Hidden options. Theory of Hidden Parameters Hidden Parameters in Quantum Mechanics pdf

Is it possible to experimentally determine whether there are unaccounted for hidden parameters in quantum mechanics?

“God does not play dice with the universe” - with these words, Albert Einstein challenged his colleagues who developed a new theory - quantum mechanics. In his opinion, the Heisenberg uncertainty principle and the Schrödinger equation introduced an unhealthy uncertainty into the microcosm. He was sure that the Creator could not allow the world of electrons to be so strikingly different from the familiar world of Newtonian billiard balls. In fact, for years, Einstein played the devil's advocate for quantum mechanics, concocting ingenious paradoxes designed to turn on the creators of quantum mechanics. new theory into a dead end. In doing so, however, he did a good deed, seriously perplexing the theoreticians of the opposite camp with his paradoxes and forcing them to think deeply about how to solve them, which is always useful when a new field of knowledge is being developed.

There is a strange irony in the fact that Einstein went down in history as the principal opponent of quantum mechanics, although he himself was originally at its origins. In particular, Nobel Prize in physics in 1921, he received not at all for the theory of relativity, but for explaining the photoelectric effect based on new quantum concepts that literally swept scientific world at the beginning of the twentieth century.

Most of all, Einstein protested against the need to describe the phenomena of the microworld in terms of probabilities and wave functions (see Quantum Mechanics), and not from the usual position of particle coordinates and velocities. That's what he meant by "dice". He recognized that the description of the motion of electrons in terms of their speeds and coordinates contradicts the uncertainty principle. But, Einstein argued, there must be some other variables or parameters, taking into account which the quantum-mechanical picture of the microworld will return to the path of integrity and determinism. That is, he insisted, it only seems to us that God is playing dice with us, because we do not understand everything. Thus, he was the first to formulate the hidden variable hypothesis in the equations of quantum mechanics. It consists in the fact that, in fact, electrons have fixed coordinates and speed, like Newton's billiard balls, and the uncertainty principle and the probabilistic approach to their definition in the framework of quantum mechanics are the result of the incompleteness of the theory itself, which is why it does not allow them for certain. define.

The theory of the latent variable can be visualized something like this: the physical justification of the uncertainty principle is that the characteristics of a quantum object, such as an electron, can be measured only through its interaction with another quantum object; the state of the measured object will change. But perhaps there is some other way to measure using tools that are not yet known to us. These instruments (let's call them "subelectrons") will probably interact with quantum objects without changing their properties, and the uncertainty principle will not apply to such measurements. Although there was no evidence to support hypotheses of this kind, they loomed ghostly on the sidelines of the main path of development of quantum mechanics - mainly, I believe, due to the psychological discomfort experienced by many scientists due to the need to abandon the established Newtonian ideas about the structure of the universe.

And in 1964, John Bell received a new and unexpected theoretical result for many. He proved that it is possible to conduct a certain experiment (details a little later), the results of which will determine whether quantum mechanical objects are really described by the probability distribution wave functions, as they are, or whether there is a hidden parameter that allows you to accurately describe their position and momentum, as at the Newtonian ball. Bell's theorem, as it is now called, shows that both in the presence of a hidden parameter in the quantum mechanical theory that affects any physical characteristic of a quantum particle, and in the absence of such, it is possible to conduct a serial experiment, the statistical results of which will confirm or disprove the presence of hidden parameters in quantum mechanical theory. Relatively speaking, in one case the statistical ratio will be no more than 2:3, and in the other - no less than 3:4.

(Here I want to parenthetically point out that the year Bell proved his theorem, I was an undergraduate student at Stanford. Red-bearded, with a strong Irish accent, Bell was hard to miss. I remember standing in the corridor of the science building of the Stanford linear accelerator , and then he came out of his office in a state of extreme excitement and publicly announced that he had just discovered a really important and interesting thing. And, although I have no proof of this, I would very much like to hope that I became an unwitting witness to his discovery that day.)

However, the experience proposed by Bell turned out to be simple only on paper and at first seemed almost impossible. The experiment was supposed to look like this: under external influence, the atom had to synchronously emit two particles, for example, two photons, and in opposite directions. After that, it was necessary to catch these particles and instrumentally determine the direction of the spin of each and do this a thousand times in order to accumulate sufficient statistics to confirm or refute the existence of a hidden parameter according to Bell's theorem (in the language of mathematical statistics, it was necessary to calculate the correlation coefficients).

The most unpleasant surprise for everyone after the publication of Bell's theorem was precisely the need to conduct a colossal series of experiments, which at that time seemed practically impossible, in order to obtain a statistically reliable picture. However, less than a decade later, experimental scientists not only developed and built the necessary equipment, but also accumulated a sufficient amount of data to statistical processing. Without going into technical details, I will only say that then, in the mid-sixties, the complexity of this task seemed so monstrous that the probability of its implementation seemed to be equal to that of someone planning to put a million trained monkeys from the proverb at typewriters in the hope of finding among the fruits of their collective labor, a creation equal to Shakespeare.

