Fundamentals of quantum mechanics. Questions for self-control

MINISTRY OF EDUCATION OF THE RUSSIAN FEDERATION

MOSCOW STATE INSTITUTE OF RADIO ENGINEERING, ELECTRONICS AND AUTOMATION (TECHNICAL UNIVERSITY)

A.A. BERZIN, V.G. MOROZOV

FUNDAMENTALS OF QUANTUM MECHANICS

Tutorial

Moscow – 2004

Introduction

Quantum mechanics appeared a hundred years ago and took shape into a coherent physical theory around 1930. Currently, it is considered the foundation of our knowledge about the world around us. For quite a long time, the application of quantum mechanics to applied problems was limited to nuclear energy (mostly military). However, after the transistor was invented in 1948

One of the main elements of semiconductor electronics, and in the late 1950s, a laser was created - a quantum light generator, it became clear that discoveries in quantum physics have enormous practical potential and serious familiarity with this science is necessary not only for professional physicists, but also for representatives of other specialties - chemists, engineers and even biologists.

As quantum mechanics increasingly began to acquire the features of not only fundamental, but also applied science, the problem arose of teaching its basics to students of non-physical specialties. The student first becomes acquainted with some quantum ideas in a course of general physics, but, as a rule, this acquaintance is limited to nothing more than random facts and their greatly simplified explanations. On the other hand, a full course of quantum mechanics taught at university physics departments is clearly redundant for those who would like to apply their knowledge not to revealing the secrets of nature, but to solving technical and other practical problems. The difficulty of “adapting” a course in quantum mechanics to the needs of teaching students of applied specialties was noticed a long time ago and has not yet been completely overcome, despite numerous attempts to create “transitional” courses focused on the practical applications of quantum laws. This is due to the specifics of quantum mechanics itself. First, to understand quantum mechanics, the student requires a thorough knowledge of classical physics: Newtonian mechanics, the classical theory of electromagnetism, the special theory of relativity, optics, etc. Secondly, in quantum mechanics, in order to correctly describe phenomena in the microworld, one has to sacrifice clarity. Classical physics operates with more or less visual concepts; their connection with experiment is relatively simple. The situation is different in quantum mechanics. As noted by L.D. Landau, who made a significant contribution to the creation of quantum mechanics, “it is necessary to understand what we can no longer imagine.” Usually, difficulties in studying quantum mechanics are usually explained by its rather abstract mathematical apparatus, the use of which is inevitable due to the loss of clarity of concepts and laws. Indeed, to learn how to solve quantum mechanical problems, you need to know differential equations, be fairly fluent in handling complex numbers, and also be able to do much more. All this, however, does not go beyond the mathematical training of a student at a modern technical university. The real difficulty of quantum mechanics is not only, or even so much, related to mathematics. The fact is that the conclusions of quantum mechanics, like any physical theory, must predict and explain real experiments, therefore, you need to learn to connect abstract mathematical constructs with measurable physical quantities and observable phenomena. This skill is developed by each person individually, mainly by independently solving problems and comprehending the results. Newton also noted: “in the study of science, examples are often more important than rules.” With regard to quantum mechanics, these words contain a great deal of truth.

The manual offered to the reader is based on many years of practice of teaching the course “Physics 4” at MIREA, dedicated to the fundamentals of quantum mechanics, to students of all specialties of the faculties of electronics and RTS and students of those specialties of the faculty of cybernetics, where physics is one of the main academic disciplines. The content of the manual and the presentation of the material are determined by a number of objective and subjective circumstances. First of all, it was necessary to take into account that the “Physics 4” course is designed for one semester. Therefore, from all sections of modern quantum mechanics, those that are directly related to electronics and quantum optics - the most promising areas of application of quantum mechanics - have been selected. However, unlike courses in general physics and applied technical disciplines, we sought to present these sections within the framework of a unified and fairly modern approach, taking into account the students’ abilities to master it. The volume of the manual exceeds the content of lectures and practical classes, since the course “Physics 4” requires students to complete coursework or individual assignments that require independent study of issues not included in the lecture plan. The presentation of these issues in textbooks on quantum mechanics, aimed at students of university physics departments, often exceeds the level of preparation of a student at a technical university. Thus, this manual can be used as a source of material for coursework and individual assignments.

An important part of the manual are the exercises. Some of them are given directly in the text, the rest are placed at the end of each paragraph. Many of the exercises include instructions for the reader. In connection with the “unusuality” of the concepts and methods of quantum mechanics noted above, performing exercises should be considered as an absolutely necessary element of studying the course.

1. Physical origins of quantum theory

1.1. Phenomena that contradict classical physics

Let's start with a brief overview of the phenomena that classical physics could not explain and which ultimately led to the emergence of quantum theory.

Spectrum of equilibrium radiation of a black body. Recall that in physics

A black body (often called an “absolute black body”) is a body that completely absorbs electromagnetic radiation of any frequency incident on it.

The blackbody is, of course, an idealized model, but it can be realized with high accuracy using a simple device

A closed cavity with a small hole, the inner walls of which are covered with a substance that absorbs electromagnetic radiation well, for example, soot (see Fig. 1.1.). If the wall temperature T is maintained constant, then thermal equilibrium will eventually be established between the substance of the walls

Rice. 1.1. and electromagnetic radiation in the cavity. One of the problems that was actively discussed by physicists at the end of the 19th century was this: how is the energy of equilibrium radiation distributed over

Rice. 1.2.

frequencies? Quantitatively, this distribution is described by the spectral radiation energy density u ω. The productu ω dω is the energy of electromagnetic waves per unit volume with frequencies in the range from ω to ω +dω. Spectral energy density can be measured by analyzing the spectrum of radiation from the opening of the cavity shown in Fig. 1.1. The experimental dependence of u ω for two temperature values ​​is shown in Fig. 1.2. With increasing temperature, the maximum of the curve shifts towards high frequencies and at a sufficiently high temperature, the frequency ω m can reach the region of radiation visible to the eye. The body will begin to glow, and with a further increase in temperature, the color of the body will change from red to violet.

