Personal site of Spassky Stanislav
Alternative: Gravity. Relativity. Quantum mechanics. Wave interpretation. (Missing chapter of physics.)
March 2008, Moscow, Zelenograd, Spassky Stanislav.
Quantum mechanics is certainly imposing. But an inner voice tells me that it is not yet the real thing. The theory says a lot, but does not really bring us any closer to the secret of the ‘old one’. I, at any rate, am convinced that He is not playing dice.
Excuse me for my English.
Hardly who would argue that the chasm is now eliminated between classical and modern physics. Suffice it to recall that a few classics of modern physics have not accepted the provisions of the probabilistic interpretation of quantum mechanics. Are we all so have grow wiser for a century that can honestly ignore this rejection? The author has endeavored not to bore colleagues by superfluous details and calculations. Recollect: "Mathematics is well and good, but nature keeps dragging us around by the nose."
The article intended for those interested in physics and not quite satisfied with a formal approach in the theory of relativity and quantum mechanics. Article proposes one of version to fill the gap between classical physics and the new physics. This article is the development of the ideas proposed by the author 5 years ago. In the previous article for a simple explanation of all SRT phenomena (including the velocity of light invariance) and the wave equation of quantum mechanics (isotropic variant) a simple model was proposed. In this model an object with "zero rest mass" was presented by fragment of "running wave" (below "RunW"), and the objects with "non-zero rest mass" were presented by "standing wave" capable of movement, (below "StW"). The article was 2 dissatisfying author points: 1)it is avoided the question (though mention was made) the very possibility of the existence of stable form of the StW type on "almost classical wave medium." 2) In the approach to quantum mechanics moving StW-object can be presented by two factors. One of them (spreading of phase) fully coincides with the wave function of de Broglie in isotropic medium. And the second multiplier was represented by spatial geometry of "standing wave" object, consisting of alternating-sign fragments. This second multiplier did problematic "desired" interference of different parts of wave object, when it "flow over" barriers. It seems these difficulties have been overcame.
In addition, the new article shows: 1) how in this " quasioptical" model "raises" the phenomenon of gravity, 2) conditional nature of "potentiality" in gravitational fields, 3) undercount in GRT possibility of changing the size of the objects in different gravitational conditions (space metric in the theories of relativity is based on the size of objects ( "on ruler")). 4) the meaning of the Schrodinger equation is clarified. 5) The author also considers important that experiment is showed that could confirm the correctness of the chosen concept in model.
A bit of history.
Very briefly, almost schematically, remember the history of physics in ending of 19th – beginning of 20th centuries to verify the position. In this simplified form should not be detected even of shadow of disrespectful attitude towards physicists of the past, remembering that we "stand on the shoulders" of these people.
At the boundary of the centuries classical physics seemed to be almost accomplished except for some aspects.
1) Electricity. It appeared impossible to "build" the physics on the basis of electricity. Moreover, the laws of electricity "suddenly" demanded at transition from one inertial system to another one the co-ordinate distortion (as spatial, as temporal, "Lorentz transformations"). From the same laws it turned out, that the velocity of electromagnetic waves is the same in all inertial systems ("C"), and, identical in all directions. It seems strange fact. Although why strange? After all, initially in all laws concerning electricity was laid invariance for all inertial systems. And since at the equations for "clean" fields (without particles and charges) there are no charges and currents, there is no binding of "C" to systems. How could it be otherwise?
Invariance of "C" was confirmed in experiments of Michelson. The young Einstein and aged Poincare expressed opinion that from a formal point of view the invariance of "C" is not a contradiction. The young Einstein, then as a adherent of Mach’s ideas, raised preservation of "C" to rank of postulates and the categories of space and time reduced to the "ruler" and "clock". It doomed the physics to the formal scanty manipulation with the "intervals". The paradox of invariance "C" in no way was explaining. It was suggested to take it as some form of manifestation of "reality". "Physical aether" abolished, "common sense" declared by remnant. Later, Einstein returned to the idea of "physical aether", noting that hypothetical aether should be tough (because of the nature of transverse electromagnetic waves) and pervasive substance. Think over, the combination of such contradictory qualities is possible only at the wave nature of the objects. Moreover, Einstein’s law of the equivalence of mass and energy is also a clear hint. But, alas, the idea of the wave nature of all objects never sounded from Einstein.