When the results of the experiments were summarized in the early 1970s, everything became crystal clear. The probability distribution wave function accurately describes the movement of particles from the source to the sensor. Therefore, the equations of wave quantum mechanics do not contain hidden variables. This is the only known case in the history of science when a brilliant theoretician proved the possibility of experimental testing of a hypothesis and gave a justification for the method of such testing, brilliant experimenters with titanic efforts carried out a complex, expensive and protracted experiment, which in the end only confirmed the already dominant theory and did not even introduce into it is nothing new, as a result of which everyone felt cruelly deceived in their expectations!

However, not all work was in vain. More recently, scientists and engineers, much to their own surprise, have found Bell's theorem to be very worthy. practical use. The two particles emitted by the Bell source are coherent (have the same wave phase) because they are emitted synchronously. And this property of theirs is now going to be used in cryptography to encrypt highly secret messages sent over two separate channels. When intercepting and attempting to decrypt a message via one of the channels, coherence is instantly broken (again, due to the uncertainty principle), and the message inevitably and instantly self-destructs at the moment when the connection between the particles is broken.

And Einstein, it seems, was wrong: God still plays dice with the universe. Perhaps Einstein should have heeded the advice of his old friend and colleague Niels Bohr, who, once again hearing the old refrain about “dice game”, exclaimed: “Albert, stop telling God what to do at last. !"

Encyclopedia of James Trefil “The nature of science. 200 laws of the universe.

James Trefil is a professor of physics at George Mason University (USA), one of the most famous Western authors of popular science books.

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Physics professor Jim Al-Khalili explores the most accurate and one of the most confusing scientific theories- quantum physics. In the early 20th century, scientists penetrated the hidden depths of matter, the subatomic building blocks of the world around us. They discovered phenomena that are different from anything seen before. A world where everything can be in many places at the same time, where reality really exists only when we observe it. Albert Einstein opposed the mere idea that the essence of nature is based on chance. Quantum physics implies that subatomic particles can interact faster than the speed of light, and this contradicts his theory of relativity.

The French physicist Pierre Simon Laplace raised the important question of whether everything in the world is predetermined by the previous state of the world, or whether a cause can cause several effects. As expected by the philosophical tradition, Laplace himself in his book “Statement of the System of the World” did not ask any questions, but said a ready-made answer that yes, everything in the world is predetermined, however, as often happens in philosophy, the picture of the world proposed by Laplace did not convince everyone and thus his answer gave rise to a discussion around that question which continues to this day. Despite the opinion of some philosophers that quantum mechanics allowed this question in favor of a probabilistic approach, however, Laplace's theory of complete predestination, or as it is otherwise called, the theory of Laplace's determinism is being discussed today.

If the initial conditions of the system are known, it is possible, using the laws of nature, to predict its final state.

In everyday life, we are surrounded by material objects whose dimensions are comparable to us: cars, houses, grains of sand, etc. Our intuitive ideas about the structure of the world are formed as a result of everyday observation of the behavior of such objects. Since we all have a life behind us, the experience accumulated over the years tells us that since everything we observe over and over again behaves in a certain way, it means that in the entire Universe, on all scales, material objects should behave in a similar way. And when it turns out that somewhere something does not obey the usual rules and contradicts our intuitive concepts of the world, this not only surprises us, but shocks us.

Hidden parameters and limits of applicability of quantum mechanics.

N.T. Saynyuk

The paper shows that a nonzero size can be used as a hidden parameter in quantum mechanics elementary particles. This made it possible to explain the fundamental physical concepts used in the theory of the de Broglie wave, wave-particle duality, and spin. The possibility of using the mathematical apparatus of the theory to describe the motion of macrobodies in a gravitational field was also shown. The existence of discrete vibrational spectra of elementary particles is predicted. The question of the equivalence of the inertial and gravitational masses is considered.

Despite the existence of quantum mechanics for almost a century, disputes about the completeness of this theory have not subsided to this day. The success of quantum mechanics in reflecting the existing regularities in the field of the subatomic world is beyond doubt. At the same time, some physical concepts used by quantum mechanics, such as wave-particle duality, the Heisenberg uncertainty relation, spin, etc., remain misunderstood and do not find proper justification within this theory. It is widely believed among scientists that the problem of substantiating quantum mechanics is closely related to hidden parameters, that is, physical quantities, which really exist, determine the results of the experiment, but for some reason cannot be detected. In this paper, based on an analogy with classical physics, it is shown that a non-zero size of elementary particles can claim the role of a hidden parameter.

Trajectory in classical and quantum physics.

