THE UNCERTAINTY RELATION FOR SOLITONS
Uzhgorod National University, 46, Pidhirna Str., 88 000, Uzhgorod, Ukraine.
The uncertainty relation is obtained in the supposition, that a photon is a soliton formation, and the photon’s length coincides with its wavelength.
In principle, it may be found out that the quantum theory is incorrect in its present form… If some time it is proved, that the uncertainty relation is incorrect, then we would have to expect the total reformation of the physical theory. J.B. Marion [1, p.609].
As it is known, the interpretation of the physical meaning of the uncertainty relation is based upon the idea concerning the impossibility of simultaneous measuring some characteristics of micro-objects, such as the coordinate and momentum of a micro-particle, or the energy of a particle and time of the particle’s energy measurement, with anyhow large exactness. Moreover, the most consistent adherents of the quantum theory even affirm that these characteristics do not exist in micro-objects simultaneously: “In reality, there is quite another situation – simply a micro-object itself cannot have both a certain coordinate and a certain appropriate momentum projection”, [2, p.35].
The world outlook conclusions, which are made from the analysis of the uncertainty relation, are extremely important for physics. In his first work already, in which the uncertainty relation was formulated, Heisenberg affirmed as one of the basic conclusions, that “the quantum mechanics definitely ascertained the invalidity of the causality law”. In the modern literature, a significantly less attention is paid to the uncertainty relation than to the relativity theory or to the probabilistic interpretation of the wave equation’s -function. In the past, however, pretensions to the interpretation of the essence of the relation were stated several times. Somerset Maugham brilliantly characterized the discussions concerning this question and the opponents’ positions in his book “Striking a Balance” (1927): “Two famous contemporary scientists accept Heisenberg’s principle with a grain of salt. Planck said that the further studies would eliminate the apparent anomaly, and Einstein called the philosophic ideas based on this principle the “literature”, I am afraid this is only a polite version of the word “junk”… Schrцdinger himself said that no final and exhaustive judgement on this problem was possible”, [3, p. 179].
Everything stated above testifies that the question cannot, however, be considered as finally solved. One can consider evident that if Planck, Schrцdinger, Einstein, de Broglie and others estimated the uncertainty relation in a similar way, then any investigation on this topic deserves to be given for readers’ discretion.
Assume that we have a quantum system, for example, an atom, in one of its excited states Е2, Fig.1. In case of the object’s transformation into a less excited state with energy Е1, its energy is changed at the quantity Е=Е2–Е1. In this case, the transformation of the free system is accompanied by the emission of a photon, whose basic characteristics are determined by Planck’s formula E= h, where Е has the meaning of the energy contained in the photon. It is assumed, that in the process of the photon’s birth, the energy conservation law is carried out, therefore, the quantity of the system’s energy change coincides with the quantity of the energy concentrated in the photon:
where Е2, Е1 and Е are the characterizations of the system;
Е and h are the characterizations of a photon.
If we write down
as a part of (2.1), then we must remember that Е is related to the quantum system, and Е – to the photon.
The quantity is the characteristic of the photon only, it determines the frequency of an oscillation process in a photon, and its period is equal to:
Fig.1. The emission of a photon by the system; the photon’s length coincides with its wavelength.
The quantity T in equation (2.3) can already be related to the equal degree both to the photon and to the system, but only in that case, if the photon’s length is equal to its wavelength . Not a wave train about a meter long or more, as it is considered in the literature, [4, 5], but only , Fig.1. The image of a photon as a specific soliton formation, limited in two coordinates, but having the ability of propagation in the third coordinate, can correspond to this condition. In this case, the oscillation process is characterized by a certain frequency and wavelength, which are connected with each other by the well-known relation:
Taking (2.3) into consideration, we can write down the relation for the photon E= h in the following form:
We have absolutely no reasons for doubting about the validity of (2.6), since it is only another writing of Planck’s relation. Though it is Planck’s formula in another representation, in its form it coincides with the well-known uncertainty relation. We are to make sure that it is not an accidental coincidence.