2)Here's another direction. Mathematician Plank absolutely formally (empirically) came to the necessity in description of the "blackbody" to determine the quantum of electromagnetic energy proportional to its frequency (with a coefficient "h", "Planck's Law" in the future). In this case, it appears long-awaited coincidence of theory with experiment. The idea was liked by Einstein, he defended it vigorously and promoted. The idea further was confirmed in several experiments. The young de Broglie purely formally applied the Planck’s law and the Einstein’s law of equivalence to the "massive" particle, thus defining its some frequency. Then, resting particle linked with some harmonic process received frequency, not only synchronous, but also inphase in the whole volume of particles resting. At movement such particles due Lorentz’s distortions (especially of time) simultaneous process was turned into a harmonic wave (de Broglie wave). The existence of such wave was later confirmed in experiments. Young Bohr used the wave model and the old model of electron movements around the nucleus. From the conditions of self-consistency on a closed trajectory "wave-particle" he had received a spectrum of possible energy levels of hydrogen, well coincided with the experience. It is strange "hit", taking into account following description of the same phenomenon with the help of the Schrodinger equation and the essential difference in the description of the processes. Schrodinger himself has withdrawn the equation called by his name and describing the behavior of the de Broglie particle-wave in a force field, on a bet with comrades and only in a few days. It is worth mentioning again, purely formally.
It is significant that in this equation physical category first "realy" is described by the complex numbers. Earlier complex numbers physicists used only as suitable instrument for intermediate description of harmonic processes.
And then Einstein began to develop SRT in the direction of description the behavior of particles (objects) with masses and without rest masses in a gravitational field. (by absolutely formal methods).
That is way the new physics was born. Basically, by use of formal methods. Moreover, the reference to "common sense" was deemed irrelevant. New theory ought to be "crazy enough".
Very brief repeating of the main aspects of the previous author’s article.
As already noted, in the previous article of author some model was proposed, in which simplistically the particle with zero rest mass is taken as fragment of running wave ("RunW"). Particles with non-zero rest mass were represented by localized standing wave ("StW" conditionally, as StW can move). Wavelength in StW is presumed corresponding to velocity of light and Planck's frequency. But in principle, it is not used in the analysis. It considered the most simple kind of StW - with the central symmetry (which does not restrict general nature of conclusions). If necessary, this StW presented by set of plane waves of all directions and cophased in the central point. What statements were made on the very possibility of stable existence of such local (energy-wise) form?
It is presumed that wave medium is not absorbing energy, almost classical, and its little distinction is presumed in the effect of wave energy on the properties of the wave medium (diminution of "C"). It was pointed out that RunW as fragment of traveling wave, in general, do not arouse feelings of rejection. You can say that the fragment "open" in one direction of RunW advance of waves, but in other directions the backward wave arises. About StW stated that the very idea of common base of StW and RunW was confirmed by experimental verification of Planck's relations for particles with a non-zero rest mass. Most likely, in StW we deal with object very close to RunW, its modification (one can say about certain "coiled" RunW form). And its complete closure in comparison with RunW could be explained by higher amplitudes of vibrations, the increase of "energy hollow" created by StW on medium. Nature of waves did not stipulate and did not associate with the electromagnetic field. The wave components in the model summarized in the simplest way.
Here is briefly that gave this model for understanding of the special theory of relativity.
1) Withdrawal of problems of light velocity invariance and "physical aether".