Let's imagine a material body with a rest mass, for example, a nucleus flying in space with a speed at a sufficiently large distance from other bodies so that their influence can be excluded. In classical physics, such a state of the body is described by a trajectory that establishes the location of its central point in space at each moment of time and is determined by the function:

How accurate is this description? As you know, any material body with a rest mass has a gravitational field that extends to infinity and which cannot be separated from the body in any way, therefore it should be considered an integral part of a material object. In classical physics, when determining the trajectory, as a rule, the potential field is neglected because of its small value. And this is the first approximation that classical physics allows. If we tried to take into account the potential field, then such a concept as a trajectory would disappear. It is impossible to attribute a trajectory to an infinitely large body, and formula (1) would lose all meaning. In addition, any material body has some dimensions and it also cannot be localized at one point. You can only talk about some volume that the body occupies in space or about its linear dimensions. And this is the second approximation that classical physics allows, endowing physical bodies with trajectories. The existence of dimensions for material bodies entails another uncertainty - the inability to accurately determine the time of location material body in space. This is due to the fact that the speed of signal propagation in nature is limited by the speed of light in vacuum, and so far there are no reliably experimentally established facts that this speed can be significantly exceeded. This can only be done with a certain accuracy required by the light signal to cover a distance equal to the linear size of the body:

Uncertainty in space and time in classical physics is of fundamental nature, it cannot be bypassed by any tricks. This uncertainty can only be neglected, which is done everywhere and for most practical engineering calculations, accuracy and without taking into account uncertainties is quite sufficient.

From the above, two conclusions can be drawn:

1. The trajectory in classical physics is not strictly justified. These concepts can be applied only when it is possible to neglect the potential field of a material object and its dimensions.

2. In classical physics, there is a fundamental uncertainty in determining the position of a body in space and time due to the presence of dimensions in material bodies and the finite speed of propagation of signals in nature.

It turns out that the Heisenberg uncertainty relation in quantum mechanics is also due to these two factors.

There is no concept of a trajectory in quantum mechanics. It would seem that in this way quantum mechanics eliminates the above listed shortcomings of classical physics and describes reality more adequately. This is only partly true, and there are some very significant nuances. Let's consider this question on the example of the electron at rest in what coordinate system. From classical physics, in particular from Coulomb's law, it is known that an electron, having an electric field, is an infinite object. And at every point in space this field is present. In quantum mechanics, such an electron is described by a wave function , which also has a non-zero value at every point in space. And in this plan, it correctly reflects the fact that the electron occupies all space. But it is explained in a different way. According to the Copenhagen interpretation, the square of the modulus of the wave function, at some point in space, is the probability density of finding an electron at that point in the process of observation. Is this interpretation correct? The answer is unequivocal - no. An electron as an infinite object cannot be instantly localized at one point. This directly contradicts special theory relativity. The collapse of an electron into a point is possible only if the speed of propagation of signals in nature was infinite. So far, no such facts have been found experimentally. In our case, the real field, quantum mechanics compares the probability of finding an electron at some point. Obviously, such an interpretation of quantum mechanics does not correspond to reality, but is only some approximation to it. And it is not surprising that when describing electric field electron, quantum mechanics faces great mathematical difficulties. The example below shows why this happens. Coulomb's law is a deterministic law, while quantum mechanics uses a probabilistic approach. In this case, classical physics is more adequate. It allows you to determine the strength of the electric field in any region of space. All that is needed for this is to indicate in the Coulomb's law the coordinates of the point at which this field is to be found. And here we are directly confronted with the question of the limits of applicability of quantum mechanics. The successes of quantum theory in various directions are so huge and the predictions are so accurate that many have wondered if there are limits to its applicability. Unfortunately there are. If there is a need to move from a probabilistic description of the world to its deterministic interpretation as it really is, then we must remember that it is at this transition that the powers of quantum mechanics end. She did an excellent job. Its possibilities are far from being exhausted, and it can still explain a lot. But it is only a certain approximation to reality, and judging by the results, it is a very successful approximation. Below we will show why this is possible.

Wave properties of particles, wave-particle duality
in quantum mechanics.

This is probably the most confusing question in quantum theory. There are countless works written on this topic and opinions expressed. The experiment unambiguously states that the phenomenon exists, but it is so incomprehensible, mythical and inexplicable that it even served as a reason for jokes that a particle, on its own whim, behaves like a corpuscle on some days of the week, and like a wave on others. Let us show that the existence of a hidden parameter of a nonzero particle size makes it possible to explain this phenomenon. Let's start with the Heisenberg uncertainty relation. It has also been repeatedly confirmed by experiment, but it does not find the proper justification within the quantum theory. Let us use the conclusions from classical physics that two factors are necessary for the emergence of uncertainty and see how these factors are implemented in quantum theory. Regarding the speed of light, we can say that it is organically built into the structures of the theory, and this is understandable, since almost all the processes that quantum mechanics deals with are relativistic. And without the special theory of relativity here simply can not do. The other factor is different. All calculations in quantum mechanics are made on the assumption that the particles it deals with are point particles, in other words, there is no second condition for the occurrence of the uncertainty relation. Let us introduce a non-zero size of elementary particles into quantum mechanics as a hidden parameter. But how to choose it? Physicists involved in the development of string theory are of the opinion that elementary particles are not point particles, but this manifests itself only at significant energies. Is it possible to use these dimensions as a hidden parameter. Most likely not, for two reasons. Firstly, these assumptions are not entirely substantiated, and on the other hand, the energies with which the developers of string theory work are so large that these ideas are difficult to verify experimentally. Therefore, it is better to look for a candidate for the role of a hidden parameter at a low-energy level accessible for experimental verification. The most suitable candidate for this is the Compton wavelength of the particle:

It is constantly in sight, is given in all reference books, although it does not find a proper explanation. Let us find an application for it and postulate that it is the Compton wavelength of a particle that determines, in some approximation, the size of this particle. Let's see if the Compton wavelength satisfies the Heisenberg uncertainty relation. It takes time to travel a distance equal to the speed of light:

Substituting (4) into (3) and taking into account that we get:

As can be seen in this case, the Heisenberg uncertainty relation is fulfilled exactly. The above reasoning cannot be considered as a justification or conclusion of the uncertainty relation. It only states the fact that the conditions for the emergence of uncertainty, both in classical physics and in quantum theory, are absolutely the same.