Let us denote the time of the system’s transformation from state Е2 into Е1 as t=t2–t1. The energy conservation law together with the causality principle require that the duration of the photon birth coincide with the duration of the change of the quantum system state. Otherwise, it may appear that the system’s energy decreases, but is not taken away by the photon in this case, or the photon has flown away already, but the system is still being “transformed”; or the system has already finished its transformation, but the photon has not yet flown away and is waiting for something, etc. If the photon’s length is equal to its wavelength , then the oscillation period Т coincides with the time of the change of the system’s state t, i.e.:
Taking (2.2) and (2.7) into consideration, one can represent the expression (2.6), true for the photon, in the form of the known relation true for the quantum system:
The physical meaning of the quantities, belonging to (2.8), we have already clarified, therefore, the expression (2.8) as a whole has the following meaning: if in the process of the transformation of the quantum system, its energy is changed at the quantity Е, then the product of the quantity of the energy change and of the transformation duration t is always constant and equal to h. The farther the energy levels are located from one another, the faster the quantum transformation, quantum jump, takes place.
The jump itself, in principle, can be represented as some non-periodical oscillation process with the frequency of =1/t. Obviously, since the quantity Е in relation (2.8) has the same meaning as in Heisenberg’s known relation, and the right part of the relation contains only Planck’s constant, then the quantity t is simply forced to have the meaning of the system’s transition time between the energetic levels.
As a matter of fact, (2.8) is another writing of Planck’s formula. For deriving (2.8), the additional supposition, that the photon’s length is equal to its wavelength, is sufficient. How this image is agreed with the interference experiments, and why the notion of a wave train appeared in physics, – it is the topic for a separate talk. Some aspects of this problem will be analyzed in our next work. Now we note that in (2.8), we cannot attribute the meaning of the quantity of the system’s energy measurement time (the duration of measurement) to the quantity t at all. Neither when analyzing Planck’s formula, nor when deriving (2.8), we had ever appealed to the problem of measuring any quantity.
On the other hand, mathematics supposes that all the quantities present in physical formulae are measured quantities, moreover, measured with infinite exactness. It means that if t is the time of the system’s transition from one state into another, then it is simultaneously the measured amount of this time. Not the duration of the system energy measurement, but the duration of its transition from one energetic level into another. This quantity cannot be less than the real duration of the process, since one cannot finish the measurements of the process duration before the finishing of the process itself. At the same time, if ideal devices (which are “used in mathematics”) are available, there is nothing to measure any more after the finishing of the process.
In real investigations, the result of measuring some quantity, or of some process duration, depends on the professional education of an investigator, on the measurement methods, and on the equipment that he has. Equation (2.8) indicates that in order to measure the time of the quantum system’s transformation from one state into another, we need not keep up with this process and interfere into it with our equipment. For this purpose, it is enough for us to measure the photon’s wavelength by means of a spectral device. Having measured the photon’s wavelength, we unequivocally determine its frequency =C/. Knowing the photon’s frequency, we determine the difference in the energies E through Planck’s formula, and through the relation t=T=1/ we determine the amount of time t – the duration of the system’s transformation from one state into another. Though it will sound strange on the background of the philosophical discussions developed around this question in the literature, but we cannot measure the quantities and t otherwise than simultaneously. Also, we cannot attribute any physical meaning, for example, that one connected with measurement processes, to the quantities belonging to (2.8).
Obviously, that relation (2.8) we cannot call the uncertainty relation at all – all quantities belonging to it have a clear determination, and the expression as a whole is a strict equality. In other words, we cannot use the sign “” or “” instead of equality sign in (2.8) at all, in order to turn it into the uncertainty relation. Note that (2.8) was obtained on the basis of supposition about carrying out the energy conservation law and causality principle.
It is known that a photon has the properties of a corpuscle, but it is hardly possible to speak seriously about the imagination, that an energy quantum exists in a system, for example, in an atom, in the form of a separate object. However, for the simplicity of the analysis, we can allow to do it. Moreover, we will assume that, when being in the system, it is at rest there, i.e. its momentum is equal to zero and begins to increase in the process of the photon emission – by analogy with the fact, how a bullet’s momentum increases in the process of its motion along the barrel. In this case, we can state that the photon’s momentum has changed at some quantity Р – from zero to Р, where Р is the photon’s momentum, whose value is determined from de Broglie’s relation:
If we assume that during the photon’s emission, the momentum conservation law is carried out, then we have a right to affirm, that the momentum of the same value, which we will denote as Р, will be obtained by the system, too, as the output momentum. On the other hand, the image of a photon, whose length coincides with its wavelength, allows us to write down х instead of , i.e. the amount of space occupied by the photon, see Fig.1. Therefore, instead of (3.1), we have:
Taking into account that numerically P=P, the left part of the equation can be related both to the system and to the photon, and the right – to the photon, therefore:
Expression (3.2) for the photon has the following physical meaning: the product of the value of the photon’s momentum and of the quantity of space occupied by the photon is equal to h for any photon. If (2.8) is Planck’s formula, but in another representation, then (3.3) is de Broglie’s relation. In comparison with de Broglie’s formula (3.1), (3.3) contains only the supposition, that х=.