In their movements StW objects should be distorted because of its wave nature. Owing to requirements for all harmonics to be co-phased in moving centre (or to retain the phase relationship in the general case) wavelengths of constituent harmonics in movement should be different. Vibrations cease to be synchronous throughout the structure. The phase distortion of vibrations along the line of movement must appear (two longitudinal components will differ slightly in frequency). But you can select such coordinate systems, connected with moving StW, with distortions coordinates and time in which the wave operator has the same view as in the stationary system. StW "looks" in these systems as it looked at rest system. If the group StW objects (may be different) moves equally, their distortions are the same. "Macro-objects" consisting of such StWs (specifically, human beings), trying to build their physics, prefer that it had the simplest kind. And they certainly (unconsciously) must come in their perceptions and formalizations to referred coordinate system (with mutual Lorentz transformations), in which everything looks, "as at rest", the most simple. But since each such system can be viewed as an initial, then all these systems are symmetrical and equivalent to each other. It is clear that the light in such coordinates will propagate always with the same velocity as in systems resting on medium. To choose the system really rested on the medium, StW objects themselves (and macro-objects) are unable. And it is clear that they will have problem with "physical aether", as they will try to impose their own wave properties to aether. Actually the aether is natural, only it is of another "naturality", not of wave one, that StW are themselves. All distortion of StW and all relations in accordance with the SRT are simply illustrated on the model.
2)Kinematics of StW. Wave movement has two important features. This is a local wave energy itself and movement (the transference, flow) of this energy. The first characteristic can lead to the notion of full wave energy of StW (connected, as it traditionally done with the notion of mass). And the second characteristic can lead to the overall movement of StW energy, associating it with the notion of StW impulse. In resting StW vectors of all components of energy movements are compensated and the overall impulse is equal to zero. In moving StW the picture in the components is changing. All of them (their impulses) receive vector increment in direction of motion (StW receives a general parameter V/C). StW energy flows cease to be compensated and StW acquires the impulse, aiming in the direction of moving StW at speed V.
Now we begin to expound new material.
It is proposed to link the notion of "gravity" with the change (decrease) of the wave velocity "C", which produced by the presence of local wave energy in space (associated with "mass"). The fact that the gravitational field bends trajectory of a photon, is well known. More correctly, the changing of the wave properties locally gives rise to changes the wave properties (and gravity) radially in the whole space. (The same happens if you weaken locally the tension of the two-dimensional membrane). Below to connect gravitation with changing of "C" we will consider how do the wave flows (StW components) behave on quasi-wave medium at the boundary between two areas (W1, W2) with different "C" (for certainty C1>C2, W2 on the right). We define the " appropriate angles" in areas W1,W2 as a pair of angles in W1 and W2, related by ray path and by the ratio Sin(a1)/C1=Sin(a2)/C2. For all those "appropriate" angles passing energy rates in both directions are identical (known theoretical fact, "mutual impedance", "reciprocity theorem"). But the area W1 "projected" by the "appropriate" lines not in the whole area W2. Beginning from critical angle of "internal reflectance" in W2 (1/C1=Sin(a2)/C2) and further the wave energy from W2 stops flowing into W1 (effect of total internal reflectance). We get one component of "drawing" effect of additional wave energy in the area with less value "C". Another one associated with the refraction of waves will be shown below. We will show, what we call "gravity" is, presumably, mere "optics" and a slight change of wave properties of the wave medium by the wave energy.
It can be assumed that except of "formed" two types of discussed objects there is also a "wave noise", which accumulate in areas with high gravity according to discussed mechanism. Perhaps, at very large accumulations of wave energy ("mass") can, in principle, an avalanche begin - an irreversible process of one-sided capture of power (of the masses) from the surrounding space.
The fact that the wave energy by its presence decreases locally the wave number "C", in addition to the phenomenon of "internal reflectance" makes it possible to produce by wave itself the corridor with the side walls for a wave without power dissipation. This could explain the stability of the wave forms such as a photon (RunW).
If such corridor to close in the ring, then there will be the example of steady form of StW type, in which wave energy may spread as in one, as in both directions simultaneously. In general, there may be very different forms of this phenomenon, from infinite linear structures of a longitudinal spread of wave energy to infinite linear wave structures with transverse wave motion along the ring-shaped cross-section.
Stability of what we call wave objects (and RunW, and StW) can be determined, probably, and as a dynamic equilibrium. Wave energy may be in this case dissipated from StW as well be absorbed from the environment.
About the phenomenon of "charges" of different signs can be assumed that at non-linear characteristics of the medium and at sufficiently large amplitudes, for standing waves can be several steady solutions: 1) 0 for average value of vibrations, 2) the average value of vibrations can be shifted as in a positive as in negative. It is logically to expect that the objects, whose average shifts are of same signs (reflecting in all the surrounding space) should have repulsive interaction, and should have attractive interaction if there are a different shifts. But in general, the problem of charges beyond these statements almost was not analyzed.
