The formation of quantum mechanics as a consistent theory with specific physical foundations is largely associated with the work of W. Heisenberg, in which it was formulated ratio (principle) of uncertainties... This fundamental position of quantum mechanics reveals the physical meaning of its equations, and also determines its relationship with classical mechanics.

The uncertainty principle postulates: an object of the microworld cannot be in states in which the coordinates of its center of inertia and momentum simultaneously take on quite definite, exact values.

Quantitatively, this principle is formulated as follows. If a ∆x - the uncertainty of the coordinate value x , but ∆p - momentum uncertainty, then the product of these uncertainties in order of magnitude cannot be less than Planck's constant:

∆x ∆ p ≥ h.

It follows from the uncertainty principle that the more accurately one of the quantities included in the inequality is determined, the less accurately the value of the other is determined. No experiment can simultaneously accurately measure these dynamic variables, and this is not due to the influence of measuring instruments or their imperfections. The uncertainty relation reflects the objective properties of the microworld, arising from its corpuscular-wave dualism.

The fact that one and the same object manifests itself both as a particle and as a wave destroys traditional ideas deprives the description of processes of the usual clarity. The concept of a particle means an object enclosed in a small area of ​​space, while a wave propagates in its extended areas. It is impossible to imagine an object possessing these qualities at the same time, and you should not try. It is impossible to construct a model that is visual for human thinking, which would be adequate to the microworld. The equations of quantum mechanics, however, do not set such a goal. Their meaning consists in a mathematically adequate description of the properties of objects of the microworld and the processes occurring with them.

If we talk about the connection between quantum mechanics and classical mechanics, then the uncertainty relation is a quantum limitation of the applicability of classical mechanics to the objects of the microworld... Strictly speaking, the uncertainty relation applies to any physical system, however, since the wave nature of macroobjects practically does not manifest itself, the coordinates and momentum of such objects can be simultaneously measured with a sufficiently high accuracy. This means that it is sufficient to use the laws of classical mechanics to describe their motion. Let us recall that the situation is similar in relativistic mechanics (special theory of relativity): at speeds of motion much lower than the speed of light, the relativistic corrections become insignificant and the Lorentz transformations pass into Galileo's transformations.

So, the uncertainty relation for coordinates and momentum reflects the wave-particle dualism of the microworld and not related to the influence of measuring instruments... A similar uncertainty relation for energyE and timet :

∆E ∆ t ≥ h.

It follows from it that the energy of the system can be measured only with an accuracy not exceeding h /∆ t, Where ∆ t - the duration of the measurement. The reason for this uncertainty lies in the very process of interaction of the system (micro-object) withmeasuring instrument... For a stationary situation, the above inequality means that the interaction energy between the measuring device and the system can be taken into account only with an accuracy of h / ∆t... In the limiting case of instantaneous measurement, the ongoing exchange of energy turns out to be completely indefinite.

If under ∆ E the uncertainty of the value of the energy of a non-stationary state is understood, then ∆ t there is a characteristic time during which the values ​​of physical quantities in the system change significantly. From this, in particular, an important conclusion follows regarding the excited states of atoms and other microsystems: the energy of the excited level cannot be strictly determined, which indicates the presence natural width this level.

The objective properties of quantum systems reflect another fundamental position of quantum mechanics - Bohr's complementarity principle, Whereby obtaining by any experimental means of information about some physical quantities describing a micro-object is inevitably associated with the loss of information about some other quantities, additional to the first.

Mutually complementary are, in particular, the coordinate of the particle and its momentum (see above - the uncertainty principle), kinetic and potential energy, electric field strength and the number of photons.

The considered fundamental principles of quantum mechanics indicate that, due to the wave-particle duality of the microworld it studies, the determinism of classical physics is alien to it. A complete departure from visual modeling of processes gives particular interest to the question of what is the physical nature of de Broglie waves. In answering this question, it is customary to "start" from the behavior of photons. It is known that when passing a light beam through a semitransparent plate S part of the light passes through it, and part is reflected (Fig. 4).

Fig. four

What happens to the individual photons in this case? Experiments with light beams of very low intensity using modern technology ( BUT- photon detector), which allows you to monitor the behavior of each photon (the so-called photon counting mode), show that the splitting of an individual photon is out of the question (otherwise the light would change its frequency). It has been reliably established that some photons pass through the plate, and some are reflected from it. It means that identical particles inthe same conditions may behave differently,i.e., the behavior of an individual photon when it meets the surface of the plate cannot be predicted unambiguously.

Reflection of a photon from a plate or passing through it are random events. And the quantitative patterns of such events are described using the theory of probability. Photon can with probability w 1 pass through the plate and with probability w 2 bounce off it. The probability that one of these two alternative events will occur to a photon is equal to the sum of the probabilities: w 1 + w 2 = 1.

Similar experiments with a beam of electrons or other microparticles also show the probabilistic nature of the behavior of individual particles. In this way, the problem of quantum mechanics can be formulated as a predictionprobabilities of processes in the microworld, in contrast to the problem of classical mechanics - predict the reliability of events in the macrocosm.

It is known, however, that the probabilistic description is also used in classical statistical physics. So what's the fundamental difference? To answer this question, let us complicate the experiment on light reflection. Using a mirror S 2 turn the reflected beam by placing the detector A registering photons in the zone of its intersection with the transmitted beam, ie, we will provide the conditions for an interference experiment (Fig. 5).

Fig. five

As a result of interference, the light intensity, depending on the location of the mirror and the detector, will periodically change over the cross section of the overlapping region of the beams over a wide range (including vanishing). How do individual photons behave in this experiment? It turns out that in this case the two optical paths to the detector are no longer alternative (mutually exclusive) and therefore it is impossible to say which path the photon passed from the source to the detector. We have to admit that he could enter the detector in two ways simultaneously, eventually forming an interference pattern. An experiment with other microparticles gives a similar result: successively passing particles create the same picture as the photon flux.

This is already a cardinal difference from classical concepts: after all, it is impossible to imagine the motion of a particle simultaneously along two different paths. However, quantum mechanics does not pose such a problem. It predicts the result that the bright fringes have a high probability of a photon appearing.

Wave optics easily explains the result of an interference experiment using the principle of superposition, according to which light waves are added taking into account the ratio of their phases. In other words, the waves are first added in amplitude, taking into account the phase difference, a periodic distribution of the amplitude is formed, and then the detector registers the corresponding intensity (which corresponds to the mathematical operation of squaring in modulus, i.e., there is a loss of information about the phase distribution). In this case, the intensity distribution is periodic:

I = I 1 + I 2 + 2 A 1 A 2 cos (φ 1 – φ 2 ),

Where BUT , φ , I = | A | 2 –amplitude,phase and intensity waves, respectively, and indices 1, 2 indicate their belonging to the first or second of these waves. It is clear that for BUT 1 = BUT 2 and cos (φ 1 – φ 2 ) = – 1 intensity value I = 0 , which corresponds to the mutual damping of light waves (with their superposition and interaction in amplitude).

To interpret wave phenomena from a corpuscular point of view, the principle of superposition is transferred to quantum mechanics, i.e., the concept is introduced probability amplitudes - by analogy with optical waves: Ψ = BUT exp (iφ ). This means that the probability is the square of this value (modulo), i.e. W = |Ψ| 2 The probability amplitude is called in quantum mechanics wave function ... This concept was introduced in 1926 by the German physicist M. Born, thereby giving probabilistic interpretation de Broglie waves. Satisfaction of the superposition principle means that if Ψ 1 and Ψ 2 - the amplitude of the probability of the passage of the particle by the first and second paths, then the amplitude of the probability when passing both paths should be: Ψ = Ψ 1 + Ψ 2 . Then, formally, the assertion that “the particle passed two paths” acquires a wave meaning, and the probability W = |Ψ 1 + Ψ 2 | 2 exhibits property interference distribution.

In this way, the quantity describing the state of a physical system in quantum mechanics is the wave function of the system under the assumption of the validity of the principle of superposition... The basic equation of wave mechanics, the Schrödinger equation, is written with respect to the wave function. Therefore, one of the main tasks of quantum mechanics is to find the wave function corresponding to a given state of the system under study.

It is essential that the description of the state of a particle by means of the wave function is probabilistic in nature, since the square of the modulus of the wave function determines the probability of finding a particle at a given time in a certain limited volume... In this, quantum theory fundamentally differs from classical physics with its determinism.

At one time, it was precisely the high accuracy of predicting the behavior of macroobjects that classical mechanics owed its triumphal march. Naturally, among scientists for a long time there was an opinion that the progress of physics and science in general will be inherently associated with an increase in the accuracy and reliability of such predictions. The uncertainty principle and the probabilistic nature of the description of microsystems in quantum mechanics radically changed this point of view.

Then other extremes began to appear. Since the uncertainty principle implies impossibility of simultaneousdetermination of position and momentum, we can conclude that the state of the system at the initial moment of time is not precisely determined and, therefore, subsequent states cannot be predicted, i.e., principle of causality.

However, such a statement is possible only with a classical view of non-classical reality. In quantum mechanics, the state of a particle is completely determined by the wave function. Its value, given for a certain point in time, determines its subsequent values. Since causality acts as one of the manifestations of determinism, it is advisable in the case of quantum mechanics to talk about probabilistic determinism based on statistical laws, i.e., ensuring the higher accuracy, the more events of the same type are recorded. Therefore, the modern concept of determinism presupposes an organic combination, dialectical unity the need and accidents.

The development of quantum mechanics has thus had a noticeable impact on the progress of philosophical thought. From an epistemological point of view, of particular interest is the already mentioned conformity principle, formulated by N. Bohr in 1923, according to which any new, more general theory, which is a development of the classical one, does not reject it completely, but includes the classical theory, indicating the limits of its applicability and passing into it in certain limiting cases.

It is easy to see that the correspondence principle perfectly illustrates the relationship of classical mechanics and electrodynamics with the theory of relativity and quantum mechanics.

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Quantum mechanics

What is Quantum Mechanics?

Quantum mechanics (QM (QM); also known as quantum physics or quantum theory), including quantum field theory, is the field of physics that studies the laws of nature that manifest themselves at small distances and at low energies of atoms and subatomic particles. Classical physics - physics that existed before quantum mechanics, follows from quantum mechanics as its passage to the limit, which is valid only at large (macroscopic) scales. Quantum mechanics differs from classical physics in that energy, momentum and other quantities are often limited to discrete values ​​(quantization), objects have characteristics of both particles and waves (particle-wave dualism), and there are limitations on the accuracy with which quantities can be determined (uncertainty principle).

Quantum mechanics consistently follows from Max Planck's 1900 black-body radiation problem (published 1859) and Albert Einstein's 1905 work that proposed a quantum theory to explain the photoelectric effect (published 1887). Early quantum theory was deeply rethought in the mid-1920s.

The revised theory is formulated in the language of specially developed mathematical formalisms. In one of them, a mathematical function (wave function) provides information about the amplitude of the probability of position, momentum, and other physical characteristics of the particle.

Important areas of application of quantum theory are: quantum chemistry, superconducting magnets, light-emitting diodes, as well as laser, transistor and semiconductor devices such as a microprocessor, medical and research imaging such as magnetic resonance imaging and electron microscopy, and explanations of many biological and physical phenomena.

History of quantum mechanics

Scientific research into the wave nature of light began in the 17th and 18th centuries, when scientists Robert Hook, Christian Huygens, and Leonard Euler proposed a wave theory of light based on experimental observations. In 1803, Thomas Young, an English general scientist, conducted the famous double slit experiment, which he later described in a paper entitled "The Nature of Light and Colors." This experiment played an important role in the general acceptance of the wave theory of light.

In 1838, Michael Faraday discovered cathode rays. These studies were followed by Gustav Kirchhoff's formulation of the blackbody radiation problem in 1859, Ludwig Boltzmann's 1877 hypothesis that the energy states of a physical system could be discrete, and Max Planck's quantum hypothesis in 1900. Planck's hypothesis that energy is emitted and absorbed by a discrete "quantum" (or energy packets) exactly matches the observed patterns of blackbody radiation.

In 1896, Wilhelm Wien empirically determined the distribution law of blackbody radiation, named in his honor, Wien's law. Ludwig Boltzmann independently arrived at this result by analyzing Maxwell's equations. However, the law only worked at high frequencies and underestimated the emission at low frequencies. Later, Planck corrected this model using a statistical interpretation of Boltzmann's thermodynamics and proposed what is now called Planck's law, which led to the development of quantum mechanics.