If the momentum conservation law is carried out in the process of the photon’s emission, then expression (3.3) should be also true for the other participant of the process – for the quantum system, however, in the quantum system, we can hardly collate the adequate quantity to quantity х as to the photon’s characteristic. The quantity х cannot represent the amount of the system’s transmission for some time, since this quantity, in addition to all the other aspects, depends on the quantum system’s mass, which does not belong to the analyzed relations at all. It cannot represent the system’s geometrical dimensions, either, since a quantum of the same wavelength can be emitted by a small hydrogen atom or by a large molecule. Besides, a photon’s wavelength, for instance, in the visible range, is about 500 nm, however, an atom’s size is about 1 nm.
By analogy with the fact, that quantities Е and t from (2.8) are measured simultaneously, the quantities Р and х from (3.3) are measured simultaneously, too, through the measurement of the photon’s wavelength.
For deriving the relation, which describes the quantum system from the viewpoint of its momentum, we use that fact, that for the photon, х=СT. Taking into consideration that, in accordance with our supposition, T= t, expression (3.3) for the system we can write down as follows:
I.e. the momentum, obtained by the quantum system during the photon’s emission, multiplied by the time of the system’s state change, is equal to h/C for any transformation and for any quantum system – for a nucleus, atom, molecule, cluster.
It is not difficult to see that if some “motion mass m” is attributed to the photon, then the momentum P=mC can be attributed to it, too, and therefore, (3.4) can be written as follows:
Since the quantities t and h in the expression (3.5) have the same meaning as in (2.8), the expression mC2 is obliged to have the same meaning as E in (2.8). Thus, we obtain the well-known relation:
and (3.5) turns into (2.8) in this case, i.e. both Planck’s formula in (2.8) and de Broglie’s relation in (3.4) reflect one and the same essence, but in different terms. Evidently, that for a photon, it is more reasonable to write down expression (3.4) in the following form:
The physical meaning of the quantities, belonging to the left parts of expressions (3.4) and (3.5), is different, since they describe different objects, but the quantities themselves are numerically equal and can be collated in their physical meaning.
Summing up what we have stated up till the present moment, one can say the following:
1. Quantity E, as the difference in the energetic states of the quantum system, is equal to the energy of the quantum E, i.e. for the quantum system and the photon, these quantities can be reciprocally collated and have a similar physical meaning for both objects. It means that in the micro-universe energy conservation law is strictly carried out.
2. In the photon, the oscillation process period ^ can be put into correspondence to the quantity t as to the duration of the process of the quantum system’s transformation from one state into another (In the micro-universe, causality is strictly carried out). Perhaps, in expression ET=h Planck’s formula reflects the physical process, occurring in the photon, more adequately than formula E=h.
3. For the quantum system, the quantity P means the output momentum, obtained by the system in the process of the photon’s emission, and this quantity is numerically equal to the momentum ^ , possessed by the photon. In the micro-universe, momentum conservation law is strictly carried out.
4. The quantity x is the space characteristic of the photon and coincides with its wavelength . In the quantum system, this quantity has no vivid interpretation; the quantity C t can be collated with it, where t is the time of the photon’s birth, or the time of the system’s transformation from one state into another.
5. In theoretical investigations all quantities are considered to be measured with infinite exactness. This circumstance is set by the mathematics essence itself.
6. The expressions, connecting the characteristics of the quantum system and the photon, have the following form:
For the system: For the photon:
or (de Broglie)
One can consider evident the incorrectness of the parallel attribution of the following meaning to the above-mentioned quantities: the meaning, which reflects the process of the measurement of these quantities (the process, which depends both on the professional education of the investigator and on the measurement equipment, which he has).
Evidently, since strict equality signs are used in these relations, we should find another term instead of the term “uncertainty”. It is quite possible that it will be the best to introduce the term “increments relation”.
Finally we can say that the uncertainty relation has not definitely determined the invalidity of the causality law, and it is clear that Heisenberg made his conclusions too early.