Let us consider the passage of a particle with velocity , which has dimensions of the Compton wavelength, through narrow gap. The time of passage of the particle through the slot is determined by the expression:

Due to its potential field, the particle will interact with the walls of the slot and experience some acceleration. Let this acceleration be small and the speed of the particle after passing through the gap, as before, can be considered equal to . The acceleration of the particle will cause a wave of perturbation of its own field, which will propagate at the speed of light. During the time the particle passes through the slit, this wave propagates over a distance:

Substituting into expression (7) expressions (3) and (6) we get:

Thus, the introduction of a non-zero particle size as a hidden parameter into quantum mechanics makes it possible to automatically obtain expressions for the de Broglie wavelength. Get what quantum mechanics was forced to take from experiment, but could not substantiate it in any way. It becomes obvious that wave properties particles are caused only by their potential field, namely, the appearance of a wave of perturbation of their own field or, as it is commonly called, a retarded potential during their accelerated motion. Based on the foregoing, it can also be argued that the expression for the de Broglie wave (8) is by no means a statistical function, but a real wave of all characteristics, which, if necessary, can be calculated based on the concepts of classical physics. Which in turn is another proof that the probabilistic interpretation quantum mechanics physical processes occurring in the subatomic world is incorrect. Now there is already an opportunity to reveal the physical essence of wave-particle duality. If the potential field of the particle is weak and can be neglected, then the particle behaves like a corpuscle and can safely be assigned a trajectory. If the potential field of particles is strong and can no longer be neglected, namely such electromagnetic fields act in atomic physics, then in this case you need to be prepared for the fact that the particle will show its wave properties in full measure. Those. one of the main paradoxes of quantum mechanics about corpuscular wave dualism turned out to be easily resolved due to the existence of a hidden parameter of a non-zero size of elementary particles.

Discreteness in quantum and classical physics.

For some reason, it is generally accepted that discreteness is characteristic only of quantum physics, while in classical physics there is no such concept. In fact, everything is not so. Any musician knows that a good resonator is tuned to only one frequency and its overtones, the number of which can also be described by integer values \u003d 1, 2, 3 ... . The same thing happens in the atom. Only in this case, instead of a resonator, there is a potential well. Moving in an atom in a closed orbit at an accelerated rate, the electron continuously generates a wave of perturbation of its own field. Under certain conditions (the distance of the orbit from the nucleus, the speed of the electron), the conditions for the emergence of standing waves can be fulfilled for this wave. An indispensable condition for the occurrence of standing waves is that an equal number of such waves fit along the length of the orbit. It is possible that Bohr was guided by such considerations when formulating his postulates regarding the structure of the hydrogen atom. This approach is based entirely on the concepts of classical physics. And he was able to explain the discrete nature energy levels in the hydrogen atom. There was more physical meaning in Bohr's ideas than in quantum mechanics. But both Bohr's postulates and the solution of the Schrödinger equation for the hydrogen atom gave exactly the same results regarding discrete energy levels. The discrepancies began when it was necessary to explain the fine structure of these spectra. In this case, quantum mechanics proved to be more than successful, and work on the development of Bohr's ideas was stopped. Why did quantum mechanics emerge victorious? The fact is that, being in a stationary orbit in conditions where the formation of standing waves is possible, the electron passes the same path many times. There is no experimental possibility to trace the motion of an electron in a bound state at the microscopic level. Therefore, the use of statistical methods here is quite justified, and the interpretation of the formation of antinodes in the orbit as the highest probability of finding an electron at these points has good grounds, which, in fact, is what quantum theory does with the help of the wave function and the Schrödinger equation. And this is the reason for the successful application of the probabilistic approach to describe physical phenomena occurring in atomic physics. Here we consider only one, the most simple example. But the conditions for the emergence of standing waves can also arise in more complex systems. And quantum mechanics does a good job with these questions as well. One can only admire the scientists who stood at the origins of quantum physics. Working in a period of destruction of familiar concepts, in conditions of a shortage of objective information, they somehow managed to feel the essence of the processes occurring at the microscopic level in some incredible way and built such a successful and beautiful theory as quantum mechanics is. It is also obvious that there are no fundamental obstacles to obtaining the same results within the framework of classical physics, because such a concept, a standing wave, is well known to it.

Quantum of minimal action in quantum mechanics and in
classical physics.