After Max Planck solved the problem of blackbody radiation in 1900 (published 1859), Albert Einstein proposed a quantum theory to explain the photoelectric effect (1905, published 1887). In 1900-1910, the atomic theory and the corpuscular theory of light first became widely recognized as a scientific fact. Accordingly, these latter theories can be viewed as quantum theories of matter and electromagnetic radiation.

Among the first to study quantum phenomena in nature were Arthur Compton, C.W. Raman, and Peter Zeeman, each of whom is named after some quantum effects. Robert Andrews Milliken investigated the photoelectric effect experimentally, and Albert Einstein developed a theory for it. At the same time, Ernest Rutherford experimentally discovered the nuclear model of the atom, according to which Niels Bohr developed his theory of the structure of the atom, which was later confirmed by the experiments of Henry Moseley. In 1913, Peter Debye expanded Niels Bohr's theory of the structure of the atom by introducing elliptical orbits, a concept also proposed by Arnold Sommerfeld. This stage in the development of physics is known as the old quantum theory.

According to Planck, the energy (E) of the radiation quantum is proportional to the radiation frequency (v):

where h is Planck's constant.

Planck cautiously insisted that this was simply a mathematical expression of the processes of absorption and emission of radiation and had nothing to do with the physical reality of radiation itself. In fact, he saw his quantum hypothesis as a mathematical trick to get the answer right, rather than a major fundamental discovery. However, in 1905, Albert Einstein gave a physical interpretation to Planck's quantum hypothesis and used it to explain the photoelectric effect, in which the illumination of certain substances with light can cause the emission of electrons from the substance. For this work, Einstein received the 1921 Nobel Prize in Physics.

Einstein then refined this idea to show that an electromagnetic wave, which is light, can also be described as a particle (later called a photon), with discrete quantum energy that depends on the frequency of the wave.

During the first half of the 20th century, Max Planck, Niels Bohr, Werner Heisenberg, Louis de Broglie, Arthur Compton, Albert Einstein, Erwin Schrödinger, Max Born, John von Neumann, Paul Dirac, Enrico Fermi, Wolfgang Pauli, Max von Laue, Freeman Dyson, David Hilbert, Wilhelm Wien, Schatiendranath Bose, Arnold Sommerfeld and others laid the foundations of quantum mechanics. Niels Bohr's Copenhagen interpretation has been widely accepted.

In the mid-1920s, the development of quantum mechanics led to its becoming the standard formulation for atomic physics. In the summer of 1925, Bohr and Heisenberg published results that closed the old quantum theory. Out of respect for their particle-like behavior in certain processes and dimensions, light quanta were called photons (1926). From a simple postulate of Einstein, a flurry of discussions, theoretical constructions and experiments arose. Thus, entire fields of quantum physics emerged, which led to its widespread acceptance at the Fifth Solvay Congress in 1927.

It was found that subatomic particles and electromagnetic waves are not just particles or waves, but have certain properties of each of them. This is how the concept of wave-corpuscle dualism arose.

By 1930, quantum mechanics had been further unified and formulated in the works of David Hilbert, Paul Dirac, and John von Neumann, which placed great emphasis on measurement, the statistical nature of our knowledge of reality, and philosophical thinking about the "observer." She subsequently penetrated many disciplines, including quantum chemistry, quantum electronics, quantum optics, and quantum information science. Her theoretical modern developments include string theory and quantum gravity theories. It also provides a satisfying explanation of many features of the modern periodic table of elements and describes the behavior of atoms in chemical reactions and the movement of electrons in computer semiconductors, and therefore plays a critical role in many modern technologies.

Although quantum mechanics was built to describe the microworld, it is also needed to explain some macroscopic phenomena such as superconductivity and superfluidity.

What does the word quantum mean?

The word quantum comes from the Latin "quantum", which means "how much" or "how much." In quantum mechanics, a quantum means a discrete unit assigned to certain physical quantities, such as the energy of an atom at rest. The discovery that particles are discrete packets of energy with wave-like properties led to the creation of a branch of physics dealing with atomic and subatomic systems, which is today called quantum mechanics. It lays the foundation for the mathematical foundation for many areas of physics and chemistry, including condensed matter physics, solid state physics, atomic physics, molecular physics, computational physics, computational chemistry, quantum chemistry, particle physics, nuclear chemistry, and nuclear physics. Some fundamental aspects of the theory are still being actively studied.

The importance of quantum mechanics

Quantum mechanics is essential for understanding the behavior of systems at atomic and smaller distance scales. If the physical nature of the atom were described exclusively by classical mechanics, then the electrons would not have to revolve around the nucleus, since the orbital electrons should emit radiation (due to circular motion) and ultimately collide with the nucleus due to the energy loss due to radiation. Such a system could not explain the stability of atoms. Instead, electrons are in vague, non-deterministic, smeared, probabilistic wave-particle orbitals near the nucleus, contrary to the traditional concepts of classical mechanics and electromagnetism.

Quantum mechanics was originally developed to better explain and describe the atom, especially the differences in the spectra of light emitted by different isotopes of the same chemical element, and the description of subatomic particles. In short, the quantum mechanical model of the atom has proven to be strikingly successful in an area where classical mechanics and electromagnetism have been helpless.

Quantum mechanics includes four classes of phenomena that classical physics cannot explain:

• quantization of individual physical properties
• quantum entanglement
• uncertainty principle
• wave-particle dualism

Mathematical Foundations of Quantum Mechanics

In a mathematically rigorous formulation of quantum mechanics developed by Paul Dirac, David Hilbert, John von Neumann, and Hermann Weil, the possible states of a quantum mechanical system are symbolized by unit vectors (called state vectors). Formally, they belong to a complex separable Hilbert space - in other words, the state space or the related Hilbert space of the system, and are determined up to the product by a complex number with unit modulus (phase factor). In other words, the possible states are points in the projective space of the Hilbert space, usually called the complex projective space. The exact nature of this Hilbert space depends on the system - for example, the state space of position and momentum is the space of square-integrable functions, while the state space for the spin of one proton is just the direct product of two complex planes. Each physical quantity is represented by a hypermaximal Hermitian (more precisely: self-adjoint) linear operator acting on the state space. Each eigenstate of a physical quantity corresponds to the eigenvector of the operator, and the associated eigenvalue corresponds to the value of the physical quantity in this eigenstate. If the spectrum of the operator is discrete, the physical quantity can only take discrete eigenvalues.

In the formalism of quantum mechanics, the state of a system at a given moment is described by a complex wave function, also called a state vector in a complex vector space. This abstract mathematical object allows you to calculate the probabilities of the outcomes of specific experiments. For example, it allows you to calculate the probability of finding an electron in a certain area around the nucleus at a certain time. Unlike classical mechanics, simultaneous predictions with arbitrary precision can never be made here for conjugate variables such as position and momentum. For example, we can assume that electrons (with some probability) are somewhere within a given region of space, but their exact location is unknown. Regions of constant probability, often called "clouds," can be drawn around the nucleus of an atom to represent where the electron is most likely to be. The Heisenberg Uncertainty Principle quantifies the inability to accurately localize a particle with a given momentum, which is a quantity conjugate to a position.

According to one of the interpretations, as a result of the measurement, the wave function containing information about the probability of the state of the system decays from a given initial state to a certain eigenstate. Possible measurement results are the eigenvalues ​​of an operator representing a physical quantity - which explains the choice of the Hermitian operator, in which all eigenvalues ​​are real numbers. The probability distribution of a physical quantity in a given state can be found by calculating the spectral decomposition of the corresponding operator. The Heisenberg uncertainty principle is represented by a formula in which the operators corresponding to certain quantities do not commute.

Measurement in quantum mechanics

The probabilistic nature of quantum mechanics thus follows from the act of measurement. This is one of the most difficult aspects of quantum systems to understand, and it was a central theme in Bohr's famous debate with Einstein, in which both scientists tried to clarify these fundamental principles through thought experiments. For decades after the formulation of quantum mechanics, the question of what constitutes a "dimension" has been extensively studied. New interpretations of quantum mechanics have been formulated to do away with the concept of "wave function collapse". The basic idea is that when a quantum system interacts with a measuring apparatus, their respective wave functions become entangled, so that the original quantum system ceases to exist as an independent entity.

The probabilistic nature of the predictions of quantum mechanics

Typically, quantum mechanics does not associate specific values. Instead, she makes predictions using a probability distribution; that is, it describes the likelihood of obtaining possible results from a measurement of a physical quantity. Often these results are deformed, like probability density clouds, by many processes. Probability density clouds are an approximation (but better than Bohr's model) in which the location of an electron is given by a probability function, wave functions corresponding to eigenvalues, such that the probability is the square of the modulus of the complex amplitude, or the quantum state of nuclear attraction. Naturally, these probabilities will depend on the quantum state at the "moment" of the measurement. Therefore, the uncertainty is introduced into the measured value. There are, however, some states that are associated with certain values ​​of a particular physical quantity. They are called eigenstates of a physical quantity ("eigen" can be translated from German as "inherent" or "inherent").

It is natural and intuitive that everything in everyday life (all physical quantities) have their own meanings. Everything seems to have a certain position, a certain moment, a certain energy, and a certain time of the event. However, quantum mechanics does not indicate the exact values ​​of the position and momentum of a particle (since these are conjugate pairs) or its energy and time (since they are also conjugate pairs); more precisely, it only provides the range of probabilities with which this particle can have a given momentum and momentum probability. Therefore, it is advisable to distinguish between states that have undefined values ​​and states that have definite values ​​(eigenstates). As a rule, we are not interested in a system in which a particle has no physical value of its own. However, when measuring a physical quantity, the wave function instantly takes on the eigenvalue (or "generalized" eigenvalue) of this quantity. This process is called the collapse of the wave function, a controversial and much discussed process in which the system under study is expanded by adding a measuring device to it. If we know the corresponding wave function immediately before the measurement, then we can calculate the probability that the wave function will pass into each of the possible eigenstates. For example, a free particle in the previous example usually has a wave function, which is a wave packet centered around some mean position x0 (having no position and momentum eigenstates). When the position of a particle is measured, it is impossible to predict the result with certainty. It is quite probable, but not certain, that it will be near x0, where the amplitude of the wave function is large. After making a measurement, having received some result x, the wave function collapses into an eigenfunction of the position operator centered at x.

Schrödinger equation in quantum mechanics

The temporal evolution of a quantum state is described by the Schrödinger equation, in which the Hamiltonian (the operator corresponding to the total energy of the system) generates the temporal evolution. The temporal evolution of wave functions is deterministic in the sense that - given what the wave function was at the initial moment in time - it is possible to make a clear prediction of what the wave function will be at any time in the future.

On the other hand, during the measurement, the change in the original wave function to another, later wave function will not be deterministic, but will be unpredictable (i.e., random). An emulation of time evolution can be seen here.

Wave functions change over time. The Schrödinger equation describes the change in wave functions in time, and plays a role similar to the role of Newton's second law in classical mechanics. The Schrödinger equation, applied to the above example of a free particle, predicts that the center of the wave packet will move through space at a constant speed (like a classical particle in the absence of forces acting on it). However, the wave packet will also blur over time, which means that the position becomes more uncertain over time. It also has the effect of converting the position eigenfunction (which can be viewed as an infinitely sharp peak of the wave packet) into an expanded wave packet that no longer represents the (defined) position eigenvalue.

Some wave functions generate probability distributions that are constant or independent of time - for example, when, in a steady state with constant energy, time disappears from the modulus of the square of the wave function. Many systems that are considered dynamic in classical mechanics are described in quantum mechanics by such "static" wave functions. For example, one electron in an unexcited atom is represented classically as a particle moving in a circular trajectory around an atomic nucleus, while in quantum mechanics it is described by a static, spherically symmetric wave function surrounding the nucleus (Fig. 1) (note, however, that only the lowest states of the orbital angular momentum, denoted as s, are spherically symmetric).

Schrödinger's equation acts on the entire amplitude of the probability, and not only on its absolute value. While the absolute value of the probability amplitude contains information about probabilities, its phase contains information about the mutual influence between quantum states. This gives rise to "wavy" behavior of quantum states. As it turns out, analytical solutions of the Schrödinger equation are possible only for a very small number of Hamiltonians of relatively simple models, such as a quantum harmonic oscillator, a particle in a box, an ion of a hydrogen molecule, and a hydrogen atom - these are the most important representatives of such models. Even a helium atom, which contains only one more electron than a hydrogen atom, has not succumbed to any attempt at a purely analytical solution.

However, there are several methods for obtaining approximate solutions. An important technique known as perturbation theory takes an analytical result from a simple quantum mechanical model and generates a result for a more complex model that differs from the simpler model (for example) by adding the energy of a weak potential field. Another approach is the "semiclassical approximation" method, which is applied to systems for which quantum mechanics is applied only to weak (small) deviations from classical behavior. These deviations can then be calculated based on the classic movement. This approach is especially important when studying quantum chaos.

