shown. According to the probabilistic interpretation, the height of the ordinate in the graph corresponds to the probability to find the particle in space with coordinate x . More concrete information about the particle we as if cannot obtain in principle. As seen from the graph, the highest probability of finding the particle is observed near the turning points – points A and B in the potential energy curve. The same curve represents the interaction of the point electron with force field (х). The scales of the axes and U are selected in such a way that the curves coincide. In contrast to the classical physics, a quantum particle, in accordance with probabilistic interpretation, can be also found with relatively high probability under interaction curves – at points with negative energy of motion (for instance, points C and D). The explanation of the cause of such a situation both in this case and in other analogous ones, is proposed the same – “specifically quantum phenomenon”.

Fig. 1. De Broglie’s wave in the force field U =kx2.

Let us draw reader’s attention, that Fig.1 is made up on the basis of the assumption that the energy conservation law is carried out, and the rectilinear appearance of the segment A–B affirms it. If A–B were not a straight line segment, it would mean that energy conservation law is not carried out. This law is introduced into wave equation through expression (2.9). On the other hand, according to probabilistic interpretation, it follows from the graph of the curve |(x)|2, that in the micro-universe, energy conservation law is not carried out – the object can be located under the potential energy curve, points C and D . What is the sense of using the curve, made up on the basis of conservation law, for analyzing a situation, if we affirm at the same time that it is not carried out?

Now let us try to suggest the interpretation of the physical sense of -function without going beyond the limits of the classical physics. The basis of equation (2.1) is formed by Newton’s equation of motion, which we always write down for a separate object, but equation (2.1) already describes not the motion of a separate object, but the wave process, the process of propagation of some disturbance. If we investigate an electromagnetic wave, for instance, a radio wave, then it is rational to bind the physical sense of -function with the intensity of electric or magnetic field:


Note that equation (2.6) is obtained by introduction of dependence (2.2) into (2.1) (dependence (2.2) represents de Broglie’s wave). De Broglie’s correlations (2.2), (2.3) and (2.11) are equally true both for photons and for “hard” particles. On the one hand, it means that it is rational to attribute the sense of intensity E of electric field to -function in equation (2.6), since de Broglie’s wave can represent a photon, and we have no doubts about its electromagnetic nature. Between the electromagnetic wave of radio range and X-ray quantum there is no principal difference – finally, both waves can be emitted (with small difference in time) by the same atom. On the other hand, the same de Broglie’s wave can also represent a “hard” particle, in particular, an electron, therefore, it is logical to understand -function as the intensity of electric field also in equations (2.14) and (2.16). Why should the sense of -function cardinally change in case of relatively insignificant transition – transition from a photon to an electron? As an additional argument one can note that an electron can be formed in pair with a positron in collision of two photons with energies of not less than 0.51 MeV (the process of the birth of pairs), and vice versa – an electron and a positron in their interaction turn into a pair of photons with energies 0.51 MeV each (annihilation). Moreover, all “hard” particles in interaction with their antiparticles can finally turn into photons. In other words, the deep essence of photons and “hard” particles is common.

The simplest conclusion that can be made from the said above is the following: all “hard” particles are wave formations limited in space and able to be at rest, to move without any loss of motion energy (inertia law is carried out), and also without any loss of energy concentrated in them. Wave formations with such properties are called solitons. Thus, in order to come back to the limits of the classical physics, it is enough to assume that photons and all “hard” particles are soliton formations of electromagnetic field.

As regards going of graph |(x)|2 beyond the limits of graph U(x), there is a very simple answer here. Curve U(x) in Fig.1 represents the dependence of the potential energy of a point electron in a force field (х), and curve |(x)|2, according to energetic interpretation, represents the distribution of the energy concentrated in a wave formation called electron. In accordance with energetic interpretation, the electron does not move from point A to point B and back – it is simply a standing wave squeezed in the force field. The cause of the discreteness of the energy value, which the electron in the potential well can perceive or return, consists in the fact that a wave object can change its state only through changing the quantity of its assemblies. If the energy of the exciting particle, for instance, of a photon running into an electron, is insufficient for increasing the quantity of assemblies in a standing wave, at least, in a unit, then superposition principle is carried out, and the system “electron + force field (х)”, i.e. the atom, turns to be transparent for the photon.