Let us consider in detail mechanism, how may appear from "nothing" the concept of "potential energy of gravitational field" (in heterogeneous wave medium).
Currently, the behavior of a photon in a gravitational field is viewed in two ways. According to first model the photon is the traveling wave fragment (RunW) that preserves itself and its energy. In accordance with Huygens principle in a field with changing of "C" the front of traveling wave turns as well as trajectory of RunW itself. In the gravitational field this RunW trajectory turning quite similar to the turning of particle trajectory by force of gravity. According to second model the photon is considered as a particle. Having the energy, the photon acquires the corresponding gravitational mass. And the mass in the gravitational potential field is moving, as is customary, with the change of kinetic energy. In this model, the properties of bodies with rest mass are widening on photons. In our model, we are doing the opposite.
Our StW-object consists of traveling wave-type flows. As each part of the traveling waves is subjected to an optical effect, which is equivalent (quite similar) to the central gravitational attraction, then, obviously and the overall optical effect on the StW is equivalent (quite similar) to the central attraction with traditional change of kinetic energy. (It becomes apparent direct link of inertial and gravitational masses). Only according to the proposed model the total «internal» (wave) power of StW with motion in a gravitational field does not change and there is no gravitational potential energy. It will be a little below about it. But it had proved convenient and had turned out historically, that the locally the behavior become be described by the «force» gravitational field. And within certain limits and extent it is consistently.
Since it is presumed that wave medium is not absorbing energy, the total energy of StW (the wave energy itself and the energy of interaction wave with medium) is conserved while crossing. Examined changing of impulse (more precisely, parameter V/C) of objects StW (particles) the macro-objects (physicists) formally will describe by the existence of potential fields, which, in essence, are absent. The changes in motion of StW occur with accordance with the laws, close to the laws of optics without changing of energy. At transition to area with lower «C» the traveling wave components are returing similarly as at acceleration of an object StW.
We come to the following conclusion. Two features, a) energy conservation of StW when crossing the boundary, and b)an increase of parameter (V/C) mean: The same object StW in a medium with less value of "C" has a stable form with less energy at rest, which presumably depends only on the value of "C". Owing to the law of energy conservation when moving to area with less value of "C", StW object compensates "difference" of rest energies by increase of Lorentz’s factor (and V/C). Considering the reverse crossing C1<C2 shows that the crossing is not always possible. But formally the macro-objects when building their physics can take (for local considerations), that not micro-objects of which they are composed are changing, but their energy changes. That is, there is a potential field.
In new physics the concept of space and time metrics is used. We can rephrase the classic’s statement that the ruler and the clock are the measures of length and time, saying that the size of StW themselves and frequency of their oscillations are measures of length and time. But we must be very cautious with it. Suppose that, with increasing of gravity the oscillation frequency in StW is preserved or increases. Then, because of the reduction of C the size of StW (and "ruler") will decrease. Then it must seem to macro-objects that in the gravitational field there will be more "space", i.e., it will be a question of "curvature of space". And here it is necessary to distinguish between when the space "has become more" because of the StW reduction, and where takes place a real curvature of space of primary wave medium. As shown below, it is hoped that the sizes of objects are preserved. Similarly, you must be very careful with Planck's constant, as the energy of objects actually is preserved, but other qualities may change. It is paradoxically, that the author of aphorism "Mathematics is well and good...", himself had built very complex and bulky construction of GR on mathematics itself.
Let's try to relate the changing of value of the wave option "C" with the potential of gravitational field (in the conventional sense) and to appreciate this relationship to a first approximation. For this let us split central-force gravitational field into the spherical zones(i) of thickness (dRi), with a value of C(i), and with the value C0 at infinite R.
where (Ci/C(i-1))= 1-|dCi|/C(i -1) is ratio, opposite to refraction coefficient in the passage from layer(i-1) to layer i.