Mathematically Equivalent Formulations of Quantum Mechanics

There are numerous mathematically equivalent formulations of quantum mechanics. One of the oldest and most frequently used formulations is the "transformation theory" proposed by Paul Dirac, which combines and generalizes the two earliest formulations of quantum mechanics - matrix mechanics (created by Werner Heisenberg) and wave mechanics (created by Erwin Schrödinger).

Given that Werner Heisenberg was awarded the Nobel Prize in Physics in 1932 for the creation of quantum mechanics, Max Born's role in the development of CM was overlooked until he was awarded the Nobel Prize in 1954. This role is mentioned in the 2005 biography of Bourne, which describes his role in the matrix formulation of quantum mechanics, as well as the use of probability amplitudes. In 1940, Heisenberg himself admits in the anniversary collection in honor of Max Planck that he learned about matrices from Born. In matrix formulation, the instantaneous state of a quantum system determines the probabilities of its measurable properties or physical quantities. Examples of quantities include energy, position, momentum, and orbital angular momentum. Physical quantities can be either continuous (for example, the position of a particle) or discrete (for example, the energy of an electron bound to a hydrogen atom). Feynman path integrals are an alternative formulation of quantum mechanics, in which the quantum mechanical amplitude is considered as the sum over all possible classical and non-classical trajectories between the initial and final states. It is a quantum mechanical analogue of the principle of least action in classical mechanics.

The laws of quantum mechanics

The laws of quantum mechanics are fundamental. It is argued that the state space of the system is Hilbert, and the physical quantities of this system are Hermitian operators acting in this space, although it is not said which are these Hilbert spaces or which are these operators. They can be chosen appropriately to quantify the quantum system. An important guideline for making these decisions is the correspondence principle, which states that the predictions of quantum mechanics are reduced to classical mechanics, when the system goes into the high-energy region or, which is the same, into the region of large quantum numbers, that is, while an individual particle possesses a certain degree of randomness, in systems containing millions of particles, averaged values ​​prevail and, when tending to the high-energy limit, the statistical probability of random behavior tends to zero. In other words, classical mechanics is simply the quantum mechanics of large systems. This "high energy" limit is known as the classical or match limit. Thus, the solution can even start with a well-established classical model of a particular system, and then try to guess the basic quantum model that would give rise to such a classical model when passing to the correspondence limit.

When quantum mechanics was originally formulated, it was applied to models whose limit of compliance was nonrelativistic classical mechanics. For example, the well-known model of a quantum harmonic oscillator uses an explicitly nonrelativistic expression for the kinetic energy of the oscillator and is thus a quantum version of the classical harmonic oscillator.

Interaction with other scientific theories

Early attempts to combine quantum mechanics with special relativity involved replacing the Schrödinger equation with covariant equations such as the Klein-Gordon equation or the Dirac equation. Although these theories were successful in explaining many experimental results, they had certain unsatisfactory qualities arising from the fact that they did not take into account the relativistic creation and annihilation of particles. Fully relativistic quantum theory required the development of quantum field theory, which uses the quantization of the field (rather than a fixed set of particles). The first full-fledged quantum field theory, quantum electrodynamics, provides a complete quantum description of electromagnetic interaction. The full apparatus of quantum field theory is often not required to describe electrodynamic systems. A simpler approach, used since the inception of quantum mechanics, is to consider charged particles as quantum mechanical objects, which are acted upon by a classical electromagnetic field. For example, the elementary quantum model of the hydrogen atom describes the electric field of the hydrogen atom using the classical expression for the Coulomb potential:

E2 / (4πε0r)

This "semiclassical" approach does not work if quantum fluctuations of the electromagnetic field play an important role, for example, when photons are emitted by charged particles.

Quantum field theories have also been developed for strong and weak nuclear forces. Quantum field theory for strong nuclear interactions is called quantum chromodynamics and describes the interactions of subnuclear particles such as quarks and gluons. Weak nuclear and electromagnetic forces have been combined in their quantized forms into a unified quantum field theory (known as electroweak theory) by physicists Abdus Salam, Sheldon Glashow, and Steven Weinberg. For this work, all three received the Nobel Prize in Physics in 1979.

It turned out to be difficult to build quantum models for the fourth remaining fundamental force - gravity. Semiclassical approximations have been made, which have led to predictions such as Hawking radiation. However, the formulation of a complete theory of quantum gravity is hampered by obvious incompatibilities between general relativity (which is the most accurate theory of gravity currently known) and some of the fundamentals of quantum theory. Resolving these incompatibilities is an area of ​​active research and theories such as string theory - one possible candidate for a future theory of quantum gravity.

Classical mechanics was also expanded into a complex field, with complex classical mechanics beginning to manifest itself like quantum mechanics.

Connection of quantum mechanics with classical mechanics

The predictions of quantum mechanics have been confirmed experimentally with a very high degree of accuracy. According to the principle of correspondence between classical and quantum mechanics, all objects obey the laws of quantum mechanics, and classical mechanics is only an approximation for large systems of objects (or statistical quantum mechanics for a large set of particles). Thus, the laws of classical mechanics follow from the laws of quantum mechanics as a statistical average when tending to a very large limiting value of the number of elements in a system or the values ​​of quantum numbers. However, chaotic systems lack good quantum numbers, and quantum chaos explores the relationship between classical and quantum descriptions of these systems.

Quantum coherence is a significant difference between classical and quantum theories, illustrated by the Einstein-Podolsky-Rosen (EPR) paradox, it was an attack on the well-known philosophical interpretation of quantum mechanics through an appeal to local realism. Quantum interference involves the addition of probability amplitudes, while classical "waves" involve the addition of intensities. For microscopic bodies, the length of the system is much less than the coherence length, which leads to entanglement at long distances and other nonlocal phenomena characteristic of quantum systems. Quantum coherence usually does not manifest itself on a macroscopic scale, although an exception to this rule can occur at extremely low temperatures (i.e., when approaching absolute zero), at which quantum behavior can manifest itself on a macroscopic scale. This is consistent with the following observations:

Many of the macroscopic properties of a classical system are a direct consequence of the quantum behavior of its parts. For example, the stability of the main part of matter (consisting of atoms and molecules, which would quickly collapse under the action of electrical forces alone), the rigidity of solids, as well as the mechanical, thermal, chemical, optical and magnetic properties of matter are the result of the interaction of electric charges in accordance with the rules of quantum mechanics.

While the seemingly "exotic" behavior of matter postulated by quantum mechanics and the theory of relativity becomes more apparent when working with particles of very small size or when traveling at speeds approaching the speed of light, the laws of classical, often called "Newtonian" physics remain accurate in predicting the behavior of the overwhelming number of "large" objects (of the order of the size of large molecules or even larger) and at speeds much lower than the speed of light.

What is the difference between quantum mechanics and classical mechanics?

Classical and quantum mechanics are very different in that they use very different kinematic descriptions.

According to the well-established opinion of Niels Bohr, the study of quantum-mechanical phenomena requires experiments with a complete description of all the devices of the system, preparatory, intermediate and final measurements. Descriptions are presented in macroscopic terms expressed in common language, supplemented by concepts of classical mechanics. The initial conditions and the final state of the system are respectively described by the position in the configuration space, for example, in the coordinate space, or in some equivalent space, such as momentum space. Quantum mechanics does not allow for a completely accurate description, both in terms of position and momentum, an accurate deterministic and causal prediction of the final state based on initial conditions or "state" (in the classical sense of the word). In this sense, promoted by Bohr in his mature writings, a quantum phenomenon is a process of transition from an initial to a final state, and not an instantaneous "state" in the classical sense of the word. Thus, there are two types of processes in quantum mechanics: stationary and transient. For stationary processes, the start and end positions are the same. For the transitional, they are different. It is obvious by definition that if only the initial condition is given, then the process is not defined. Taking into account the initial conditions, the prediction of the final state is possible, but only at the probabilistic level, since the Schrödinger equation is deterministic for the evolution of the wave function, and the wave function describes the system only in the probabilistic sense.

In many experiments it is possible to take the initial and final states of the system as a particle. In some cases, it turns out that there are potentially several spatially distinguishable paths or trajectories along which a particle can go from the initial to the final state. An important feature of the quantum kinematic description is that it does not allow one to unambiguously determine which of these paths makes the transition between states. Only the initial and final conditions are defined, and, as indicated in the previous paragraph, they are only defined as precisely as the spatial configuration or its equivalent would permit. In every case that requires a quantum kinematic description, there is always a good reason for this limitation of kinematic accuracy. The reason is that for the experimental finding of a particle in a certain position, it must be stationary; for the experimental finding of a particle with a certain momentum, it must be in free motion; these two requirements are logically incompatible.

Initially, classical kinematics does not require an experimental description of its phenomena. This allows you to fully accurately describe the instantaneous state of the system by a position (point) in phase space - the Cartesian product of the configuration and momentum spaces. This description simply assumes, or imagines, the state as a physical entity, without worrying about its experimental measurability. Such a description of the initial state, together with Newton's laws of motion, makes it possible to accurately make a deterministic and causal prediction of the final state together with a certain trajectory of the evolution of the system. For this, Hamiltonian dynamics can be used. Classical kinematics also allows you to describe the process, similar to the description of the initial and final states used by quantum mechanics. Lagrangian mechanics allows you to do this. For processes in which it is necessary to take into account the magnitude of the action of the order of several Planck constants, classical kinematics is not suitable; it requires the use of quantum mechanics.

General theory of relativity

Even though the defining postulates of general relativity and Einstein's quantum theory are unconditionally supported by rigorous and repetitive empirical evidence, and although they do not contradict each other theoretically (at least in their primary statements), they have proved extremely difficult to integrate into one coherent , a single model.

Gravity can be neglected in many areas of particle physics, so the unification between general relativity and quantum mechanics is not a pressing issue in these particular applications. However, the lack of a correct theory of quantum gravity is an important issue in physical cosmology and physicists' search for an elegant "Theory of Everything" (TV). Therefore, solving all the inconsistencies between both theories is one of the main goals for physics of the 20th and 21st centuries. Many prominent physicists, including Stephen Hawking, have labored over the years to try to discover the theory behind everything. This TV will not only combine different models of subatomic physics, but also deduce the four fundamental forces of nature - strong interaction, electromagnetism, weak interaction and gravity - from one force or phenomenon. While Stephen Hawking originally believed in TV, after considering Gödel's incompleteness theorem, he concluded that such a theory was not feasible, and stated this publicly in his lecture "Gödel and the End of Physics" (2002).

Basic theories of quantum mechanics

The quest to unify fundamental forces through quantum mechanics is still ongoing. Quantum electrodynamics (or "quantum electromagnetism"), which is currently (at least in perturbative mode) the most accurate proven physical theory in competition with general relativity, successfully combines weak nuclear interactions into electroweak interactions, and work is currently underway on combining electroweak and strong interactions into electro-strong interactions. Current forecasts state that around 1014 GeV, the three aforementioned forces merge into a single unified field. Aside from this "grandiose unification", it is assumed that gravity can be combined with the other three gauge symmetries, which is expected to occur at around 1019 GeV. However - and while special relativity is carefully incorporated into quantum electrodynamics - extended general relativity, currently the best theory describing the forces of gravity is not fully incorporated into quantum theory. One of those who develop a consistent theory of everything, Edward Witten, a theoretical physicist, formulated M-theory, which is an attempt to expound supersymmetry based on superstring theory. M-theory assumes that our visible 4-dimensional space is actually an 11-dimensional space-time continuum containing ten space dimensions and one time dimension, although 7 space dimensions at low energies are completely "condensed" (or infinitely curved) and not easy to measure or research.

Another popular theory, Loop quantum gravity (LQG), is a theory first proposed by Carlo Rovelli that describes the quantum properties of gravity. It is also a theory of quantum space and quantum time, since in general relativity the geometric properties of space-time are a manifestation of gravity. LQG is an attempt to combine and adapt standard quantum mechanics and standard general relativity. The main result of the theory is a physical picture in which space is grainy. The graininess is a direct consequence of quantization. It has the same graininess of photons in the quantum theory of electromagnetism or discrete energy levels of atoms. But here the space itself is discrete. More precisely, space can be viewed as an extremely thin fabric or net "woven" from finite loops. These loop networks are called spin networks. The evolution of a spin network over time is called spin foam. The predicted size of this structure is the Planck length, which is approximately 1.616 x 10-35 m. According to theory, there is no point in a shorter length than this. Consequently, LQG predicts that not only matter, but space itself, has an atomic structure.