On the contrary, if the energy of the photon is sufficient for increasing the quantity of assemblies in a unit, an inelastic interaction takes place. In this case, the photon appears to be absorbed, and the electron (and the whole atom together with it) turns into a qualitatively new state. According to Fig.1, it simply increases its dimensions, and in a number of cases, it may be in this state quite long – thus, we deal with metastable excited states of atoms.

In collision physics, it is assumed as automatically understood, that the minimum section of the interaction is determined by the geometrical dimensions of the colliding object. On the basis of this assumption and the appropriate experimental data, one makes a conclusion that an electron is almost a point object, ~10-13 m. This conclusion, which is true for hard corpuscles (if they really existed), is not obligatory for wave objects – for example, a window glass with dimensions of 22 meters for photons practically does not exist, however, we do not make from it a conclusion that the geometrical dimensions of the glass are ~10-13 m.

The fact stated here concerning probabilistic interpretation can be illustrated by using the example from the macro-universe. In railway practice, the following cases are known: when a train has gone down a wide sorting hill, it overcomes the higher hill, if it does not match the second hill plateau completely, Fig.2. At first sight, the situation looks like the emission of -particles from a nucleus, when the energy of a point object is insufficient for overcoming the barrier, however, this object gets behind the barrier (the -decay of nuclei).

Fig.2. Overcoming the sorting hill by a long train.

For a long train, the height of the second hill on the interaction curve (curve 2) is lower than the height, which the train goes down, h3 is the geometrical height climbed by the mass center of the train.

In order to obtain the interaction curve 2, the train should not necessarily pass the hills, as it is depicted in Fig. 2. We can assume that the train moves at a straight line, but the distribution of the potential of the force field, acting upon the cars along the axis x, has such a form that the potential curve provided the appropriate selection of the division value of the axis y coincides with the geometrical shape of the hills. For both cases, the potential energy curve will be the same, however, in the second case, the analogy with the potential energy curve in Fig. 1 will be more understandable. For a point object (also for a separate car with a good approximation), the potential energy curve coincides with the shape of the potential distribution curve, i.e. with the hills shape. However, for a body system (for a train), the situation is significantly different. Both the mathematical expressions and the diagrams corresponding to them (in our case, the potential energy curve), we make up only for the objects centers. It is seen from Fig.2 that if a separate car begins to go down the hill at the distance x2 , then the train center is located at point x1 , but the train as a whole system also begins to go down, but at the energetic curve. It means that for a long train, the potential energy curve should be made up from point x1 , curve 2. As we see, the interaction curve for a long train essentially differs from the interaction curve of a separate car.

If the train does not match the second hill plateau completely, then the energetic height mgh3 of the second hill will decrease for this train and may even turn to be less that the energetic height of the first hill, curve 2 in Fig. 2. It means that the train may overcome a geometrically higher hill than that one, which it has gone down. There is no paradox here at all. The energetic situation must be analyzed by using energetic curve, but the latter must be made up correctly. Just the misunderstanding of the fact, that the curve of the potential can significantly differ in shape from the potential energy curve in case, when an object cannot be considered as a point one (both curves have their common name – potential curve), is the cause of many deceptions.

Fig.3. Oscillation motion of the train in a symmetrical potential well.

The train center, located on the oscillation level ^ AB , is not transmitted to the left from point A and to the right from point B , but the wave process is spread at axis x to the left from point хА and to the right from point хВ .