We'll use the picture of changes in structure of harmonics in our "standard" StW with central symmetry (consisting of set of all flat harmonics, which are synchronous in the "centre") at acquisition V/C from the rest in isotropic medium. Then we’ll estimate the change of parameter V/C at boundary in our concrete case at acceleration from the rest by analysis of harmonics. In isotropic medium at the acquisition of a some speed from rest the total energy of StV increases by β=1/√(1-Vi²/Ci²). At that, from viewpoint of the rest observer the former "transverse components" turn forward by a corner ArcSin(V/C). The reason is that in this component the points of wave fronts simultaneously must move in the direction of travel with velocity of StW V, to be "from viewpoint of the StW" moving transversely.
In the case of StW transition from layer with C(i-1) to layer of Ci transverse component turns forward to "critical angle of internal reflectance". We identify this with change of parameter V/C (that is, changing of the total energy in the conventional sense, although in fact here this energy does not change). "The angle of internal reflectance" A is counted from the plane of the boundary.
For "A" 1-dCi/C(i -1)=Sin(π/2-A)=Cos(A) Cos(A)= √(1-Sin²(A))=√(1-Vi²/Ci²) , counting StW accelerating from rest. So we have:
The result is:
Let's analyze the expression qualitatively. Note that C(R) stands in the left side and the right one under the sign of integral. While C(R) changes slightly, on the right side it can be taken C(R) = C0. Then C(R)= С0*e^(-(GM/R) / С0²). when (GM/R) becomes to some extent commensurate with the C0², it begins very sharp "collapse" of C(R) to 0 from the fact that decreasing C(R) is on the right side in the denominator of fraction, and this fraction is exponent. That is, there is a very sharp change (reduced to 0) of C, at some value of (GM/R)/C0².
Then let’s take a logarithm and differentiate:
When R = 2*GM/ C0² C(R) becomes 0. That is, we get so-called "gravitational radius".
Since our simplified model of behavior of StW in the gravitational field has given an acceptable result, let’s try to use it in the question of how StW distorts in the gravitational field. So, the rest (almost) StW crosses the boundary according to 1-|dCi|/C(i-1)=√(1-Vi²/Ci²) =1/β . In doing so, transverse component in almost resting StW because of the refraction is converted into a component turned forward. 1) First, remember the case when StW accelerates from rest in isotropic medium. Transverse components are turned forward, while frequency of these components increase by β. This is because of the increment of impulse vector by forward vector. Transverse component of impulse remains unchanged, as well as all cross sizes of StW. 2) In the case of a passage in the region with another C the turned transverse component preserves the frequency. So, if you will stop StW after its passage through the boundary StW, the frequency of transverse components (and others) will be in β times less. But under our assumptions C in the new area will decrease in β times. As the frequency and C vary equally, the size of StW should conserve. This is an interesting result (if it is trustworthy). So, spatial metric should not be changed by change of "rule length". But the local "running of time" should vary according with change of C(R).
If to use received above (1) C’(R)/C(R) = (GM/R²)/С(R)²), as well as the result of conservation for StW size (proportionality of C(R) and ω(R) ), it could be rewritten:
Or you may so, knowing C(R) (2):
At what R will the photon move on closed circular trajectory? To the wave front turns because of the difference of C, depending on R, and for this reason the photon would remain at the same radius, the condition must hold C(R)'=C(R)/R.
Using C'(R)(2) we get that it is possible at
We analyzed the behavior of StW objects in a gravitational field. And how will behave itself the BegW object in space with changing C? It also is likely to consist of harmonics of all directions. But unlike StW, RunW object is represented almost by one harmonic. So the approach used for StW for RunV can be applied very restrictedly. Let's remember that we are determined RunW (simplistically), as a piece of "running wave".
It is expected that such an object with strongly expressed one component crosses the "boundary" with very little change in frequency. If it is accepted the concept of conservation of main component frequency in RunW, then in terms of StW (and macro-objects), whose frequency decreases with increasing gravity, frequency of RunW objects should be increased. It seems, is confirmed by experiments.
This result does not enter in contradiction with the fact that the hydrogen atom, located in a different gravitational field emits different photons. Owing to reduction of C, and therefore slowing down all processes while conservation all sizes this atom should emit a photon of standard size, but reduced frequency and reduced energy. At the same time a photon, coming here and "created" outside of gravity should become smaller in length and increased frequency in comparison with "native" photon.
On quantum mechanics.