The quantum of minimal action was first used by Planck in 1900 to explain the radiation of a black body. Since then, the constant introduced by Planck into physics, later named after the author as Planck's constant, has firmly taken its place of honor in subatomic physics and is found in almost all mathematical expressions which are used here. Perhaps this was the most significant blow to classical physics and determinists, who could not do anything to counter it. Indeed, there is no such concept as a minimum quantum of action in classical physics. Does this mean that it cannot be there in principle and that this is the domain of only the microworld? It turns out that for macrobodies with a potential field you can also use the minimum action quantum, which is defined by the expression:

(9)

where is the body weight

Diameterthis body

speed of light

Expression (9) is postulated in this paper and requires experimental verification. The use of this quantum of action in the Schrödinger equation makes it possible to show that the orbits of the planets solar system are also quantized, as are the orbits of an electron in atoms. In classical physics it is no longer necessary to take the value of the minimal action quantum from experiment. Knowing the mass and dimensions of the body, its value can be unambiguously calculated. Moreover, expression (9) is also valid for quantum mechanics. If in formula (9) instead of the diameter of the macrobody we substitute the expression that determines the size of the microparticle (3), then we get:

Thus, the value of Planck's constant, which is used in quantum mechanics, is just a special case of expression (9) used in the macrocosm. In passing, we note that in the case of quantum mechanics, expression (9) contains a hidden parameter, the particle size. Perhaps that is why Planck's constant was not understood in classical physics, and quantum mechanics could not explain what it is, but simply used its value taken from the experiment.

Quantum effects in gravity.

Introduction to quantum mechanics as a hidden parameter, a non-zero size of elementary particles, made it possible to determine that the wave properties of particles are due exclusively to the potential field of these particles. Macrobodies with rest mass also have a potential gravitational field. And if the conclusions drawn above are correct, then quantum effects should also be observed in gravity. Using the expression for the minimum quantum of action (9), we formulate the Schrödinger equation for a planet that moves in the gravitational field of the Sun. It looks like:

wherem is the mass of the planet;

M is the mass of the Sun;

G is the gravitational constant.

The procedure for solving equation (10) is no different from the procedure for solving the Schrödinger equation for the hydrogen atom. This makes it possible to avoid cumbersome mathematical calculations and solutions (10) can be immediately written out:

Where

Since the presence of trajectories for planets moving in orbit around the Sun is beyond doubt, it is convenient to transform expression (11) and represent it in terms of the quantum radii of the planets' orbits. Let us take into account that in classical physics the energy of a planet in orbit is determined by the expression:

(12 );

Where is the average radius of the planet's orbit.

Equating (11) and (12) we get:

(13 );

Quantum mechanics does not make it possible to unequivocally answer in what excited state a bound system can be. It only allows you to find out all possible states and the probabilities of being in each of them. Formula (13) shows that for any planet there is an infinite number of discrete orbits in which it can be located. Therefore, one can try to determine the main quantum numbers of the planets by comparing the calculations made by formula (13) with the observed radii of the planets. The results of this comparison are presented in Table 1. The data on the observed values of the parameters of the orbits of the planets are taken from .

Table 1.

*Planet*	*Actual orbit radius* R *million km*	*Result* *computing* *million km*	n	*Mistake* *million km*	*Relative error* %
*Mercury*	57.91	58.6		0.69
*Venus*	108.21	122.5		14.3	13.2
*Earth*	149.6	136.2		13.4
*Mars*	227.95	228.2		0.35	0.15
*Jupiter*	778.34	334.3
*Saturn*	1427.0
*Uranus*	2870.97	2816		54.9
*Neptune*	4498.58	4888.4
*Pluto*	5912.2	5931		18.8

As can be seen from Table 1, each planet can be assigned a certain main quantum number. And these numbers are quite small compared to those that could be obtained if in the Schrödinger equation, instead of the minimum action quantum determined by formula (9), Planck's constant, usually used in quantum mechanics, would be used. Although the discrepancy between the calculated values and the observed radii of the orbits of the planets is quite large. Perhaps this is due to the fact that the derivation of formula (11) did not take into account the mutual influence of the planets, leading to a change in their orbits. But it is shown that the main orbits of the planets of the solar system are quantized, just as it takes place in atomic physics. The given data unambiguously testify that quantum effects also take place in gravitation.

There are also experimental confirmations of this. V. Nesvizhevsky with colleagues from France managed to show that neutrons moving in a gravitational field are detected only at discrete heights. This is a precision experiment. The difficulty of conducting such experiments is that the wave properties of the neutron are due to its gravitational field, which is very weak.

Thus, it can be argued that the creation of a theory of quantum gravity is possible, but it should be taken into account that elementary particles have a non-zero size, and the minimum quantum of action in gravity is determined by expression (9).

Particle spin in quantum mechanics and classical physics.

In classical physics, every rotating body has an internal angular momentum, which can take on any value.