Philosophical aspects of quantum mechanics

Since its inception, many of the paradoxical aspects and results of quantum mechanics have caused violent philosophical debate and many interpretations. Even fundamental questions, such as Max Born's ground rules about probability amplitude and probability distribution, took decades to be appreciated by society and many leading scientists. Richard Feynman once said, “I think I can safely say that nobody understands quantum mechanics. According to Steven Weinberg,“ right now, in my opinion, there is no absolutely satisfactory interpretation of quantum mechanics.

The Copenhagen interpretation - largely thanks to Niels Bohr and Werner Heisenberg - has remained the most acceptable among physicists for 75 years after its proclamation. According to this interpretation, the probabilistic nature of quantum mechanics is not a temporary feature that will eventually be replaced by a deterministic theory, but should be seen as a final rejection of the classical idea of ​​"causality". In addition, it is believed that any clearly defined applications of the quantum mechanical formalism should always make reference to the experimental design, due to the conjugate nature of the evidence obtained in various experimental situations.

Albert Einstein, as one of the founders of quantum theory, himself did not accept some of the more philosophical or metaphysical interpretations of quantum mechanics, such as the rejection of determinism and causation. His most quoted famous response to this approach is, "God does not play dice." He rejected the concept that the state of a physical system depends on an experimental measurement setup. He believed that natural phenomena occur according to their own laws, regardless of whether they are observed and how. In this regard, it is supported by the currently accepted definition of a quantum state, which remains invariant with an arbitrary choice of the configuration space for its representation, that is, the observation method. He also believed that quantum mechanics should be based on a theory that carefully and directly expresses a rule that rejects the principle of action at a distance; in other words, he insisted on the principle of locality. He considered, but theoretically reasonably rejected the private concept of hidden variables in order to avoid uncertainty or lack of causality in quantum mechanical measurements. He believed that quantum mechanics was at that time valid, but not the final and unshakable theory of quantum phenomena. He believed that its future replacement would require deep conceptual advances, and that this would not happen so quickly and easily. Bohr-Einstein's discussions provide a vivid criticism of the Copenhagen interpretation from an epistemological point of view.

John Bell showed that this EPR paradox led to experimentally testable differences between quantum mechanics and theories that rely on the addition of hidden variables. Experiments have been carried out to confirm the accuracy of quantum mechanics, thus demonstrating that quantum mechanics cannot be improved by adding hidden variables. Alain Aspect's initial experiments in 1982 and many subsequent experiments since then have conclusively confirmed quantum entanglement.

Entanglement, as Bell's experiments showed, does not violate causality, since no information is transmitted. Quantum entanglement forms the basis of quantum cryptography, which is proposed for use in highly secure commercial applications in banking and government.

Everett's many-worlds interpretation, formulated in 1956, believes that all the possibilities described by quantum theory arise simultaneously in a multiverse, consisting mainly of independent parallel universes. This is not achieved by introducing some "new axiom" into quantum mechanics, but, on the contrary, is achieved by removing the axiom of wave packet decay. All possible sequential states of the measured system and the measuring device (including the observer) are present in the real physical - and not only in the formal mathematical, as in other interpretations - quantum superposition. This superposition of successive combinations of states of different systems is called an entangled state. While the multiverse is deterministic, we perceive non-deterministic behavior, of a random nature, since we can only observe the universe (i.e., the contribution of a compatible state to the aforementioned superposition) in which we, as observers, inhabit. Everett's interpretation fits perfectly with John Bell's experiments and makes them intuitive. However, according to the theory of quantum decoherence, these "parallel universes" will never be available to us. Inaccessibility can be understood as follows: as soon as the measurement is made, the measured system becomes entangled both with the physicist who measured it and with a huge number of other particles, some of which are photons flying away at the speed of light to the other end of the universe. To prove that the wave function did not decay, it is necessary to bring all these particles back and measure them again along with the system that was originally measured. Not only is this completely impractical, but even if theoretically it could be done, then any evidence that the original measurement took place (including the memory of the physicist) would have to be destroyed. In light of these Bell experiments, Kramer formulated his transactional interpretation in 1986. In the late 1990s, relational quantum mechanics emerged as a modern derivative of the Copenhagen interpretation.

Quantum mechanics has had tremendous success in explaining many of the features of our universe. Quantum mechanics is often the only tool available that can reveal the individual behavior of subatomic particles that make up all forms of matter (electrons, protons, neutrons, photons, etc.). Quantum mechanics has greatly influenced string theory, a contender for the theory of everything (and the Theory of Everything).

Quantum mechanics is also critical to understanding how individual atoms create covalent bonds to form molecules. The application of quantum mechanics to chemistry is called quantum chemistry. Relativistic quantum mechanics can, in principle, describe most of chemistry mathematically. Quantum mechanics can also provide a quantitative understanding of the processes of ionic and covalent bonding, clearly showing which molecules are energetically suitable for other molecules and at what values ​​of energy. In addition, most of the calculations in modern computational chemistry rely on quantum mechanics.

In many industries, modern technology operates at a scale where quantum effects are significant.

Quantum physics in electronics

Many modern electronic devices are designed using quantum mechanics. For example, a laser, a transistor (and thus a microchip), an electron microscope, and magnetic resonance imaging (MRI). The study of semiconductors led to the invention of the diode and transistor, which are indispensable components of modern electronic systems, computer and telecommunication devices. Another application is a light emitting diode, which is a highly efficient light source.

Many electronic devices are powered by quantum tunneling. It is even present in a simple switch. The switch would not work if electrons could not quantum tunnel through the oxide layer on the metal contact surfaces. Flash memory chips, the main part of USB storage devices, use quantum tunneling to erase information in their cells. Some negative differential resistance devices, such as a resonant tunneling diode, also use the quantum tunneling effect. Unlike classical diodes, the current in it flows under the action of resonant tunneling through two potential barriers. Its mode of operation with negative resistance can only be explained by quantum mechanics: as the energy of the state of bound carriers approaches the Fermi level, the tunneling current increases. With distance from the Fermi level, the current decreases. Quantum mechanics is vital to understanding and designing these types of electronic devices.

Quantum cryptography

Researchers are currently looking for reliable methods for directly manipulating quantum states. Efforts are being made to fully develop quantum cryptography, which theoretically will guarantee the secure transfer of information.

Quantum computing

A more distant goal is to develop quantum computers that are expected to perform certain computational tasks exponentially faster than classical computers. Instead of classical bits, quantum computers use qubits, which can be in a superposition of states. Another active research topic is quantum teleportation, which deals with methods of transmitting quantum information over arbitrary distances.

Quantum Effects

While quantum mechanics is primarily applied to atomic systems with less matter and energy, some systems exhibit quantum mechanical effects on a large scale. Superfluidity, the ability of a fluid flow to move without friction at temperatures near absolute zero, is one well-known example of such effects. Closely related to this phenomenon is the phenomenon of superconductivity - an electron gas flow (electric current) moving without resistance in a conducting material at sufficiently low temperatures. The fractional quantum Hall effect is a topologically ordered state that corresponds to long-range quantum entanglement models. States with different topological orders (or different configurations of long-range entanglement) cannot change states into each other without phase transformations.

Quantum theory

Quantum theory also contains accurate descriptions of many previously unexplained phenomena, such as blackbody radiation and the stability of orbiting electrons in atoms. She also provided insight into the workings of many different biological systems, including olfactory receptors and protein structures. Recent research on photosynthesis has shown that quantum correlations play an important role in this fundamental process in plants and many other organisms. However, classical physics can often provide good approximations to the results obtained by quantum physics, usually under conditions of large numbers of particles or large quantum numbers. Since classical formulas are much simpler and easier to compute than quantum formulas, the use of classical approximations is preferable when the system is large enough to make the effects of quantum mechanics negligible.

Free particle motion

For example, consider a free particle. In quantum mechanics, wave-particle duality is observed, so that the properties of a particle can be described as properties of a wave. Thus, a quantum state can be represented as a wave of arbitrary shape and extending in space as a wave function. The position and momentum of a particle are physical quantities. The uncertainty principle states that position and momentum cannot be accurately measured at the same time. However, it is possible to measure the position (without measuring momentum) of a moving free particle by creating an eigenstate of position with a wave function (Dirac delta function) that is very large at a certain position x, and zero at other positions. If you measure the position with such a wave function, then the result x will be obtained with a probability of 100% (that is, with full confidence, or with full accuracy). This is called the eigenvalue (state) of the position or, in mathematical terms, the eigenvalue of the generalized coordinate (eigendistribution). If a particle is in its own state of position, then its momentum is absolutely undetectable. On the other hand, if a particle is in its own momentum state, then its position is completely unknown. In the eigenstate of an impulse, the eigenfunction of which has the form of a plane wave, it can be shown that the wavelength is h / p, where h is Planck's constant and p is the impulse of the eigenstate.

Rectangular potential barrier

It is a model of the quantum tunneling effect, which plays an important role in the production of modern technological devices such as flash memory and scanning tunneling microscope. Quantum tunneling is the central physical process in superlattices.

Particle in a one-dimensional potential box

A particle in a one-dimensional potential box is the simplest mathematical example in which spatial constraints lead to quantization of energy levels. A box is defined as the presence of zero potential energy everywhere within a certain area and infinite potential energy everywhere outside this area.

The ultimate potential pit

A finite potential well is a generalization of the problem of an infinite potential well with a finite depth.

The problem of a finite potential well is mathematically more complicated than the problem of a particle in an infinite potential box, since the wave function does not vanish on the walls of the well. Instead, the wave function must satisfy more complex mathematical boundary conditions, since it is nonzero in the region outside the potential well.

Surely you have heard many times about the unexplained mysteries of quantum physics and quantum mechanics... Its laws fascinate with mysticism, and even physicists themselves admit that they do not fully understand them. On the one hand, it is curious to understand these laws, but on the other hand, there is no time to read multivolume and complex books on physics. I really understand you, because I also love knowledge and the search for truth, but there is sorely not enough time for all the books. You are not alone, so many curious people type in the search line: “quantum physics for dummies, quantum mechanics for dummies, quantum physics for beginners, quantum mechanics for beginners, the basics of quantum physics, the basics of quantum mechanics, quantum physics for children, what is quantum Mechanics". This publication is for you..

You will understand the basic concepts and paradoxes of quantum physics. From the article you will learn:

• What is quantum physics and quantum mechanics?
• What is interference?
• What is Quantum Entanglement (or Quantum Teleportation for Dummies)? (see article)
• What is the Schrödinger's Cat thought experiment? (see article)

Quantum mechanics is part of quantum physics.

Why is it so difficult to understand these sciences? The answer is simple: quantum physics and quantum mechanics (part of quantum physics) study the laws of the microworld. And these laws are absolutely different from the laws of our macrocosm. Therefore, it is difficult for us to imagine what is happening with electrons and photons in the microcosm.

An example of the difference between the laws of macro- and microworlds: in our macroworld, if you put a ball in one of 2 boxes, then one of them will be empty, and the other - a ball. But in the microcosm (if instead of a ball there is an atom), an atom can be simultaneously in two boxes. This has been confirmed experimentally many times. Isn't it hard to get it in your head? But you can't argue with facts.

One more example. You photographed a fast speeding red sports car and in the photo you saw a blurred horizontal stripe, as if the car at the moment of the photo was from several points in space. Despite what you see in the photo, you are still sure that the car was in one particular place in space... In the micro world, it's not like that. An electron that revolves around the nucleus of an atom does not actually revolve, but is located simultaneously at all points of the sphere around the nucleus of an atom. Like a loose ball of fluffy wool. This concept in physics is called "Electronic cloud" .

A small excursion into history. For the first time, scientists started thinking about the quantum world when, in 1900, the German physicist Max Planck tried to find out why metals change color when heated. It was he who introduced the concept of a quantum. Before that, scientists thought that light was spreading continuously. The first to take Planck's discovery seriously was the then unknown Albert Einstein. He realized that light is not just a wave. Sometimes it behaves like a particle. Einstein received the Nobel Prize for his discovery that light is emitted in portions, quanta. A quantum of light is called a photon ( photon, Wikipedia) .

In order to make it easier to understand the laws of quantum physics and mechanics (Wikipedia), it is necessary in a sense to abstract from the laws of classical physics that are familiar to us. And imagine that you dived, like Alice, into the rabbit hole in Wonderland.

And here is a cartoon for children and adults. Describes the fundamental experiment of quantum mechanics with 2 slits and an observer. Lasts only 5 minutes. Check it out before we dive into the basic questions and concepts of quantum physics.

Quantum physics for dummies video... In the cartoon, pay attention to the "eye" of the observer. He became a serious enigma for physicists.

What is interference?