When analyzing the passing of a narrow slit by a photon, the attention should be paid, that this phenomenon is well observed in case, when the photon’s wavelength can be collated with the slit’s width. In optical spectral diffraction devices, intended for working in the range of 500 – 8000 Е, the most frequently the lattices of 1200 str/mm are used, and it corresponds to the step d of the thread, approximately 8000 Е (more exactly, 8333.(3) Е). If one supposes that the slit's width is approximately a half of the thread’s step, then it follows that photons, having a wavelength twice as much as the slit’s width, cannot “break” through the slit already. To a significant extent, the lattice’s surface has already the properties of a mirror for them.
The process of passing a narrow slit by a photon can be relatively well illustrated vividly by means of the model of an oscillating dumb-bell, in which two elastic balls are connected with each other with a spring, Fig.2. Depending on the phase, in which the dumb-bell will come to the slit, with what aiming distance it will enter the slit, the dumb-bell can pass the slit without changing the direction of its translational motion, or with changing at some angle in the figure’s plane to the one or the other side.
Fig.2. Passing the narrow slit by an oscillating dumb-bell. The aiming distance is equal to zero.
We assume that the photon can interact with the slit’s wall by analogy with the fact, that the dumb-bell can be repelled from the wall. The main difference consists in the fact, that the dumb-bell can be repelled at an arbitrary angle, but the photon – only at a discrete one. It will look quite plausible, if we assume that within the limits of the slit, in the photon, as in a wave object, there may be formed a transverse harmonic, whose properties are described by de Broglie’s relation:
The number of the assemblies in this harmonic (in the standing wave limited by the dimensions x) can be changed only discretely. It means that the momentum, which can be attributed to the harmonic, can be changed only discretely, too, proportionally to the momentum P from the relation (4.1):
where n = 0, 1/2, 1, 3/2, corresponding to the quantity of semi-waves in the harmonic.
In case of the “attempt to radiate this harmonic”, the photon obtains the momentum Px in the transverse direction and changes the direction of its motion at some angle . It is known that in a similar interaction, the energy, contained in the photon, does not change (the wavelength does not change). Therefore, the photon’s momentum will not change, either, since these quantities are connected by de Broglie’s relation. It means, that only the direction of the momentum can change. At the figure, this fact is depicted in such a way, that the end of vector P describes a circle. As seen from the figure,
Taking into consideration (4.1), (4.2), (4.3) and de Broglie’s formula h = P, we have:
Let us multiply the left and the right part of (4.4) by 2:
Taking into account that 2х = d (Fig.2), and also that k = 2n = 0, 1, 2, 3, we have:
Formula (4.6) is a well-known formula of the diffraction lattice, where k is the number of the spectral order.
It is known from the optical tools theory that the “quantum of light must be at least as large as the biggest objectives; and since it is impossible that the volume of the quantum depends on the size of our tools, we can imagine it to be much larger”, Lorentz, [4, p.81]. Schrцdinger asked a similar question: “If the unit light pulse will not possess at least the 2.5 m wide wave front, then the resolution of a large 2.5 m telescope-reflector of the Mount-Wilson observatory won’t be better than that of the smallest telescope”, [5, p.16]. Now Schrцdinger would speak about 8.4 m – that is the record in constructing optical mirror telescopes.
Fig.3. Formation of interference circles in the telescope objective.
When we say that the photon must have cross-sectional dimensions not less than the telescope objective diameter, we mean the following. It is known that the image of a point object, e.g., of a star, is drawn by a collecting lens in the form of a finite-size point surrounded by a series of concentric circles of less intensity, Fig.3. Since the source emits the photons chaotically, not coherently, the appearance of the interference circles is explained by the interaction of the parts of the wave within the same photon. In this case, the radius of the central spot can be defined from the following formula:
where is the photon wavelength, D is the objective diameter, f is the focal distance.
The quantity r restricts the angular distance (in radians), which can be resolved by this objective (with Rayleigh’s criterion being taken into account):
If we assume that the transverse dimensions of the photon are about , then we should answer the following question: why does the resolution of the objective depend on its diameter, why does the lens direct the lion’s share of the photons to the central maximum (rays 5 and 2), and cannot direct the insignificant part (rays 3 and 4)?