Now let us analyze the behavior of the train in the symmetrical potential well. Assume that at the moment, when the train was located (and rested) at point A on the potential energy curve, the brakes were released. If we neglect friction of any kind, then the train center will climb to point B of the opposite potential wall, then it will come back to point A etc., to infinity, i.e. we have an oscillation process of some kind. Now if we answer the question, how in this process, for instance, the quantity of weight (or mass) of the train at x axis changes, or how the vertical constituent of the cars’ velocity changes, and represent it graphically, then we will obtain some vivid depicting of this process in a wave form, curve 2 at Fig.3 (in this figure, the dependence of the square of the velocity’s vertical constituent is represented, so that the analogy will be as full as possible). As seen from the figure, the last car in the extreme left position may move to point x1 , and the first one in the extreme right position – to point x4 , but in this case, can we speak about the penetration of the object (in our case – the train) into the range of the negative values of motion energy? Note that curve 1 in Fig. 3 is made up with taking into consideration, that the investigated object is not a point one, and, in spite of it, curve 2 goes beyond the limits of curve 1 all the same, however, it could be also applied on the point object interaction curve (curve 3), as it is made in literature, Fig. 1. In this case, the train would “tunnel” still deeper under the potential curve.

There is a high analogy degree between Fig 1 and 3b. In both figures, the potential energy curves are depicted, and the graphs of the oscillation processes of the objects located at the definite energetic levels, are applied on these curves. Obviously, if at the point with coordinate хА one releases not the whole train, but a separate car, which we will consider as a point, then the oscillation process curve will not be spread to the left from хА and to the right from хВ , and we will obtain a picture close to that one, which we expected (but did not obtain) for an electron. From that fact that the oscillation process graph in Fig. 3b goes beyond the limits of the potential curve we do not make any far-going conclusion at all, since we made up this graph ourselves, and we understand that it concerns not a point object, not a separate car, but a system of bodies, i.e. the whole train. Analyzing curve 2 in Fig. 3, we can only say that the oscillation process is not spread to the left from х1 and to the right from х4 , but in this case, the center of the train changes its position only between xA and xB .

Now we will use the obtained result for analyzing Fig. 1. Into Schrödinger’s equation we have put the information about the form of the potential curve for a point object (mathematical expression for curve U(x)), and also the information, that the object is some wave with properties of a soliton – through de Broglie’s relations (2.11). After solving the equation, we obtained the possibility to make up graph |(x)|2 , which appeared to be analogous to graph 2 in Fig. 3 (the analogy consists in the fact, that a part of the graph |(x)|2 also goes under the potential energy curve – to points x1 and x2 , Fig. 1). A logical and the simplest conclusion, which can be made in this situation, consists in the fact, that in reality, an electron is not a point object, and curve |(x)|2 in Fig. 1 illustrates some oscillation process occurring in an electron located in a force field. In other words, we deal not with a separate “car”, but with a more or less “long train”. In this case, the electron dimensions cannot be point already. In any case, Schrödinger’s equation “understands” the situation just in such a way.

If an electron is considered as a soliton electromagnetic formation (though a specific one, but a wave formation all the same, and some oscillation process with a definite frequency an wavelength can be collated to it), then in the obtained result, there is nothing unusual – all meditations are within the limits of wave mechanics as a part of classical physics. Schrödinger imagined the situation approximately as follows: “…important theoretical and experimental results lead us to the following conclusion: only waves exist in general. Both the light and all else considered earlier to be the particles are, in fact, the waves. Thus, no particles exist in general; and the matter which earlier was thought of as consisting of the particles, now should be considered as made of systems of waves. This would promote sufficiently the achievement of the unity of our universe’s pattern", [2, p. 17].

Let us draw reader’s attention that curve 1 in Fig. 2, in contrast to curve 3 in Fig. 3, has no bottom – the left potential wall turns into the right one at once. It is explained by the fact that in the given example, the train length coincides with the bottom dimensions on curve 1. The train begins to climb the right hill at the same moment, when it finishes going down the left hill. If the train were a bit shorter, then a bottom would occur on the curve. On the contrary, if the train were still longer, then the depth of the potential well would decrease.