At the beginning of the cartoon, using the example of a liquid, it was shown how waves behave - alternating dark and light vertical stripes appear on the screen behind a plate with slits. And in the case when discrete particles (for example, pebbles) are "shot" at the plate, they fly through 2 slots and hit the screen directly opposite the slots. And only 2 vertical stripes “draw” on the screen.

Light interference- This is the "wave" behavior of light, when a lot of alternating bright and dark vertical stripes are displayed on the screen. Still those vertical stripes called the interference pattern.

In our macrocosm, we often observe that light behaves like a wave. If you put your hand in front of the candle, then on the wall there will be not a clear shadow from the hand, but with blurry contours.

So, it's not all that difficult! Now it is quite clear to us that light has a wave nature and if 2 slits are illuminated with light, then on the screen behind them we will see an interference pattern. Now let's look at the 2nd experiment. This is the famous Stern-Gerlach experiment (which was carried out in the 1920s).

The installation described in the cartoon was not shone with light, but "shot" with electrons (as separate particles). Then, at the beginning of the last century, physicists all over the world believed that electrons are elementary particles of matter and should not have a wave nature, but the same as pebbles. After all, electrons are elementary particles of matter, right? That is, if they are "thrown" into 2 slots, like pebbles, then on the screen behind the slots we should see 2 vertical stripes.

But ... The result was stunning. Scientists saw an interference pattern - a lot of vertical stripes. That is, electrons, like light, can also have a wave nature, can interfere. On the other hand, it became clear that light is not only a wave, but also a particle - a photon (from the historical background at the beginning of the article, we learned that Einstein received the Nobel Prize for this discovery).

You may remember that at school we were told in physics about "Particle-wave dualism"? It means that when it comes to very small particles (atoms, electrons) of the microworld, then they are both waves and particles

Today we are so smart and understand that the two experiments described above - shooting with electrons and illuminating the slits with light - are the same thing. Because we are shooting quantum particles at the slits. Now we know that both light and electrons are of a quantum nature, they are both waves and particles at the same time. And at the beginning of the 20th century, the results of this experiment were a sensation.

Attention! Now let's move on to a more subtle question.

We shine on our slits with a stream of photons (electrons) - and we see an interference pattern (vertical stripes) behind the slits on the screen. It is clear. But we are interested in seeing how each of the electrons travels through the slot.

Presumably, one electron flies to the left slot, the other to the right. But then 2 vertical stripes should appear on the screen directly opposite the slots. Why is there an interference pattern? Maybe the electrons somehow interact with each other already on the screen after flying through the slits. And the result is such a wave pattern. How can we track this?

We will throw electrons not in a beam, but one at a time. Let's drop, wait, drop the next one. Now, when the electron flies alone, it will no longer be able to interact on the screen with other electrons. We will register each electron on the screen after the throw. One or two, of course, will not "paint" a clear picture for us. But when we send a lot of them into the slots one at a time, we will notice ... oh, horror - they again "painted" an interference wave pattern!

We begin to slowly go crazy. After all, we expected that there would be 2 vertical stripes opposite the slots! It turns out that when we threw photons one at a time, each of them passed, as if through 2 slits at the same time, and interfered with itself. Fiction! Let's return to the explanation of this phenomenon in the next section.

What are spin and superposition?

We now know what interference is. This is the wave behavior of micro particles - photons, electrons, other micro particles (let's call them photons for the sake of simplicity from now on).

As a result of the experiment, when we threw 1 photon into 2 slits, we realized that it seemed to fly through two slits at the same time. How else to explain the interference pattern on the screen?

But how to imagine a picture that a photon flies through two slits at the same time? There are 2 options.

• 1st option: a photon, like a wave (like water) "floats" through 2 slits at the same time
• 2nd option: a photon, like a particle, flies simultaneously along 2 trajectories (not even along two, but along all at once)

In principle, these statements are equivalent. We arrived at the "path integral". This is Richard Feynman's formulation of quantum mechanics.

By the way, exactly Richard Feynman the well-known expression belongs that we can confidently assert that no one understands quantum mechanics

But this expression of him worked at the beginning of the century. But now we are smart and know that a photon can behave both as a particle and as a wave. That he can, in some incomprehensible way for us, fly simultaneously through 2 slots. Therefore, it will be easy for us to understand the following important statement of quantum mechanics:

Strictly speaking, quantum mechanics tells us that this behavior of a photon is the rule, not the exception. Any quantum particle is, as a rule, in several states or in several points in space at the same time.

Objects of the macrocosm can be located only in one specific place and in one specific state. But a quantum particle exists according to its own laws. And she doesn't care if we don't understand them. This is the point.

We just have to admit, as an axiom, that the "superposition" of a quantum object means that it can be on 2 or more trajectories at the same time, at 2 or more points at the same time

The same applies to another parameter of a photon - spin (its own angular momentum). Spin is a vector. A quantum object can be thought of as a microscopic magnet. We are used to the fact that the vector of the magnet (spin) is either directed up or down. But an electron or a photon again tells us: “Guys, we don't care what you are used to, we can be in both spin states at once (vector up, vector down), just like we can be on 2 trajectories at the same time, or at 2 points at the same time! ".

What is “measurement” or “collapse of the wave function”?

There is not much left for us - to understand what a “measurement” is and what a “collapse of the wave function” is.

Wave function Is a description of the state of a quantum object (our photon or electron).

Suppose we have an electron, it flies to itself in an indefinite state, its spin is directed both up and down at the same time... We need to measure his condition.

Let's measure with the help of a magnetic field: electrons, whose spin was directed in the direction of the field, will deflect in one direction, and electrons, whose spin is directed against the field, in the other. Photons can also be directed into a polarizing filter. If the spin (polarization) of the photon is +1, it passes through the filter, and if -1, then it does not.

Stop! Here you will inevitably have a question: before the measurement, the electron did not have any specific spin direction, right? He was in all states at the same time?

This is the trick and sensation of quantum mechanics.... Until you measure the state of a quantum object, it can rotate in any direction (have any direction of the vector of its own angular momentum - spin). But at the moment when you measured his state, he seems to be deciding which spin vector to take.

This is such a cool quantum object - it decides on its own state. And we cannot predict in advance what decision he will make when he flies into the magnetic field in which we measure him. The probability that he decides to have a spin up or down vector is 50-50%. But as soon as he decided - he is in a certain state with a specific direction of the spin. The reason for his decision is our "dimension"!

This is called “ collapse of the wave function "... The wave function before measurement was undefined, i.e. the electron spin vector was located simultaneously in all directions, after measurement the electron fixed a certain direction of its spin vector.

Attention! An excellent example of an association from our macroworld to understand:

Spin the coin on the table like a whirligig. While the coin is spinning, it has no specific meaning - heads or tails. But as soon as you decide to “measure” this value and slap the coin with your hand, this is where you get a specific state of the coin - heads or tails. Now imagine that it is a coin that decides what value to "show" you - heads or tails. The electron behaves approximately the same way.

Now remember the experiment shown at the end of the cartoon. When photons were sent through the slits, they behaved like a wave and showed an interference pattern on the screen. And when scientists wanted to fix (measure) the moment of flight of photons through the slit and put an "observer" behind the screen, the photons began to behave, not like waves, but like particles. And "drew" 2 vertical stripes on the screen. Those. at the moment of measurement or observation, quantum objects themselves choose what state they should be in.

Fiction! Is not it?

But that's not all. Finally we got to the most interesting.

But ... it seems to me that there will be an overload of information, so we will consider these 2 concepts in separate posts:

• What ?
• What is a thought experiment.

Now, do you want the information to be sorted out on the shelves? Watch a documentary produced by the Canadian Institute for Theoretical Physics. In it, in 20 minutes, very briefly and in chronological order, you will be told about all the discoveries of quantum physics, since the discovery of Planck in 1900. And then they will tell you what practical developments are being carried out now on the basis of knowledge in quantum physics: from the most accurate atomic clocks to super-fast computations of a quantum computer. I highly recommend watching this movie.

See you!

I wish you all inspiration for all your plans and projects!

P.S.2 Write your questions and thoughts in the comments. Write, what other questions on quantum physics are of interest to you?

P.S.3 Subscribe to the blog - a form to subscribe under the article.

QUANTUM MECHANICS, a section of theoretical physics, which is a system of concepts and a mathematical apparatus necessary to describe physical phenomena due to the existence in nature of the smallest quantum of action h (Planck's constant). The numerical value h = 6.62607 ∙ 10ˉ 34 J ∙ s (and another, often used value ħ = h / 2π = 1.05457 ∙ 10ˉ 34 J ∙ s) is extremely small, but the fact that it is, of course, fundamentally distinguishes quantum phenomena from all others and determines their main features. Quantum phenomena include radiation processes, phenomena of atomic and nuclear physics, condensed matter physics, chemical bonding, etc.

The history of the creation of quantum mechanics. Historically, the first phenomenon, for the explanation of which the concept of the quantum of action h was introduced in 1900, was the radiation spectrum of an absolutely black body, i.e., the dependence of the intensity of thermal radiation on its frequency v and the temperature T of a heated body. Initially, the connection of this phenomenon with the processes taking place in the atom was not clear; at that time, the very idea of ​​the atom was not generally recognized, although even then observations were known that indicated a complex intra-atomic structure.

In 1802 W. Wollaston discovered narrow spectral lines in the solar radiation spectrum, which were described in detail by J. Fraunhofer in 1814. In 1859, G. Kirchhoff and R. Bunsen established that each chemical element has an individual set of spectral lines, and the Swiss scientist I. Ya.Balmer (1885), the Swedish physicist J. Rydberg (1890) and the German scientist W. Ritz (1908) found certain patterns in their location. In 1896, P. Zeeman observed the splitting of spectral lines in a magnetic field (the Zeeman effect), which H. A. Lorentz explained the next year by the motion of an electron in an atom. The existence of the electron was experimentally proved in 1897 by J.J. Thomson.

The existing physical theories turned out to be insufficient to explain the laws of the photoelectric effect: it turned out that the energy of electrons emitted from a substance when it is irradiated with light depends only on the frequency of light v, and not on its intensity (A.G. Stoletov, 1889; F. von Lenard, 1904). This fact completely contradicted the generally accepted wave nature of light at that time, but was naturally explained under the assumption that light propagates in the form of energy quanta E = hv (A. Einstein, 1905), later called photons (G. Lewis, 1926).

Within 10 years after the discovery of the electron, several models of the atom were proposed, which were not supported, however, by experiments. In 1909-11, E. Rutherford, studying the scattering of α-particles by atoms, established the existence of a compact positively charged nucleus, in which practically all the mass of an atom is concentrated. These experiments became the basis of the planetary model of the atom: a positively charged nucleus around which negatively charged electrons revolve. Such a model, however, contradicted the fact of the stability of the atom, since it followed from classical electrodynamics that after a time of the order of 10 -9 s, the rotating electron would fall onto the nucleus, losing energy to radiation.

In 1913 N. Bohr suggested that the stability of the planetary atom is explained by the finiteness of the quantum of action h. He postulated that there are stationary orbits in the atom in which the electron does not radiate (Bohr's first postulate), and singled out these orbits from all possible by the quantization condition: 2πmυr = nh, where m is the electron mass, υ is its orbital velocity, r is the distance to the kernel, n = 1,2,3, ... are integers. From this condition, Bohr determined the energies E n = -me 4 / 2ħ 2 n 2 (e is the electric charge of an electron) of stationary states, as well as the diameter of a hydrogen atom (about 10 -8 cm) - in full accordance with the conclusions of the kinetic theory of matter.

Bohr's second postulate asserted that radiation occurs only when electrons pass from one stationary orbit to another, and the radiation frequency v nk of transitions from the E n state to the E k state is equal to v nk = (E k - E n) / h (see Atomic physics ). Bohr's theory naturally explained the patterns in the spectra of atoms, but its postulates were in obvious contradiction with classical mechanics and the theory of the electromagnetic field.

In 1922, A. Compton, studying the scattering of X-rays by electrons, found that the incident and scattered X-ray energy quanta behave like particles. In 1923, Ch. TR Wilson and DV Skobel'tsyn observed a recoil electron in this reaction and thereby confirmed the corpuscular nature of X-rays (nuclear γ-radiation). This, however, contradicted the experiments of M. Laue, who as early as 1912 observed the diffraction of X-rays and thereby proved their wave nature.

In 1921, the German physicist K. Ramsauer discovered that at a certain energy, electrons pass through gases, practically without scattering, like light waves in a transparent medium. This was the first experimental evidence of the wave properties of the electron, the reality of which in 1927 was confirmed by the direct experiments of C.J.Davisson, L. Jermer and J.P. Thomson.