Our assumption consists in the fact, that if a beam of parallel photons falls on the objective lens (from the star), then only those photons, which did not contact with the objective’s mounting, will be collected into the point, whose geometrical dimensions are about the transverse dimension of the photon. It is not difficult to estimate the quantity of photons contacting with the mounting. The transverse section of the photon can be considered to be equal to:
The photons, contacting with the mounting, enter the circle with the diameter of D and the length of . The area of this circle is equal to:
The quantity of photons, contacting with the lens, is equal to the ratio of the areas S2 and S1:
The total quantity of photons passed through the objective is equal to the ratio of the objective’s area to the photon’s area S1:
The degree of the spot’s diffuseness, which we denote by symbol k, can be determined as the ratio of the quantity of photons, participating in the diffusing of the point’s image, to the total quantity of photons:
Therefore, we have obtained the formula, which reflects the dependence of the objective’s resolution on its diameter and of the photon’s wavelength with coefficient exactness:
The photons’ interaction process with the mounting has probabilistic character – some photons will go aside, others will not, some photons will be absorbed by the mounting, others will not, etc. It is seen from Fig.3 that, in case of the given focal distance and wavelength, the smaller the objective’s diameter is, the larger the relative number of the photons interacting with the mounting is. For the given objective’s diameter, the larger the focal distance is, the larger the radius of the interference circles is – rays 2, 3 and 4 will have time to move apart at large distances.
It is a common practice in literature to think that a series of experiments bear witness in favor of a long photon – the so-called wave train. The results of interference observation in Michelson's device or in the experiments with the LummerHercke plate are meant here. Since in a series of cases the interference is observed for the difference in the lengths of the paths of interacting beams of about 1 m (or several million wavelengths), and the phenomenon is observed in the non-coherent light (gas discharge, fire), [6, p. 143], one has to assume that only the "odds and ends" of the same photon may interfere. The photon must be partitioned into these "odds and ends" when it falls onto the flat-parallel plate of the interferometer or when it exits from it, i.e. on the two optical media interface. This is Lorentz's thought concerning this question: "…the consideration of the simplest interference phenomena, e.g. Newton's rings, indicates that at least the quanta must be partible because the beams are divided into two parts, which move in different ways and finally reach the place of their interference", [4, p.81].
When scrutinizing Michelson's interferometer performance one may find that in reality the image of the photon as a long train makes more complications rather than clarifies the problem. We will proceed from the assumption, that photons are strictly subordinated to Planck’s formula. It is known that photons’ beams are divided at the borderline of the optical environments – some part of the beam is reflected, another part penetrates inside the other environment. In this case, if separate photons undergo division, then, according to Planck’s law, the formed parts of the photon must increase their wavelength. When division into two parts takes place, the wavelength must be doubled. It is known from the experiment, that the light beams’ color is not changed, i.e. the photons’ wavelength is not changed. Therefore, one can make only two conclusions: either the photon in a similar situation does not undergo division, or it undergoes, but the division products are not subordinated to Planck’s formula. Certainly, we prefer the first conclusion.
The situation with the interference of the photon’s odds and ends is not better, either. Let the arms of the interferometer be of equal length (0.5 m each). Assume further that the train length is 1 m (you may find in literature the allusions to even longer trains). It is seen from Fig. 4, that, irrespective of the motion direction of the train’s first part (the first part is denoted by figure 1, the second part – by figure 2), at their meeting at the plate the second part will catch up the first one, and the quantum will renew its length. It means that the first part of the train cannot influence the state of the second one. The same result will also occur in case, when the end 1 penetrates through the semitransparent plate, and the end 2 turns to the left mirror. Obviously, that if the train does not undergo division into parts on the silvered plate, then interference is out of the question.
Fig.4. Passing a long wave train through Michelson’s interferometer. The photon is divided by the semitransparent plate into parts 1 and 2.
Thus, we come to the conclusion, that by the division of a light quantum (a quantum as a long wave train) on the semitransparent plate, it is impossible to explain the occurrence of the interferential pattern in the device with equal arms. However, if the arms are not equal, for instance, the left arm is 50 % longer, then the ends will come to the output simultaneously, but only in case, when part 1 always turns to the left, and if the photon is halved on the plate. At the given stage, we have no reasons for advancing similar requirements.
For explaining the mechanism of the formation of the interferential pattern in Michelson’s device with the usage of the image of a photon, whose length is equal to its wavelength, we will make use of that fact, that for the large difference in the device’s arms, the interference is observed only when gas discharge sources are used, . It is known, that gas discharges are the working body in a series of lasers. It means, that from such a source, even when a resonator is absent, there may be radiated the “odds and ends of the induced radiation” – relatively long “photon trains”, where the separate photon corresponds to each wavelength. Not a photon, as a long wave train, but a train, as a long chain of photons.