3. Norming Condition

It is known that in a standing electromagnetic wave, the maxima of the electric component and the magnetic one are shifted with respect to one another on ¼ of the wavelength in space and on ¼ of oscillation period in time. In this case, for a wave as a whole, a half of the energy is contained in the electric component, and a half – in the magnetic one. The volume density u of the energy, contained in the electric component, can be represented as [7]:


If corresponds to intensity Е , the quantity Е2 can be calculated as the product of and * , when =1 (* is complexly conjugated with ):


Multiplying this quantity by the elementary volume dV, we obtain the quantity of energy dW, concentrated in the volume dV :


The integral from this quantity in the whole volume will determine the energy contained in the electric component of the particle, i.e. a half of mC2 :


or: (3.5)

As seen, in spite of the fact that, from the physical viewpoint, the energetic interpretation and probabilistic one differ more than simply essentially, from the mathematical viewpoint, the suggested condition of norming differs from the known one only by the coefficient before the integral, i.e. by the division value of the scale on the axis of quantity . It means that all calculations made for probabilistic interpretation are, in fact, valid for the energetic one, too. The successes of quantum theory are explained just by this fact. The graphical depiction of the dependence (х)2 in different interpretations differs only by the division value of the ordinate.

4. Dependence of the Form of Wave Function on the Form of Potential Well

Now let us analyze the question of the dependence of the wave function’s form on the form of the potential well, in which the wave object is located. The information about the form of the potential well is put into the wave equation through the expression for a particle’s potential energy U . If instead of U one substitutes into the wave equation the expression B/r , which accounts the action of the electric field of a nucleus, in particular, of a proton or positron, upon an electron, then the solution of the wave equation can be obtained analytically, but only for definite, discretely taken values of Еn coinciding exactly enough with the values of energetic levels in a hydrogen atom. Note that in this case, the potential well is not of full value – only the right potential wall is available, and the left wall and the bottom of the potential well are absent, Fig. 4, a. Ordinate U is only the mathematical abstraction and cannot carry any force load, i.e. it cannot be considered as the left potential wall, and the depicting of energetic levels cannot be finished on it, as it is made in literature.

Fig.4. The potential energy of the electron in the field of the attracting center:

a) only attraction force acts upon the electron, the left wall and the bottom are absent, b) both attraction and repulsion forces act upon the electron simultaneously.

Still better coincidence of the theory with the experiment is observed for a hydrogen atom in case, when a wave object (an electron) is put into the potential well of the following form:


In Fig. 4, b, the graphical depicting of this dependence is represented. As seen, in this case, the potential well appears to be of full value, and for n=2 ,the solution of the wave equation can be obtained analytically. If n=2, the quantity A /rn can be stated as the result of the action of centrifugal forces. The quantity of energetic levels of the electron in the potential well and their position on the energetic scale depend on the form and depth of the potential well.

The question concerning the dependence of the form of wave function on the steepness of potential walls was probably not investigated in a goal-directed way in literature. The imagination about it can be obtained in analyzing the results of investigating the form of a wave function of an electron in a potential well like (4.1) or in the potential well U=kx2/2 , describing the change of a body’s potential energy in a linear harmonic oscillator. In the latter case, both walls of the potential well go to infinity symmetrically. It is impossible to put the electron into this well from outside. Theoretically it should be born there and stay forever. The energetic levels in such a well are located equidistantly. The dependence of the form of -function (more exactly, the square of its module) is shown in Fig. 5, [8, 9]. As the number of the level increases, the steepness of the walls of the potential well also increases. It is seen from the figures that in this case, increases the relative height of the peaks, corresponding the “probability of finding an electron near the turning points”, i.e. the points of crossing the energetic levels with the potential energy curve. If for n=2 all peaks have approximately equal height, then for n=10 the extreme peaks are already almost twice higher than the central one. It is quite logical to suppose that this tendency will be also carried out for large values of n , i.e. when potential walls are practically vertical already.

Fig.5. Distribution n(x)2 for n = 0,1,2,3 and 10, [8, p.533]. The coordinates of the turning points, which are located at the places of crossing the interaction curve U(x) with the position of the corresponding energetic level, are depicted by dashed vertical lines.