In subatomic physics, experimental studies also confirm the fact that particles have an internal angular momentum called spin. It is believed, however, that in quantum mechanics the spin cannot be expressed in terms of coordinates and momentum, since for any allowable particle radius, the speed on its surface will exceed the speed of light and, therefore, such a representation is unacceptable. Introduction to quantum physics of non-zero particle size allows us to somewhat clarify this issue. To do this, we use the concepts of string theory and imagine a particle whose diameter is equal to the Compton wavelength as a string closed in three-dimensional space, along which a stream of some field circulates at the speed of light. Since any field has energy and momentum, it is possible with with good reason to attribute to this field a momentum related to the mass of this particle:

Considering that the radius of the field circulation around the center is , we obtain the expression for the spin:

Expression (15) is valid only for fermions and cannot be considered a justification for the existence of spin in elementary particles. But it allows us to understand why particles with different rest masses can have the same spin. This is due to the fact that when the particle mass changes, the Compton wavelength changes accordingly, and expression (15) remains unchanged. This did not find an explanation in quantum mechanics and the values for the particle spin were taken from the experiment.

Vibrational spectra of elementary particles.

In the previous chapter, when considering the issue of spin, a particle with a size equal to the Compton wavelength was represented as a string closed in three-dimensional space. This representation makes it possible to show that discrete vibrational spectra can be excited in elementary particles.

Let us consider the interaction of two identical closed strings with rest masses moving towards each other with a speed . From the beginning of the collision to the complete stop of the strings, some time will pass, due to the fact that the speed of momentum transfer inside the strings cannot exceed the speed of light. During this time, the kinetic energy of the strings will be converted into potential energy due to their deformation. At the moment the string stops, its total energy will consist of the sum of the rest energy and the potential energy stored during the collision. Later, when the strings begin to move in the opposite direction, part of the potential energy will be spent on excitation of the natural vibrations of the strings. The simplest form of vibration at low energies that can be excited in strings can be represented as harmonic vibrations. The potential energy of the string when deviating from the equilibrium state by a value has the form.

k - coefficient of elasticity of the string

We write the Schrödinger equation for stationary states of a harmonic oscillator in the form:

The exact solution of equation (17) leads to the following expression for discrete values :

Where 0, 1, 2, … (18)

In formula (18) unknown coefficient elasticity of elementary particles k . It can be approximately calculated based on the following considerations. When particles collide at the moment they stop, all kinetic energy is converted into potential energy. Therefore, we can write the equality:

If the momentum inside the particle is transmitted at the maximum possible speed equal speed light, then from the moment the collision begins to the moment the particles diverge time will pass necessary for the impulse to propagate along the diameter of the entire particle, equal to the Compton wavelength:

During this time, the deviation of the string from the equilibrium state due to deformation can be:

Taking into account (21), expression (19) can be written as:

Substituting (23) into (18) we obtain an expression for possible values suitable for practical calculations:

Where , 1, 2, … (24)

Tables (2, 3) present the values for the electron and proton calculated by formula (24). The tables also indicate the energies released during the decay of excited states during transitions and the total energies of particles in an excited state. All experimental values of particle rest masses are taken from .

Table 2. Vibrational spectrum of electron e (0.5110034 MeV.)

Quantum number n

Table 3. Vibrational spectrum of proton P (938.2796 MeV)

Quantum number n

"God does not play dice with the universe."

With these words, Albert Einstein challenged his colleagues who were developing a new theory - quantum mechanics. In his opinion, the Heisenberg uncertainty principle and the Schrödinger equation introduced an unhealthy uncertainty into the microcosm. He was sure that the Creator could not allow the world of electrons to be so strikingly different from the familiar world of Newtonian billiard balls. In fact, for many years, Einstein played the role of the devil's advocate in relation to quantum mechanics, inventing ingenious paradoxes designed to lead the creators of a new theory into a dead end. In doing so, however, he did a good deed, seriously perplexing the theoreticians of the opposite camp with his paradoxes and forcing them to think deeply about how to solve them, which is always useful when a new field of knowledge is being developed.

There is a strange irony of fate in the fact that Einstein went down in history as a principled opponent of quantum mechanics, although initially he himself stood at its origins. In particular, he received the Nobel Prize in Physics in 1921 not at all for the theory of relativity, but for explaining the photoelectric effect on the basis of new quantum concepts that literally swept the scientific world at the beginning of the 20th century.

Most of all, Einstein protested against the need to describe the phenomena of the microworld in terms of probabilities and wave functions ( cm. Quantum mechanics), and not from the usual position of coordinates and particle velocities. That's what he meant by "dice". He recognized that the description of the motion of electrons in terms of their speeds and coordinates contradicts the uncertainty principle. But, Einstein argued, there must be some other variables or parameters, taking into account which the quantum mechanical picture of the microworld will return to the path of integrity and determinism. That is, he insisted, it only seems to us that God is playing dice with us, because we do not understand everything. Thus, he was the first to formulate hypotheses for a latent variable in the equations of quantum mechanics. It consists in the fact that, in fact, electrons have fixed coordinates and speed, like Newton's billiard balls, and the uncertainty principle and the probabilistic approach to their determination in the framework of quantum mechanics are the result of the incompleteness of the theory itself, which is why it does not allow them determine for sure.