In 1923, L. de Broglie introduced the concept of waves of matter: each particle with mass m and velocity υ can be associated with a wave with a length λ = h / mυ, just as each wave with a frequency v = c / λ can be associated with a particle with energy E = hv. A generalization of this hypothesis, known as wave-particle dualism, has become the foundation and universal principle of quantum physics. Its essence lies in the fact that the same objects of study manifest themselves in two ways: either as a particle or as a wave, depending on the conditions of their observation.

The relationships between the characteristics of a wave and a particle were established even before the creation of quantum mechanics: E = hv (1900) and λ = h / mυ = h / p (1923), where frequency v and wavelength λ are characteristics of the wave, and energy E and mass m, velocity υ and momentum p = mυ are the characteristics of the particle; the connection between these two types of characteristics is carried out through the Planck constant h. The duality relations are most clearly expressed in terms of the circular frequency ω = 2πν and the wave vector k = 2π / λ:

E = ħω, p = ħk.

A clear illustration of the wave-particle dualism is shown in Figure 1: the diffraction rings observed in the scattering of electrons and X-rays are practically identical.

Quantum mechanics - the theoretical basis of all quantum physics - was created in less than three years. In 1925 W. Heisenberg, relying on Bohr's ideas, proposed matrix mechanics, which by the end of the same year acquired the form of a complete theory in the works of M. Born, the German physicist P. Jordan and P. Dirac. The main objects of this theory are matrices of a special type, which in quantum mechanics represent the physical quantities of classical mechanics.

In 1926, E. Schrödinger, proceeding from L. de Broglie's ideas about waves of matter, proposed wave mechanics, where the main role is played by the wave function of a quantum state, which obeys a second-order differential equation with given boundary conditions. Both theories were equally good at explaining the stability of the planetary atom and made it possible to calculate its main characteristics. In the same year, M. Born proposed a statistical interpretation of the wave function, Schrödinger (and also independently W. Pauli and others) proved the mathematical equivalence of the matrix and wave mechanic, and Born, together with N. Wiener, introduced the concept of an operator of a physical quantity.

In 1927 W. Heisenberg discovered the uncertainty relation, and N. Bohr formulated the principle of complementarity. The discovery of the electron spin (J. Uhlenbeck and S. Goudsmit, 1925) and the derivation of the Pauli equation that takes into account the electron spin (1927) completed the logical and design schemes of nonrelativistic quantum mechanics, and P. Dirac and J. von Neumann presented quantum mechanics as complete conceptually an independent theory based on a limited set of concepts and postulates, such as operator, state vector, probability amplitude, superposition of states, etc.

Basic concepts and formalism of quantum mechanics. The main equation of quantum mechanics is the Schrödinger wave equation, the role of which is similar to the role of Newton's equations in classical mechanics and Maxwell's equations in electrodynamics. In the space of variables x (coordinate) and t (time), it has the form

where H is the Hamilton operator; its form coincides with the Hamilton operator of classical mechanics, in which the coordinate x and momentum p are replaced by the operators x and p of these variables, that is,

where V (x) is the potential energy of the system.

In contrast to Newton's equation, from which the observed trajectory x (t) of a material point moving in the field of potential forces V (x) is found, from the Schrödinger equation an unobservable wave function ψ (x) of a quantum system is found, with which, however, one can calculate the values ​​of all measurable quantities. Immediately after the discovery of the Schrödinger equation, M. Born explained the meaning of the wave function: | ψ (x) | 2 is the probability density, and | ψ (x) | 2 · Δx is the probability of detecting a quantum system in the range of Δx values ​​of the x coordinate.

Each physical quantity (dynamic variable of classical mechanics) in quantum mechanics is associated with an observable a and the corresponding Hermitian operator J, which in the chosen basis of complex functions | i> = f i (x) is represented by the matrix

where f * (x) is the function complex conjugate to the function f (x).

The orthogonal basis in this space is the set of eigenfunctions | n) = f n (x)), n = 1,2,3, for which the action of the operator В is reduced to multiplication by a number (the eigenvalue a n of the operator В):

The basis of the functions | n) is normalized by the condition for n = n ’, for n ≠ n’.

and the number of basis functions (in contrast to the basis vectors of the three-dimensional space of classical physics) is infinite, and the index n can vary both discretely and continuously. All possible values ​​of the observable a are contained in the set (a n) of the eigenvalues ​​of the corresponding operator Â, and only these values ​​can become the measurement results.

The main object of quantum mechanics is the state vector | ψ), which can be expanded in terms of eigenfunctions | n) of the chosen operator В:

where ψ n is the probability amplitude (wave function) of the state | n), and | ψ n | 2 is equal to the weight of the state n in the expansion | ψ), and

that is, the total probability of finding a system in one of the quantum states n is equal to unity.

In Heisenberg quantum mechanics, the operators  and the corresponding matrices obey the equations

where | Â, Ĥ | = ÂĤ - Ĥ is the commutator of the operators  and. In contrast to the Schrödinger scheme, where the wave function ψ depends on time, in the Heisenberg scheme the time dependence is related to the operator J. Both of these approaches are mathematically equivalent, but in numerous applications of quantum mechanics, Schrödinger's approach has proved to be preferable.

The eigenvalue of the Hamilton operator Ĥ is the total energy of the system E, independent of time, which is found as a solution of the stationary Schrödinger equation

Its solutions are divided into two types depending on the type of boundary conditions.

For a localized state, the wave function satisfies the natural boundary condition ψ (∞) = 0. In this case, the Schrödinger equation has a solution only for a discrete set of energies Е n, n = 1,2,3, ..., to which the wave functions ψ n ( r):

An example of a localized state is a hydrogen atom. Its Hamiltonian Ĥ has the form

where Δ = ∂ 2 / ∂х 2 + ∂ 2 / ∂у 2 + ∂ 2 / ∂z 2 is the Laplace operator, e 2 / r is the interaction potential of the electron and the nucleus, r is the distance from the nucleus to the electron, and the energy eigenvalues Е n, calculated from the Schrödinger equation, coincide with the energy levels of the Bohr atom.

The simplest example of a non-localized state is the free one-dimensional motion of an electron with momentum p. It corresponds to the Schrödinger equation

the solution of which is a plane wave

where in the general case С = | С | exp (iφ) is a complex function, | С | and φ is its modulus and phase. In this case, the electron energy E = p 2 / 2m, and the index p of the solution ψ p (x) takes on a continuous series of values.

The coordinate and momentum operators (and any other pair of canonically conjugate variables) obey the permutation (commutation) relation:

There is no common basis for the eigenfunctions for pairs of such operators, and the physical quantities corresponding to them cannot be determined simultaneously with arbitrary accuracy. From the commutation relation for the operators x̂ and p̂, there follows a restriction on the accuracy Δх and Δр of determining the coordinate x and its conjugate momentum p of a quantum system (Heisenberg's uncertainty relation):

Hence, in particular, the conclusion immediately follows about the stability of the atom, since the relation Δх = Δр = 0, corresponding to the fall of an electron onto the nucleus, is forbidden in this scheme.

The set of simultaneously measurable quantities characterizing a quantum system is represented by a set of operators

commuting with each other, that is, satisfying the relations А̂В̂ - В̂А̂ = А̂С̂ - С̂А̂ = В̂С̂ - С̂В̂ = ... = 0. For a nonrelativistic hydrogen atom, such a set consists, for example, of the operators: Ĥ (total energy operator), (the square of the operator moment) and (z-component of the moment operator). The state vector of an atom is defined as a set of common eigenfunctions ψ i (r) of all operators

which are numbered by a set (i) = (nlm) of quantum numbers of energy (n = 1,2,3, ...), orbital angular momentum (l = 0,1, ..., n - 1) and its projection onto the z-axis (m = -l, ..., - 1,0,1, ..., l). Functions | ψ i (r) | 2 can be conventionally regarded as the shape of an atom in various quantum states i (the so-called White silhouettes).

The value of a physical quantity (observable quantum mechanics) is defined as the average value Ā of the corresponding operator Â:

This relationship is valid for pure states, i.e., for isolated quantum systems. In the general case of mixed states, we always deal with a large collection (statistical ensemble) of identical systems (for example, atoms), the properties of which are determined by averaging over this ensemble. In this case, the mean value Ā of the operator J takes the form

where р nm is the density matrix (LD Landau; J. von Neumann, 1929) with the normalization condition ∑ n ρ пп = 1. The formalism of the density matrix allows us to combine quantum mechanical averaging over states and statistical averaging over the ensemble. The density matrix also plays an important role in the theory of quantum measurements, the essence of which always consists in the interaction of quantum and classical subsystems. The concept of a density matrix is ​​the basis of quantum statistics and the basis for one of the alternative formulations of quantum mechanics. Another form of quantum mechanics, based on the concept of the path integral (or path integral), was proposed by R. Feynman in 1948.

Compliance principle... Quantum mechanics has deep roots in both classical and statistical mechanics. Already in his first work, N. Bohr formulated the correspondence principle, according to which quantum relations should transform into classical ones for large quantum numbers n. P. Ehrenfest in 1927 showed that, taking into account the equations of quantum mechanics, the average value Ā of the operator  satisfies the equation of motion of classical mechanics. Ehrenfest's theorem is a special case of the general correspondence principle: in the limit h → 0, the equations of quantum mechanics go over into the equations of classical mechanics. In particular, the Schrödinger wave equation in the limit h → 0 turns into the equation of geometric optics for the trajectory of a light beam (and any radiation) without taking into account its wave properties. Representing the solution ψ (x) of the Schrödinger equation in the form ψ (x) = exp (iS / ħ), where S = ∫ p (x) dx is an analogue of the classical action integral, one can verify that in the limit ħ → 0 the function S satisfies the classical the Hamilton - Jacobi equation. In addition, in the limit h → 0, the operators x̂ and p̂ commute and the corresponding values ​​of the coordinate and momentum can be determined simultaneously, as is assumed in classical mechanics.

The most significant analogies between the relations of classical and quantum mechanics for periodic motions can be traced on the phase plane of canonically conjugate variables, for example, the coordinates x and momentum p of the system. Integrals of the type ∮p (x) dx taken along a closed trajectory (Poincaré integral invariants) are known in the prehistory of quantum mechanics as the adiabatic Ehrenfest invariants. A. Sommerfeld used them to describe quantum laws in the language of classical mechanics, in particular for the spatial quantization of an atom and the introduction of quantum numbers l and m (it was he who introduced this term in 1915).

The dimension of the phase integral ∮pdx coincides with the dimension of the Planck constant h, and in 1911 A. Poincaré and M. Planck proposed to consider the quantum of action h as the minimum volume of the phase space, the number n of cells of which is a multiple of h: n = ∮pdx / h. In particular, when an electron moves along a circular trajectory with a constant momentum p, the Bohr quantization condition immediately follows from the relation n = ∮p (x) dx / h = p ∙ 2πr / h: mυr = nħ (P. Debye, 1913).

However, in the case of one-dimensional motion in the potential V (x) = mω 2 0 x 2/2 (harmonic oscillator with natural frequency ω 0), from the quantization condition ∮р (х) dx = nh follows a series of energy values ​​Е n = ħω 0 n, while the exact solution of the quantum equations for the oscillator leads to the sequence Е n = ħω 0 (n + 1/2). This result of quantum mechanics, first obtained by W. Heisenberg, is fundamentally different from the approximate one by the presence of zero vibration energy E 0 = ħω 0/2, which has a purely quantum nature: the state of rest (x = 0, p = 0) is forbidden in quantum mechanics, since it contradicts the uncertainty relation Δх ∙ Δр ≥ ħ / 2.

The principle of superposition of states and probabilistic interpretation. The main and visual contradiction between the corpuscular and wave pictures of quantum phenomena was eliminated in 1926, after M. Born proposed to interpret the complex wave function ψ n (x) = | ψ n (x) | exp (iφ n) as the amplitude probability of state n, and the square of its modulus | ψ n (х) | 2 - as the probability density of detecting state n at point x. A quantum system can be in different, including alternative, states, and its probability amplitude is equal to a linear combination of the probability amplitudes of these states: ψ = ψ 1 + ψ 2 + ...