The situation depicted in Fig.5 is not, certainly, a pure illustration of the dependence of the wave function form on the steepness of the potential walls, since in this case, the distance between the turning points also changes. Let us emphasize especially, that everything that we had to do is to substitute the expression U = kx2 /2 instead of U into Schrödinger’s equation, and then the equation “knows itself” already, in what values of x for a given level the turning points are located, what form the -function for a given level has, till what limits it is spread, etc. In literature, this result is disregarded for some reason, when solving the question, what form the -function will have, if a particle is put into a potential well with vertical walls of infinite height.

Analytically this problem cannot be solved in principle, since analytically it is impossible to select such a form of a potential well, whose wall would be vertical. At first sight, it seems that it is possible to investigate a wave object’s behavior in the potential well with linear oblique walls, where the equations of sidewalls have the form U=kx , and then to investigate the behavior of the -function for k→ . Unfortunately, a potential well of such a form cannot be represented analytically, either, i.e. the problem cannot be solved in such a way, either.

In literature, this problem is considered solved, and it is “solved” in the following way, [8]. It is assumed that the potential energy of a particle in some force field corresponds to the following condition:


denoting as  , Schrödinger’s equation (2.14) can be represented in the following form:


Within the limits of the potential well, U = 0 , therefore, the relation  has a concrete, finite value


The solution of the problem of finding the form of -function is now reduced to integrating this equation at the initial conditions (0) = 0 , (l) = 0 . The own functions of the equation (4.4) have the following form:


In Fig. 6, the graphs of the own functions (4.5) and the squares of their modules for the levels with numbers n =1, 2, 3 and 4 are represented. If one means the tendency represented at the graphs of Fig. 5 (where the amplitude increased as the steepness of the wall increased), then the obtained result strongly differs from that one, which we expected to obtain – for all values of n , graphs of own functions have the form of a usual sinusoid, only the quantity of assemblies changes.

Fig.6. Graphs of own functions and probability density for a particle located into a potential well with vertical and infinitely high walls, [8, p.514].

In reality, there is nothing strange in it, since, in contrast to a harmonic oscillator, in the given case, we did not put any information about the form of the potential well into the wave equation. Although it is not emphasized anywhere, equation (4.4) is obtained from the wave equation in supposition that, in fact, a particle is free (we substituted U = 0 into equation (4.3)). We attributed zero values to the ordinates of the graphs of own functions at turning points (on the potential walls) through the initial conditions, i.e. forcedly. Thus, graphs of own functions, like those depicted in Fig. 6, are nothing else than de Broglie’s waves, which characterize free micro-particles. As it is known, de Broglie’s wavelength depends on the velocity of a particle’s motion – the higher is the velocity, the lower is the wavelength. The dependence of own functions graphs in Fig. 6 on the number of the energetic level is determined just by this fact.

The verticality of the wall on the potential energy curve means that at the given place, there is a potential jump (infinitely high intensity of the force field), and also that the object is point or non-deformable (infinitely hard). However, if an object is considered deformable, and one must not neglect its dimensions, then the wall on the potential energy curve must have a slant within the limits of the object’s deformability. It should take place even in that case, when, for some simplification, a field is characterized by infinitely high intensity. In this case, the slant of the walls is conditioned by the object’s deformation. The height of the potential walls is determined by the strength characteristics of the object and field – the interaction curve is finished, when the object overcomes the field or damages itself.

Note that in literature, one can also find the results of investigations, when potential walls are vertical, but not of infinite height, [10]. The verticality of the wall of finite height means that the material of the physical wall, with which the investigated object interacts, is infinitely hard, but fragile. As it is assumed in [10], in this case, the wave functions go quite far inside the potential walls, by analogy with that, as shown in Fig. 1. Physically it means that a quantum object knows in advance what strength of the walls is, and behaves very ill-bred, and allows itself “tunneling” into the range of negative energies of motion at quite large distances, without deforming or damaging the wall itself in this case. On the contrary, the same quantum object in a well with infinitely walls (infinitely hard and infinitely strong material of the physical wall) behaves very modestly – only seldom it approaches the wall, but does not touch the wall, even if its energy is significant, Fig. 6. Taking into consideration the stated above concerning the energetic interpretation of a wave function, we consider the deeper analysis of this question excessive. We add only that perhaps the energy in a wave equation plays the role of a parameter not without purpose.