And so, in 1964, John Bell obtained a new and unexpected theoretical result for many. He proved that it is possible to conduct a certain experiment (details a little later), the results of which will determine whether quantum mechanical objects are really described by the probability distribution wave functions, as they are, or whether there is a hidden parameter that allows you to accurately describe their position and momentum, as at the Newtonian ball. Bell's theorem, as it is now called, shows that, as if there is a hidden parameter in quantum mechanical theory that affects any physical characteristic of a quantum particle, and in the absence of such a serial experiment, the statistical results of which will confirm or disprove the presence of hidden parameters in quantum mechanical theory. Relatively speaking, in one case the statistical ratio will be no more than 2:3, and in the other - no less than 3:4.

(Here I want to parenthetically point out that I was an undergraduate student at Stanford the year Bell proved his theory to him. Red-bearded and with a thick Irish accent, Bell was hard to miss. I remember standing in the science building corridor of Stanford Linear accelerator, and then he came out of his office in a state of extreme excitement and publicly announced that he had just discovered a really important and interesting thing.And although I have no evidence of this, I would very much like to hope that I that day I became an unwitting witness to its discovery.)

However, the experience proposed by Bell turned out to be simple only on paper and at first seemed almost impossible. The experiment was supposed to look like this: under external influence, the atom had to synchronously emit two particles, for example, two photons, and in opposite directions. After that, it was necessary to capture these particles and instrumentally determine the direction of the spin of each and do this a thousandfold in order to accumulate sufficient statistics to confirm or refute the existence of a hidden parameter according to Bell's theorem (in the language of mathematical statistics, it was necessary to calculate correlation coefficients).

The most unpleasant surprise for everyone after the publication of Bell's theorem was precisely the need to conduct a colossal series of experiments, which at that time seemed practically impossible, in order to obtain a statistically reliable picture. However, less than a decade later, experimental scientists not only developed and built the necessary equipment, but also accumulated a sufficient amount of data for statistical processing. Without going into technical details, I will only say that then, in the mid-sixties, the complexity of this task seemed so monstrous that the probability of its implementation seemed to be equal to that of someone planning to put a million trained monkeys from the proverb at typewriters in the hope of finding among the fruits of their collective labor, a creation equal to Shakespeare.

When the results of the experiments were summarized in the early 1970s, everything became crystal clear. The probability distribution wave function accurately describes the movement of particles from the source to the sensor. Therefore, the equations of wave quantum mechanics do not contain hidden variables. This is the only known case in the history of science when a brilliant theorist proved possibility experimental verification of the hypothesis and gave justification method such a test, brilliant experimenters with titanic efforts carried out a complex, expensive and protracted experiment, which in the end only confirmed the already dominant theory and did not even introduce anything new into it, as a result of which everyone felt cruelly deceived in their expectations!

However, not all work was in vain. More recently, scientists and engineers, much to their own surprise, have found Bell's theorem to be of very worthy practical application. The two particles emitted by the Bell source are coherent(have the same wave phase), since they are emitted synchronously. And this property of theirs is now going to be used in cryptography to encrypt highly secret messages sent over two separate channels. When intercepting and attempting to decrypt a message via one of the channels, coherence is instantly broken (again, due to the uncertainty principle), and the message inevitably and instantly self-destructs at the moment when the connection between the particles is broken.

HIDDEN OPTIONS- hypothetical. add. variables unknown at the present time, the values of which should fully characterize the state of the system and determine its future more completely than quantum mechanics. state vector. It is believed that with the help of S. p. from statistical. descriptions of micro-objects, you can go to dynamic. regularities, at to-rykh unequivocally connected in time themselves physical. values, not their statistics. distribution (see Causality). With. n. are usually considered decomp. fields or coordinates and momenta of smaller, component parts of quantum particles. However, after the discovery (of the composite particles of hadrons), it turned out that their behavior is subordinate, like the behavior of the hadrons themselves.

According to von Neumann's theorem, no theory with quantum mechanics can reproduce all the consequences of quantum mechanics, however, as it turned out later, J. von Neumann's proof was based on assumptions, generally speaking, optional for any model S. p. A weighty argument in favor of the existence of S. p. put forward A. Einstein (A. Einstein), B. Podolsky (V. Podolsky) and N. Rosen (N. Rosen) in 1935 (the so-called. Einstein - Podolsky - Rosen paradox), the essence of which is that certain characteristics of quantum particles (in particular, spin projections) can be measured without exposing the particles to force. A new incentive to experiment. verification of the Einstein-Podolsky-Rosen paradox became proven in 1951 Bell inequality, which made it possible to direct experiments. verification of the hypothesis about S. p. These inequalities demonstrate the difference between the predictions of quantum mechanics and any theories of S. p., which do not allow the existence of physical. processes propagating at superluminal speeds. Experiments carried out in a number of laboratories around the world confirmed the predictions of quantum mechanics about the existence of stronger correlations between particles than any local theories of S.p. predict. According to these theories, the results of an experiment conducted on one of the particles are determined only by this experiment itself and do not depend on the results experiment, which can be carried out on another particle that is not associated with the first force interactions.