Let us advert to quantum mechanics. Many amazed by similarity of mathematical reflection of two phenomena: of picture of two opposite classical waves with near wavelength values (in the case of moving StW this may correspond to the two longitudinal harmonics) and of the wave function freely moving particle according to the de Broglie (or Schrodinger equation). The picture of "Pair of harmonics" is representable by the product of two wave functions:
Cos((ω+dω)t - ((ω+dω)/C*x) + Cos((ω-dω)t + ((ω-dω)/C*x)= 2 * Cos(ωt-ω/((ω/dω)*C)*x) * Cos(dω*t–(ω/C)*x) In the case of coincidence of both harmonic wavelengths (d?=0) 1st of factors is a harmonic function of the time of main frequency. 2nd of factors is a spatial pattern of standing waves - the sequence of alternating -sign half-wave areas. If the wavelengths of two harmonics begin to differ, the 2nd factor is modified in the wave shifting at a speed V ( =(dω/ω)*C), close to the x-component of "standing wave". And first factor modifies in frequency function that now is no longer synchronous at all parts, but having a kind of wave of "establishing of phase", similar in the wavelength and in speed to the de Broglie wave. Its speed is more than "C" ( =C*(C/V)= C*( ω/dω)).
Agree, the waves, corresponding to Schrodinger equation, is of very strange character. On the one hand, it is presumed, they obey the principle of superposition, and that in "reality" they represent an object in the form of "wave packet" (a set of waves close in frequency and direction). This means that it is implied high enough their mutual independence. On the other hand, at the same frequency these waves can have different lengths, depending on the speed of the body V, which they represent. Very strange waves, if not to assume that they simply reflect the "external manifestation" of hidden under them another process.
In this paper, particle was presented not by "packet" of waves, but by "bouquet" (i.e., by set of waves, directed to all sides) with properties that are very close to classical.
We came to the some physics of a wave objects that possess qualities of "material objects". It is logical to think that the parts of the StW objects have similar qualities and are to some extent autonomous objects of StW type with similar properties. The behavior of any part of the StW object is determined by "field" (picture of C), created by the other remaining parts. It is such self-consistent wave object. Moving to a neighbor region of space, with another value of "C" StW fragment on corresponding laws alters its parameter V/C, or for the same reasons, has no right to move into the adjacent area at all. It looks like some kind of "fluid". Physics of these objects (and their parts) is boundary between the physics of waves and physics of "massive" bodies.
Assume there is static self-consistent solution for StW object. If in such solution to consider the wave energy flows along lines of these streams, then these lines should be a closed lines, or lines, closed to infinity. Let's try to prove that at any point of energy flow line along it "the wave of establishing of vibration phase" spreads, which speed depends on the local value of parameter V/C at this point (=C*(C/V).
Consider a fragment of the wave object with the center in above point of line, which has the unbalanced flow of wave energy. Select "inertial systems", in which the piece is balanced on the energy flow. In this system the dispersion of frequencies has to reduced (or, even will be zeroized). The fragment is (more precisely, close due to some scattering) a fragment of "standing wave" with the co phased vibrations. It is clear that returning to the initial system the vibrations due to distortions of "time" will cease to be co phased and there will be a wave of the establishing of phase, with corresponding parameters ω and V/C at this point. The length of this wave can be related to the local value of the energy flow at this point (volume).
Schrodinger equation, or its variety, reflects the movement of the wave energy with ignoring the structure of "primary waves", close in frequency. It may well be interpreted as a balance of the wave energy per unit volume. One could, if necessary, to consider the meaning of each of the three members of the equation.
We considered behavior of parts of StW in entire StW structure, which connects all the wave energy in some form. And what happens to StW in real force fields, for example, an electron in the field of proton? StW is the wave energy linked into a single object by a certain structure, at the same time with binding energy much less than the wave energy. First try to imagine what would happen with the wave energy, if it is closed in some elastic frame, which could change the shape and together with this wave energy starts to move with the acceleration (say, attracted). Frequency spectrum will change. Being found itself in the actual field proton, an electron wave energy of the (with charge, an assumption about the nature of which was mentioned), "the structure, holding the whole" of StW "is trying" "to fit itself in" the potential field of the proton. Parts of StW while obey the same laws as in the existing autonomously StW, and therefore, Schrodinger equation. In discussed just above energy flow lines one problem becomes apparent, the difference of frequencies along the flow lines of two major opposite waves in a stationary decision must change. This contradiction can be explained by the fact that structures, linking StW in a whole may change to some extent frequency and phase of the opposite waves along the flow of energy.