The probability density of the resulting state is equal to the square of the sum of the probability amplitudes, and not the sum of the squares of the amplitudes, as is the case in statistical physics:

This postulate - the principle of superposition of states - is one of the most important in the system of concepts of quantum mechanics; it has many observable consequences. One of them, namely the passage of an electron through two closely spaced slits, is discussed more often than others (Fig. 2). The electron beam falls on the left, passes through the slits in the partition, and then is recorded on the screen (or photographic plate) on the right. If we alternately close each of the slits, then on the screen on the right we will see an image of an open slit. But if you open both slits at the same time, then instead of two slits we will see a system of interference fringes, the intensity of which is described by the expression:

The last term in this sum represents the interference of two probability waves arriving at a given point of the screen from different slots in the partition, and depends on the phase difference of the wave functions Δφ = φ 1 - φ 2. In the case of equal amplitudes | ψ 1 | = | ψ 2 |:

that is, the intensity of the image of the slits at different points of the screen varies from 0 to 4 | ψ 1 | 2 - in accordance with the change in the phase difference Δφ from 0 to π / 2. In particular, in this case it may turn out that with two open slits in place of the image of a single slit, we will not detect any signal, which is absurd from the corpuscular point of view.

It is essential that this picture of the phenomenon does not depend on the intensity of the electron beam, that is, it is not the result of their interaction with each other. An interference pattern arises even in the limit when electrons pass through the slits in the partition one by one, i.e., each electron interferes with itself. This is impossible for a particle, but it is quite natural for a wave, for example, when it is reflected or diffracted by an obstacle whose dimensions are comparable to its length. In this experiment, the wave-particle dualism is manifested in the fact that the same electron is registered as a particle, but propagates as a wave of a special nature: this is a wave of probability to find an electron at any point in space. In such a picture of the scattering process, the question is: "Through which of the slits did the electron-particle pass?" loses its meaning, since the corresponding probability wave passes through both slots at once.

Another example illustrating the probabilistic nature of the phenomena of quantum mechanics is the transmission of light through a semitransparent plate. By definition, the light reflectance is equal to the ratio of the number of photons reflected from the plate to the number of incident photons. However, this is not the result of averaging a large number of events, but a characteristic inherent in each photon.

The principle of superposition and the concept of probability made it possible to carry out a consistent synthesis of the concepts of "wave" and "particle": each of the quantum events and its registration are discrete, but their distribution is dictated by the law of propagation of continuous waves of probability.

Tunneling effect and resonant scattering. The tunnel effect is perhaps the most famous phenomenon in quantum physics. It is due to the wave properties of quantum objects and only in the framework of quantum mechanics received an adequate explanation. An example of the tunneling effect is the decay of a radium nucleus into a radon nucleus and an α-particle: Ra → Rn + α.

Figure 3 shows a diagram of the α-decay potential V (r): the α-particle vibrates with frequency v in the “potential well” of the nucleus with a charge Z 0, and after leaving it moves in the repulsive Coulomb potential 2Ze 2 / r, where Z = Z 0 -2. In classical mechanics, a particle cannot leave a potential well if its energy E is less than the height of the potential barrier V max. In quantum mechanics, due to the uncertainty relation, a particle with a finite probability W penetrates into the sub-barrier region r 0< r < r 1 и может «просочиться» из области r < r 0 в область r >r 1 is similar to how light penetrates into the region of the geometric shadow at distances comparable to the length of the light wave. Using the Schrödinger equation, we can calculate the coefficient D of the passage of an α-particle through the barrier, which in the semiclassical approximation is equal to:

Over time, the number of radium nuclei N (t) decreases according to the law: N (t) = N 0 exp (-t / τ), where τ is the average lifetime of a nucleus, N 0 is the initial number of nuclei at t = 0. Probability α- decay W = vD is related to the lifetime by the relation W = l / τ, whence the Geiger - Nettol law follows:

where υ is the velocity of the α-particle, Z is the charge of the formed nucleus. Experimentally, this dependence was discovered as early as 1909, but it was not until 1928 that G. Gamow (and independently the English physicist R. Gurney and the American physicist E. Condon) first explained it in the language of quantum mechanics. Thus, it was shown that quantum mechanics describes not only radiation processes and other phenomena of atomic physics, but also the phenomena of nuclear physics.

In atomic physics, the tunneling effect explains the phenomenon of field emission. In a uniform electric field of strength E, the Coulomb potential V (r) = -е 2 / r of the attraction between the nucleus and the electron is distorted: V (r) = - е 2 / r - eEr, the energy levels of the atom E nl m are shifted, which leads to to a change in the frequencies ν nk of transitions between them (Stark effect). In addition, this potential becomes qualitatively similar to the α-decay potential, as a result of which a finite probability of electron tunneling through the potential barrier arises (R. Oppenheimer, 1928). When critical values ​​of E are reached, the barrier decreases so much that the electron leaves the atom (the so-called avalanche ionization).

Alpha decay is a special case of the decay of a quasi-stationary state, which is closely related to the concept of quantum mechanical resonance and allows us to understand additional aspects of non-stationary processes in quantum mechanics. The time dependence of its solutions follows from the Schrödinger equation:

where E is the eigenvalue of the Hamiltonian Ĥ, which is valid for the Hermitian operators of quantum mechanics, and the corresponding observable (total energy E) does not depend on time. However, the energy of nonstationary systems depends on time, and this fact can be formally taken into account if the energy of such a system is presented in a complex form: E = E 0 - iΓ / 2. In this case, the time dependence of the wave function has the form

and the probability of detecting the corresponding state decreases exponentially:

which coincides in form with the α-decay law with the decay constant τ = ħ / Г.

In the reverse process, for example, in the collision of deuterium and tritium nuclei, which results in the formation of helium and a neutron (thermonuclear fusion reaction), the concept of the reaction cross section σ is used, which is defined as a measure of the reaction probability with a single flux of colliding particles.

For classical particles, the cross section of scattering on a ball of radius r 0 coincides with its geometric cross section and is equal to σ = πr 0 2. In quantum mechanics, it can be represented in terms of the scattering phases δl (k):

where k = p / ħ = √2mE / ħ is the wave number, l is the orbital momentum of the system. In the limit of very low collision energies, the quantum scattering cross section σ = 4πr 0 2 is 4 times larger than the geometric cross section of the ball. (This effect is one of the consequences of the wave nature of quantum phenomena.) In the vicinity of the resonance at Е ≈ Е 0, the scattering phase behaves as

and the scattering cross section is

where λ = 1 / k, W (E) is the Breit - Wigner function:

At low scattering energies l 0 ≈ 0, and the de Broglie wavelength λ is much larger than the dimensions of the nuclei, therefore, at E = E 0, the resonant cross sections of the nuclei σres ≈ 4πλ 0 2 can exceed their geometric cross sections πr 0 2 by a factor of thousands and millions. In nuclear physics, the operation of nuclear and thermonuclear reactors depends on these cross sections. In atomic physics, this phenomenon was first observed by J. Frank and G. Hertz (1913) in experiments on the resonant absorption of electrons by mercury atoms. In the opposite case (δ 0 = 0), the scattering cross section is anomalously small (the Ramsauer effect, 1921).

The function W (E) is known in optics as the Lorentzian profile of the emission line and has the form of a typical resonance curve with a maximum at E = E 0, and the resonance width G = 2∆E = 2 (E - E 0) is determined from the relation W (E 0 ± ΔΕ) = W (E 0) / 2. The function W (E) has a universal character and describes both the decay of a quasi-stationary state and the resonance dependence of the scattering cross section on the collision energy E, and in radiation phenomena determines the natural width Г of the spectral line, which is related to the lifetime τ of the emitter by the relation τ = Г / Г ... This ratio also determines the lifetime of elementary particles.

From the definition of τ = ħ / Г, taking into account the equality Г = 2∆Е, the uncertainty relation for energy and time follows: ∆Е ∙ ∆t ≥ ħ / 2, where ∆t ≥ τ. In form, it is similar to the ratio ∆х ∙ ∆р ≥ ħ / 2, however, the ontological status of this inequality is different, since in quantum mechanics time t is not a dynamic variable. Therefore, the relation ∆Е ∙ ∆t ≥ ħ / 2 does not follow directly from the basic postulates of stationary quantum mechanics and, strictly speaking, makes sense only for systems whose energy changes over time. Its physical meaning is that during the time ∆t the energy of the system cannot be measured more accurately than the value ∆E, determined by the relation ∆E ∙ ∆t ≥ ħ / 2. A stationary state (ΔЕ → 0) exists for an infinitely long time (Δt → ∞).

Spin, particle identity and exchange interaction. The concept of "spin" was established in physics by the works of W. Pauli, the Dutch physicist R. Kronig, S. Goudsmit and J. Uhlenbeck (1924-27), although experimental evidence of its existence was obtained long before the creation of quantum mechanics in the experiments of A. Einstein and W. J. de Haaz (1915), as well as O. Stern and the German physicist W. Gerlach (1922). The spin (own mechanical moment of a particle) for an electron is S = ħ / 2. This is the same important characteristic of a quantum particle as charge and mass, which, however, has no classical counterparts.

The spin operator Ŝ = ħσИ / 2, where σИ = (σИ х, σИ у, σИ z) are two-dimensional Pauli matrices, is defined in the space of two-component eigenfunctions u = (u +, u -) of the operator Ŝ z of the spin projection on the z axis: σИ zu = σu, σ = ± 1/2. The intrinsic magnetic moment μ of a particle with mass m and spin S is equal to μ = 2μ 0 S, where μ 0 = еħ / 2mс is the Bohr magneton. The operators Ŝ 2 and Ŝ z commute with the set Ĥ 0 L 2 and L z of operators of the hydrogen atom and together they form the Hamiltonian of the Pauli equation (1927), the solutions of which are numbered by the set i = (nlmσ) of quantum numbers of the eigenvalues ​​of the set and commuting operators Ĥ 0, L 2, L z, Ŝ 2, Ŝ z. These solutions describe the most subtle features of the observed spectra of atoms, in particular, the splitting of spectral lines in a magnetic field (normal and anomalous Zeeman effect), as well as their multiplet structure as a result of the interaction of the electron spin with the orbital moment of the atom (fine structure) and the spin of the nucleus (hyperfine structure ).

In 1924, even before the creation of quantum mechanics, W. Pauli formulated the exclusion principle: an atom cannot have two electrons with the same set of quantum numbers i = (nlmσ). This principle made it possible to understand the structure of the periodic table of chemical elements and explain the periodicity of changes in their chemical properties with a monotonic increase in the charge of their nuclei.

The exclusion principle is a special case of a more general principle that establishes a relationship between the spin of a particle and the symmetry of its wave function. Depending on the spin value, all elementary particles are divided into two classes: fermions - particles with half-integer spin (electron, proton, μ-meson, etc.) and bosons - particles with zero or integer spin (photon, π-meson, K -meson, etc.). In 1940, Pauli proved a general theorem on the connection between spin and statistics, from which it follows that the wave functions of any system of fermions have negative parity (change sign when they are pairwise permuted), and the parity of the wave function of a system of bosons is always positive. Accordingly, there are two types of particle energy distributions: the Fermi - Dirac distribution and the Bose - Einstein distribution, a particular case of which is the Planck distribution for a system of photons.

One of the consequences of the Pauli principle is the existence of the so-called exchange interaction, which manifests itself already in a system of two electrons. In particular, it is this interaction that provides the covalent chemical bond of atoms in the molecules Н 2, N 2, О 2, etc. Exchange interaction is an exclusively quantum effect, there is no analogue of such interaction in classical physics. Its specificity is explained by the fact that the probability density of the wave function of a system of two electrons | ψ (r 1, r 2) | 2 contains not only the terms | ψ n (r 1) | 2 | ψ m (r 2) | 2, where n and m are the quantum states of electrons of both atoms, but also the "exchange terms" ψ n * (r 1) ψ m * (r 1) ψ n (r 2) ψ m (r 2), arising as a consequence of the principle superposition, which allows each electron to be simultaneously in different quantum states n and m of both atoms. In addition, by virtue of the Pauli principle, the spin part of the wave function of a molecule should be antisymmetric with respect to the permutation of electrons, i.e., the chemical bond of atoms in a molecule is carried out by a pair of electrons with oppositely directed spins. The wave function of complex molecules can be represented as a superposition of wave functions corresponding to various possible configurations of the molecule (theory of resonance, L. Pauling, 1928).

The calculation methods developed in quantum mechanics (the Hartree - Fock method, the molecular orbital method, etc.) make it possible to calculate on modern computers all the characteristics of stable configurations of complex molecules: the order of filling of electron shells in an atom, the equilibrium distances between atoms in molecules, the energy and direction of chemical bonds , the arrangement of atoms in space, and build potential surfaces that determine the direction of chemical reactions. This approach also makes it possible to calculate the potentials of interatomic and intermolecular interactions, in particular the van der Waals forces, to estimate the strength of hydrogen bonds, etc. Thus, the problem of chemical bonding is reduced to the problem of calculating the quantum characteristics of a system of particles with Coulomb interaction, and from this point of view, structural chemistry can be considered as one of the branches of quantum mechanics.