We suppose that the question about the dependence of the form of own functions on the form of the potential well can be conducted more or less correctly only for cases, when a wave equation is succeeded to solve analytically, i.e. for a harmonic oscillator or a particle in a potential well, as described in equation (4.1), Fig. 4. Varying by the constants A and B , one can obtain potential wells of different width and depth and investigate this question in such a way correctly enough.

As for the explanation of some experimental data (-decay of nuclei, cold emission of electrons, predissociation of molecules) in terms of classical physics (in literature, the explanation of these data is available only within the limits of probabilistic interpretation), then we hope to state it in one of our next works. These explanations are stated quite in detail in [11].

5. Conclusions

The main argument against the probabilistic interpretation of the physical sense of -function consists in the fact that it leads to the conclusion about the violation of the causality principle and energy conservation law in the micro-universe. Since the macro-universe consists of a large number of micro-universes, it is logical to expect that the macro-universe laws should act also in the micro-universe.

The energetic interpretation, proposed in this work, is based upon the supposition that photons and “hard” elementary particles are soliton formations of electromagnetic field. Obviously, this supposition does not contradict the conception of the light-carrying ether and returns Schrödinger’s wave mechanics to the lap of classical physics, since it does not contradict to the laws of causality and energy conservation, and the quantum properties of microobjects (quantum jumps) obtain a simple explanation through the wave properties of solitons – standing waves change their state only discretely, in accordance with the change of the assemblies’ number. It is a very important question from the viewpoint of physics philosophy. Schrödinger said the following in his discussion with Bohr concerning quantum jumps: “If we are going to preserve these damned quantum jumps, then I feel sorry in general, that I dealt with the atomic theory!”, [12].

Since some types of solitons have also the properties of corpuscles, the corpuscular-wave dualism of quantum objects obtains a simple explanation within the limits of classical physics, which follows common sense philosophy. In this case, the explanation appears to be so simple, that it does not need in detailed statement already.

Schrödinger was, at least, as well as Einstein, stubborn in his conservative attitude towards quantum mechanics: but he rejected not only its statistical interpretation, but also insisted in the fact, that his wave mechanics means the return to the classical thinking.

M. Born, [3, p.68].


1. Schrödinger E. Selected Works on Quantum Mechanics. – Moscow: Nauka. – 424 p.

2. Schrödinger E. New Ways in Physics. – Moscow: Nauka. 1971. – 424 p.

3. Einstein's Digest 1972. – Moscow: Nauka. 1974. – 390 p.

4. Philippow A.T. Multifaceted Soliton. – Moscow: Nauka, 1990. – 288 p.

5. Chavarga. N. Relative Motion of Solitons in the Light-Carrying Ether.// Uzhgorod University Scientific Herald. Series Physics. Issue 7, 2000. – p. 174–194. (See also www.chavarga.iatp.org.ua)

6. Akhiezer A.I. Atomic Physics. Handbook. – Kiev: Naukova Dumka. 1988. – 268 p..

7. Landsberg G.S. Optics. – Moscow: Nauka, 1976. – 928 p.

8. Shpolsky E. Atomic Physics. V. II. – Moscow: Nauka, 1984. – 440 p.

9. Herzberg G. Molecular Spectra and Molecular Structure. 1.Diatomic Molecule. – N.Y., 1939. – 404 p.

10. Marion J.B. Physics and the Physical Universe. – N.Y. – London – Sydney – Toronto, 1971. – 624 p.

11. Chavarga N. Problem of Rational and Irrational in Physics. – Uzhgorod: Patent, 2000. – 236 p.

12. Kuznetsov B.G. Einstein. – Moscow: Nauka, 1967. – 428 p.


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