Lit.: 1) Sudbury A., Quantum mechanics and elementary particles, trans. from English, M., 1989; 2) A. A. Grib, Bell’s Inequalities and Experimental Verification of Quantum Correlations at Macroscopic Distances, UFN, 1984, vol. 142, p. 619; 3) Spassky B. I., Moskovsky A. V., On nonlocality in quantum physics, UFN, 1984, vol. 142, p. 599; 4) Bom D., On the possibility of interpreting quantum mechanics on the basis of ideas about "hidden" parameters, in: Questions of causality in quantum mechanics, M., 1955, p. 34. G. Ya. Myakishev.

In quantum mechanics

The theory of hidden variables (TST) is a traditional, but not the only basis for constructing various types of Bell's theorem. The starting point can also be the recognition of the existence of a positive definite probability distribution function. Based on this assumption, without resorting to additional assumptions, Bell's paradoxes of various types are formulated and proved in the work. A specific example shows that a formal quantum calculation sometimes gives negative values appearing in the proof of joint probabilities. An attempt has been made to clarify physical sense this result, and an algorithm for measuring negative joint probabilities of this type is proposed.

Since the laws of quantum theory predict the results of an experiment, generally speaking, only statistically, then, based on classical point of view, it might be assumed that there are hidden parameters which, being unobservable in any ordinary experiment, actually determine the result of the experiment, as has always been considered previously in accordance with the principle of causality. Therefore, an attempt was made to invent such parameters within the framework of quantum mechanics.

In a narrow sense, applicable in quantum mechanics and theoretical physics of the microcosm, where the determinism of the laws of macroscopic physics ceases to operate, the theory of hidden parameters has served as an important tool of knowledge.

But the significance of the approach to the theory of hidden parameters, undertaken in the framework of the study of the microworld and quantum mechanical paradoxes, is not limited to this range of phenomena. Perhaps a broader, truly philosophical interpretation of the reasons why this phenomenon takes place in our world.

In the philosophy of knowledge

However, the raised question about hidden parameters is related not only to narrow physical problems. It is related to the general methodology of knowledge. A small excerpt from a treatise on understanding written by A. M. Nikiforov helps to understand the essence of this phenomenon:

To begin with, let's try to understand what understanding is at the usual everyday level. We can say that understanding is the process of reducing the incomprehensible to the understandable. That is, by means of available logical manipulations, we build a representation (model) of what was previously incomprehensible to us from representations that are understandable to us. […] There is another approach to understanding, when the presence of a certain entity or substance is declared, which has the necessary properties that ensure the existence of the phenomenon of interest to us ... It should be noted that this approach underlies the theory of relativity and quantum mechanics, which declare how, but not explain why. […] I must say that if the first approach is more rigorous and clear, then the second one is more powerful, universal and simple ... The first approach is widely used in science, and it can be considered dominant, but the second one is also used. An example of this is "hidden parameter theory"[highlighted by the author], according to which the discrepancy between theory and experiment is removed by introducing some hypothetical object. The parameters of this object are substituted into the formula, and it begins to coincide with the experiment.

In quantum mechanics, this theory has a significant scope, although it is not generally accepted.

Historical example

For many centuries, Euclid's geometry was considered the unshakable rock of science. For a long time before the beginning of physical research of the microworld and astrophysical measurements, there were no grounds to consider it incomplete. However, the situation changed in the first decade of the 20th century. A conceptual crisis was growing in physics, which Albert Einstein was able to resolve. Together with the resolution of particular problems - the coordination of observations with the predictions of the theories of that time ("saving the phenomenon") - in his work together with Niels Bohr, Einstein managed to draw a brilliant conclusion regarding the possibility of the influence of masses on the geometry of space and the speed of a moving object - at speeds commensurate with light, - for the course of local time for the given object.

In geometry, this was an epoch-making theoretical and practical discovery for cosmology, although it echoed the theoretical premises postulated by Hermann Minkowski, but it occupied a special place in modern cosmology.

The effect of the real influence of gravity on the geometry of space can be considered a "hidden parameter" in the classical theory of Euclid, but revealed in the theory of Einstein. Reasoning from the point of view of the methodology of cognition: in one conceptual (theoretical) system, a certain parameter can be hidden, and in another - become disclosed, in demand and theoretically justified. In the first case, its "non-disclosure" does not mean the absence of this parameter in nature as such. It's just that this parameter was not significant, and therefore was not found, was not introduced by any of the scientists into the "fabric" of this theory.

This situation quite clearly reveals the property of such "hidden parameters". This is not a denial of the predecessor theory, but a finding of objective limitations for its predictions. In the case considered above, the physical space is indeed Euclidean with high accuracy in the case of insufficiently strong gravitational fields acting within the given space (which is also the terrestrial field), but more and more it ceases to be so with a huge increase in the gravitational potential. The latter, in the observed nature, can manifest itself only in extraterrestrial space objects such as black holes and some other "exotic" space objects.

Notes

Links

I. Z. Tsekhmistro, V. I. Shtanko et al. "THE CONCEPT OF INTEGRITY" - CHAPTER 3 THE CONCEPT OF INTEGRITY AND EXPERIMENT: causality and non-locality in quantum physics (L. E. Pargamanik)

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