The exchange interaction essentially depends on the type of potential interaction between the particles. In particular, in some metals, it is thanks to it that the state of pairs of electrons with parallel spins is more stable, which explains the phenomenon of ferromagnetism.

Applications of quantum mechanics. Quantum mechanics is the theoretical basis of quantum physics. It made it possible to understand the structure of the electron shells of atoms and the patterns in their radiation spectra, the structure of nuclei and the laws of their radioactive decay, the origin of chemical elements and the evolution of stars, including explosions of new and supernova stars, as well as the source of the Sun's energy. Quantum mechanics explained the meaning of the periodic table of elements, the nature of the chemical bond and the structure of crystals, the heat capacity and magnetic properties of substances, the phenomena of superconductivity and superfluidity, etc. Quantum mechanics is the physical basis of numerous technical applications: spectral analysis, laser, transistor and computer, nuclear reactor and atomic bombs, etc.

The properties of metals, dielectrics, semiconductors and other substances in the framework of quantum mechanics also receive a natural explanation. In crystals, atoms perform small vibrations near equilibrium positions with a frequency ω, which are associated with the vibrational quanta of the crystal lattice and the corresponding quasi-particles - phonons with energy E = ħω. The heat capacity of a crystal is largely determined by the heat capacity of the gas of its phonons, and its thermal conductivity can be interpreted as the thermal conductivity of a phonon gas. In metals, conduction electrons are a gas of fermions, and their scattering by phonons is the main reason for the electrical resistance of conductors, and also explains the similarity of the thermal and electrical properties of metals (see Wiedemann-Franz law). In magnetically ordered structures, quasiparticles appear - magnons, which correspond to spin waves, quanta of rotational excitation - rotons appear in quantum liquids, and the magnetic properties of substances are determined by the spins of electrons and nuclei (see Magnetism). The interaction of the spins of electrons and nuclei with a magnetic field is the basis for practical applications of the phenomena of electron paramagnetic and nuclear magnetic resonances, in particular, in medical tomographs.

The ordered structure of crystals gives rise to additional symmetry of the Hamiltonian with respect to the shift x → x + a, where a is the period of the crystal lattice. Taking into account the periodic structure of a quantum system leads to the splitting of its energy spectrum into allowed and forbidden bands. This structure of energy levels underlies the operation of transistors and all electronics based on them (TV, computer, cell phone, etc.). At the beginning of the 21st century, significant advances were made in the creation of crystals with specified properties and structure of energy bands (superlattices, photonic crystals and heterostructures: quantum dots, quantum threads, nanotubes, etc.).

With a decrease in temperature, some substances pass into the state of a quantum liquid, the energy of which at temperature T → 0 approaches the energy of zero-point vibrations of the system. In some metals, at low temperatures, Cooper pairs are formed - systems of two electrons with opposite spins and momenta. In this case, the electron gas of fermions is transformed into a gas of bosons, which entails Bose condensation, which explains the phenomenon of superconductivity.

At low temperatures, the de Broglie wavelength of thermal motions of atoms becomes comparable to the interatomic distances and a correlation of the phases of the wave functions of many particles arises, which leads to macroscopic quantum effects (Josephson effect, quantization of magnetic flux, fractional quantum Hall effect, Andreev reflection).

Based on quantum phenomena, the most accurate quantum standards of various physical quantities have been created: frequency (helium-neon laser), electric voltage (Josephson effect), resistance (quantum Hall effect), etc., as well as devices for various precision measurements: squids, quantum clock, quantum gyroscope, etc.

Quantum mechanics arose as a theory to explain the specific phenomena of atomic physics (it was originally called that: atomic dynamics), but gradually it became clear that quantum mechanics also forms the basis of all subatomic physics, and all its basic concepts are applicable to describe the phenomena of nuclear physics and elementary particles. The original quantum mechanics was nonrelativistic, that is, it described the motion of systems with speeds much less than the speed of light. The interaction of particles in this theory was still described in classical terms. In 1928, P. Dirac found the relativistic equation of quantum mechanics (Dirac's equation), which, while retaining all its concepts, took into account the requirements of the theory of relativity. In addition, the formalism of secondary quantization was developed, which describes the creation and annihilation of particles, in particular, the creation and absorption of photons in radiation processes. On this basis, quantum electrodynamics arose, which made it possible to calculate with great accuracy all the properties of systems with electromagnetic interaction. Later, it developed into a quantum field theory that unites in a single formalism particles and fields through which they interact.

To describe elementary particles and their interactions, all the basic concepts of quantum mechanics are used: the wave-particle dualism remains valid, the language of operators and quantum numbers is preserved, the probabilistic interpretation of the observed phenomena, etc. In particular, to explain the interconversion of three types of neutrinos: v e, ν μ and ν τ (neutrino oscillations), as well as neutral K-mesons, the principle of superposition of states is used.

Interpreting quantum mechanics... The validity of the equations and conclusions of quantum mechanics has been repeatedly confirmed by numerous experiments. The system of its concepts, created by the works of N. Bohr, his students and followers, known as the "Copenhagen interpretation", is now generally accepted, although a number of creators of quantum mechanics (M. Planck, A. Einstein and E. Schrödinger, etc.) until the end of their lives remained convinced that quantum mechanics was an unfinished theory. The specific difficulty of perceiving quantum mechanics is due, in particular, to the fact that most of its basic concepts (wave, particle, observation, etc.) are taken from classical physics. In quantum mechanics, their meaning and range of applicability are limited due to the finiteness of the quantum of action h, and this, in turn, required a revision of the established provisions of the philosophy of knowledge.

First of all, the meaning of the concept of "observation" has changed in quantum mechanics. In classical physics, it was assumed that the disturbances of the system under study caused by the measurement process can be correctly taken into account, after which it is possible to restore the initial state of the system, independent of the means of observation. In quantum mechanics, the uncertainty relation sets a fundamental limit on this path that has nothing to do with the skill of the experimenter and the subtlety of the observation methods used. The quantum of action h defines the boundaries of quantum mechanics, like the speed of light in the theory of electromagnetic phenomena or absolute zero temperatures in thermodynamics.

The reason for the rejection of the uncertainty relation and the way of overcoming the difficulties of perceiving its logical consequences was proposed by N. Bohr in the concept of complementarity (see Complementarities principle). According to Bohr, a complete and adequate description of quantum phenomena requires a pair of additional concepts and a corresponding pair of observables. To measure these observables, two different types of instruments with incompatible properties are required. For example, to accurately measure a coordinate, a stable, massive device is needed, and to measure an impulse, on the contrary, a light and sensitive one. Both of these devices are incompatible, but they are complementary in the sense that both quantities they measure are equally necessary for the complete characterization of a quantum object or phenomenon. Bohr explained that “phenomenon” and “observation” are additional concepts and cannot be defined separately: the process of observation is already a certain phenomenon, and without observation, a phenomenon is a “thing-in-itself”. In reality, we are always dealing not with the phenomenon itself, but with the result of observing the phenomenon, and this result depends, among other things, on the choice of the type of device used to measure the characteristics of a quantum object. Quantum mechanics explains and predicts the results of such observations without any arbitrariness.

An important difference between quantum equations and classical equations is that the wave function of a quantum system itself is not observable, and all quantities calculated with its help have a probabilistic meaning. In addition, the concept of probability in quantum mechanics is fundamentally different from the usual understanding of probability as a measure of our ignorance of the details of processes. Probability in quantum mechanics is an intrinsic property of an individual quantum phenomenon, inherent in it initially and independently of measurements, and not a way of presenting the results of measurements. Accordingly, the principle of superposition in quantum mechanics refers not to probabilities, but to the amplitudes of probability. In addition, due to the probabilistic nature of events, the superposition of quantum states can include states that are incompatible from the classical point of view, for example, the states of reflected and transmitted photons at the boundary of a semitransparent screen or alternative states of an electron passing through any of the slits in the famous interference experiment.

The rejection of the probabilistic interpretation of quantum mechanics gave rise to a lot of attempts to modify the basic provisions of quantum mechanics. One of such attempts is the introduction of hidden parameters into quantum mechanics, which change in accordance with strict laws of causality, and the probabilistic nature of the description in quantum mechanics arises as a result of averaging over these parameters. The proof of the impossibility of introducing hidden parameters into quantum mechanics without violating the system of its postulates was given by J. von Neumann back in 1929. A more detailed analysis of the system of postulates of quantum mechanics was undertaken by J. Bell in 1965. Experimental verification of the so-called Bell's inequalities (1972) once again confirmed the generally accepted scheme of quantum mechanics.

Nowadays, quantum mechanics is a complete theory that always gives correct predictions within the limits of its applicability. All known attempts to modify it (about ten of them are known) did not change its structure, but laid the foundation for new branches of science about quantum phenomena: quantum electrodynamics, quantum field theory, electroweak interaction theory, quantum chromodynamics, quantum theory of gravity, theory of strings and superstrings, etc. ...

Quantum mechanics is among such scientific achievements as classical mechanics, the theory of electricity, the theory of relativity and kinetic theory. No physical theory has explained such a wide range of physical phenomena in nature: of the 94 Nobel Prizes in physics awarded in the 20th century, only 12 are not directly related to quantum physics. The importance of quantum mechanics in the entire system of knowledge about the surrounding nature goes far beyond the theory of quantum phenomena: it created a language of communication in modern physics, chemistry and even biology, led to a revision of the philosophy of science and the theory of knowledge, and its technological consequences still determine the direction development of modern civilization.

Lit .: Neiman I. Mathematical foundations of quantum mechanics. M., 1964; Davydov A.S. Quantum mechanics. 2nd ed. M., 1973; Dirac P. Principles of Quantum Mechanics. 2nd ed. M., 1979; Blokhintsev D.I., Fundamentals of Quantum Mechanics. 7th ed. SPb., 2004; Landau L.D., Lifshitz E.M., Quantum mechanics. Non-relativistic theory. 5th ed. M., 2004; Feynman R., Leighton R., Sands M. Quantum Mechanics. 3rd ed. M., 2004; Ponomarev L.I. Under the Sign of Quantum. 2nd ed. M., 2007; Fock VA Beginnings of quantum mechanics. 5th ed. M., 2008.

Quantum mechanics is understood as the physical theory of the dynamic behavior of forms of radiation and matter. This is the foundation of the modern theory of physical bodies, molecules and elementary particles. At all, quantum mechanics was created by scientists who sought to understand the structure of the atom. For many years, the legendary physicists have studied the features and directions of chemistry and followed the historical time of the development of events.

A concept like quantum mechanics, emerged over the years. In 1911, scientists N. Bohr proposed a nuclear model of the atom, which resembled Copernicus's model with his solar system. After all, the solar system had a core at its center, around which the elements revolved. On the basis of this theory, calculations began on the physical and chemical properties of some substances, which were built from simple atoms.

One of the important questions in such a theory as quantum mechanics is the nature of the forces that bound the atom. Thanks to Coulomb's law, E. Rutherford showed that this law is valid on a huge scale. Then it was necessary to determine how the electrons move in their orbit. Helped at this point

Actually, quantum mechanics often contradicts concepts such as common sense. Along with the fact that our common sense works and shows only such things that can be taken from everyday experience. And, in turn, everyday experience deals only with the phenomena of the macrocosm and large objects, while material particles at the subatomic and atomic level behave in a completely different way. For example, in the macrocosm, we are easily able to determine the location of any object using measuring instruments and methods. And if we measure the coordinates of an electron microparticle, then it is simply unacceptable to neglect the interaction of the measurement object and the measuring device.

In other words, we can say that quantum mechanics is a physical theory that establishes the laws of motion of various microparticles. From classical mechanics that describes the movement of microparticles, quantum mechanics differs in two indicators:

The probable nature of some physical quantities, for example, the speed and position of a microparticle, cannot be accurately determined; only the probability of their values ​​can be calculated;

A discrete change, for example, the energy of a microparticle has only certain certain values.

Quantum mechanics is still associated with such a concept as quantum cryptography, which is a fast-growing technology that can change the world. Quantum cryptography aims to protect communications and information privacy. This cryptography is based on certain phenomena and considers cases where information can be transferred using an object of quantum mechanics. It is here, with the help of electrons, photons and other physical means, that the process of receiving and sending information is determined. Thanks to quantum cryptography, it is possible to create and design a communication system that can detect eavesdropping.

To date, there are a lot of materials where it is proposed to study such a concept as quantum mechanics fundamentals and directions, as well as activities of quantum cryptography. To gain knowledge in this complex theory, it is necessary to thoroughly study and delve into this area. After all, quantum mechanics is far from an easy concept that has been studied and proven by the greatest scientists for many years.