Then we should assume, that the photons in the train are connected in such a way, that, when falling on the silvered layer of the interferometer’s plate, the train is divided as follows. The photons with odd numbers turn to one side (for example, left), and the photons with even ones – to another side (for example, go straight), Fig. 5. For vividness, the photons with odd numbers are depicted in the form of dashes, and the photons with even ones – in the form of ovals. Therefore, we have the depiction of the train in the form of the peculiar “chain”.
At the last stage of passing the interferometer by the train, the even and odd photons come together on the other side of the silvered layer. In this case, if for one of the photons, the process of reflection from the silvered layer occurs, then, for the second one, the process of “going” out from the plate occurs at the same time. The device’s geometry forces them to move further in one direction, taking one and the same position in space in this case. If two photons could really move in one direction, coinciding in this case, and preserving the same wavelength, then, in fact, we would obtain a new photon, whose wavelength would remain the same, but the energy, contained in it, would be twice as much as before. Obviously, that such a photon is not subordinated to Planck’s formula – in accordance with this formula, it should halve its wavelength. In the experiment, the change of the light beam’s color is not observed. It means, that the photons on the plate surface should somehow interact with each other, influence each other in such a way, that they would not occupy the same position in space, when moving further. Perhaps they simply have no other way out, as to repel each other from the common way and change the direction of their motion, Fig. 5. The latter we just perceive as the interferential pattern. From the proposed viewpoint, as a matter of fact, we should understand the “diffraction of photons on photons” as the interference.
Fig.5. The motion of the wave train in Michelson’s interferometer:
a) 1 – falling train of photons, the photons with odd numbers are depicted by dashes, the photons with even numbers – by ovals, b) 2 – the photons with odd numbers changed the direction of their motion, 3 – the photons with even numbers penetrated through the plate, c) the interacted even and odd photons repelled each other from the common way.
Practically any new supposition allows us to make the conclusion about its possible additional experimental testing. The idea, that in Michelson’s interferometer, not parts of separate photons, but separate photons as the odds and ends of a photon train interact, can be tested, if the light source, which guarantees the emission of separate photons, i.e. guarantees the absence of trains, is used for the experiments. Nowadays it is not difficult to make such a source. For this purpose, it is necessary to organize (in a closed space) the process of atoms excitement by an electronic hit in conditions of the singleness of collisions. It is easy to provide these conditions, if the pressures are about 10-3 mm Hg and electronic currents are about 100 A/m2. It is easy to obtain such current densities by means of conventional electronic guns, which are used in investigations of processes of electronic-atomic collisions. As the atoms, it is the simplest to use an inert gas with heavy atoms – for reducing the influence of Doppler’s effect.
If everything stated above concerning the photon’s image adequately reflects the reality, then in the proposed experiment, the interference pattern should not be observed in any ratios in the length of the device’s arms – a photon cannot be divided into parts, therefore, there is nobody to interact. Note that it will be impossible to explain the absence of an interferential pattern by the low signal level, since in the above-mentioned conditions, the collision range is well observed visually, i.e. the quantity of photons is quite sufficient, and interference can be also observed in case of a weaker signal: “It is well known, that interferential strips can be photographed in case of extremely weak intensities (daily expositions)”, [7, p.48]. In case of sufficient monoenergeticalness of the electronic beam, one can provide such a fact, that only one energetic atomic level will be excited, i.e. the investigated radiation will appear to be monochromatic, so that it will become significantly easier to observe the interferential pattern.
As for Doppler’s effect, then in the proposed photon source, its influence upon the photon’s wavelength is not higher than in the electric discharge, but sufficiently lower, since the excitement by the ionic hit is absent.
As quite a strong argument against the image of a photon as a long wave train, one can give that fact, that nowadays the laser impulses, whose duration is ~ 410-15 sec, are known already, . It means that the light impulse occupies ~1.2·103 Е in space, which is already comparable with the photon’s wavelength.
Finally we can say that the uncertainty relation has not definitely determined the invalidity of the causality law, and it is clear that Heisenberg made his conclusions too early.
If long searches have taught me anything, then their result is as follows: we go considerably over from the understanding of elementary processes than most contemporaries assume, and noisy triumphs do not correspond to the modern situation.
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