Consider, how the pair of oposite waves proves in the "flat" "potential well" with "0" at the edges. In principle, the stationary solution is possible with the same frequency of both opposite waves. That is, in this case there is a standard standing wave, which corresponds to the resting object. Most likely, this solution is not stable. We have to seek stationary solution in the form of "motion" in both directions. The movement to one side is represented by a pair of opposite waves with slightly different frequency. A general stationary solution in the form of two opposite streams can be presented by four waves, which can be grouped into two opposite wave of one frequency and two opposite waves of slightly different frequency. A pair of two opposite waves of one frequency is the standing wave, which is represented on the x-axis by set of half-wave alternating-sign zones. When the decision is represented by two such standing waves of slightly different frequencies, along the well (along the axis x) takes place alternately two qualitatively different pictures. The cycle of their repetition coincides with a cycle of half de Broglie wave. Type of qualitative picture depends on the mutual position of "nodes" of two standing waves on x-axis. At the very edge of well the positions of the nodes of two standing waves coincide. A correlation of phases is changing over time because of the difference of frequencies. Therefore, on the edge the picture of "pulsation" is produced with the difference frequency. Next to zone of "pulsation" a zone follows in which oscillating loops and nodes of two standing waves do not match and compensate each other. Here there is a picture of small vibrations without pulsation. According to the assumed boundary conditions on the edges of "well" should be packed whole number of half-waves that we identify with half-waves of de Broglie wave.
The question arises whether there are experiments in which one can identify the difference of pictures on the edge of "well" according to both models, where the "classic wave of de Broglie" should have zero amplitude, while the considered wave model gives "pulsation" of large amplitude.
Consider the angular dependence of the wave function when it is (there is non-zero moment of rotation). In the first model of Bohr particle-wave upon proper trajectory should have been packed whole number of times on a circle according to the condition of self-consistency. The analogue to such angular function in considered model occurs when there are two opposing wave motion with different frequency (different wavelength along a circle). We have two opposite waves, each of which must be self-consistent along a circle and one of which has several cycles more than other. That is, self-consistency on closed trajectory of the wave "of establishing phase" is equivalent to self-consistency on closed trajectory of two opposite waves. Considered model easily explains the "no radiation" of the electron in this case.
Can it be clarified in the experiment, whether inside of the structures of de Broglie waves there is a primary wave structure? It is hoped that yes. This is possible in the experiment with passing of electrons through a thin crystal lattice. Consider how StW would show itself in this case if it was an electron. The front part of the structure, consisting mainly of two longitudinal components, having gone through the lattice, begins to form its direction for further spreading. By its very nature StW chooses one of the directions, favorable for phases of as direct as backward longitudinal components. Consider the picture of the phase relationship of StW parts after lattice, starting from the main axis in both waves. Wavelengths in direct and backward waves are slightly different. First the both pictures of maxima-minima alternation go on both harmonics together. Then they diverge and go as opposite and then go together again. One such cycle corresponds to the picture of de Broglie wave diffraction. In available now samples of diffraction of electrons, we may not notice a primary picture of alternated maxima and zeros due to weak resolution and weak frequency spread, but see the "total" picture corresponding to de Broglie wave. Necessary resolution, even if it is possible in principle, is achievable at the present level of technology. The primary wave for electron (L) should be about 1/400A. Lattice slits located at a distance of approximately 1A. So, in order to distinguish the close to the axis of symmetry (the main maximum) structure of primary waves requires resolution in angle on "target" of order 1/800 radian. In principle, it may be hoped... The electrons to be taken of high speed to diminish the number of maxima of assumed waves within the angular range of de Broglie wave. If electron would be more lightweight...
[There was a mistake that was corrected in the next article. The primary wavelength for an electron is 1/40A.]
Let me once again to disquiet the peace of the classic by his aphorism: "God is subtle but he is not malicious."
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