So far we have talked about experimental data. Interest in the spectrum of black body radiation was caused by the fact that the function u ω can be accurately calculated using the methods of classical statistical physics and Maxwell's electromagnetic theory. According to classical statistical physics, in thermal equilibrium the energy of any system is distributed evenly over all degrees of freedom (Boltzmann's theorem). Each independent degree of freedom of the radiation field is an electromagnetic wave with a certain polarization and frequency. According to Boltzmann's theorem, the average energy of such a wave in thermal equilibrium at temperature T is equal to k B T, where k B = 1. 38· 10− 23 J/ K is Boltzmann's constant. That's why

where c is the speed of light. So, the classical expression for the equilibrium spectral radiation density has the form

This formula is the famous Rayleigh-Jeans formula. In classical physics it is accurate and, at the same time, absurd. In fact, according to it, in thermal equilibrium at any temperature there are electromagnetic waves of arbitrarily high frequencies (i.e. ultraviolet radiation, x-rays and even gamma radiation, which is fatal to humans), and the higher the frequency of radiation, the more energy falls on him. The obvious contradiction between the classical theory of equilibrium radiation and experiment received an emotional name in the physical literature - ultraviolet

catastrophe Let us note that the famous English physicist Lord Kelvin, summing up the development of physics in the 19th century, called the problem of equilibrium thermal radiation one of the main unsolved problems.

Photo effect. Another “weak point” of classical physics turned out to be the photoelectric effect - the knocking out of electrons from a substance under the influence of light. It was completely incomprehensible that the kinetic energy of electrons does not depend on the intensity of light, which is proportional to the square of the amplitude of the electric field

V light wave and is equal to the average energy flux incident on the substance. On the other hand, the energy of emitted electrons depends significantly on the frequency of light and increases linearly with increasing frequency. It's also impossible to explain

V within the framework of classical electrodynamics, since the energy flow of an electromagnetic wave, according to Maxwell’s theory, does not depend on its frequency and is completely determined by the amplitude. Finally, the experiment showed that for each substance there is a so-called the red border of the photoelectric effect, i.e., the minimum

frequency ω min at which electron knockout begins. Ifω< ω min , то свет с частотойω не выбьет ни одного электрона, независимо от интенсивности.

Compton effect. Another phenomenon that classical physics could not explain was discovered in 1923 by the American physicist A. Compton. He discovered that when electromagnetic radiation (in the X-ray frequency range) is scattered by free electrons, the frequency of the scattered radiation is less than the frequency of the incident radiation. This experimental fact contradicts classical electrodynamics, according to which the frequencies of incident and scattered radiation must be exactly equal. To verify this, you don't need complex mathematics. It is enough to recall the classical mechanism of scattering of an electromagnetic wave by charged particles. Scheme

electron), which changes over time with the same frequency ω as the field in the incident wave. According to classical electrodynamics, a charge moving with acceleration emits electromagnetic waves. This is scattered radiation. If acceleration changes with time according to a harmonic law with frequency ω, then waves with the same frequency are emitted. The appearance of scattered waves with frequencies lower than the frequency of incident radiation clearly contradicts classical electrodynamics.

Atomic stability. In 1912, a very important event for the entire further development of natural sciences occurred - the structure of the atom was clarified. The English physicist E. Rutherford, conducting experiments on the scattering of alpha particles in matter, established that the positive charge and almost the entire mass of the atom are concentrated in the nucleus with dimensions of the order of 10−12 - 10−13 cm. The dimensions of the nucleus turned out to be negligible compared to the dimensions the atom itself (approximately 10−8 cm). To explain the results of his experiments, Rutherford hypothesized that the atom was structured similarly to the solar system: light electrons move in orbits around a massive nucleus, just as the planets move around the Sun. The force that holds electrons in their orbits is the Coulomb force of attraction of the nucleus. At first glance, such a “planetary model” seems very

1 The symbol everywhere denotes a positive elementary charge = 1.602· 10− 19 C.

attractive: it is clear, simple and quite consistent with Rutherford’s experimental results. Moreover, based on this model, it is easy to estimate the ionization energy of a hydrogen atom containing only one electron. The estimate gives good agreement with the experimental value of the ionization energy. Unfortunately, taken literally, the planetary model of the atom has an unpleasant drawback. The fact is that, from the point of view of classical electrodynamics, such an atom simply cannot exist; he is unstable. The reason for this is quite simple: the electron moves in its orbit with acceleration. Even if the electron's velocity does not change, there is still an acceleration towards the nucleus (normal or "centripetal" acceleration). But, as noted above, a charge moving with acceleration must emit electromagnetic waves. These waves carry away energy, so the energy of the electron decreases. The radius of its orbit decreases and eventually the electron must fall onto the nucleus. Simple calculations, which we will not present, show that the characteristic “lifetime” of an electron in orbit is approximately 10−8 seconds. Thus, classical physics is unable to explain the stability of atoms.

The above examples do not exhaust all the difficulties that classical physics encountered at the turn of the 19th and 20th centuries. We will consider other phenomena where its conclusions contradict experiment later, when the apparatus of quantum mechanics is developed and we can immediately give the correct explanation. Gradually accumulating contradictions between theory and experimental data led to the realization that “not everything is in order” with classical physics and completely new ideas are needed.

1.2. Planck's hypothesis about the quantization of oscillator energy

December 2000 marked the centenary of quantum theory. This date is associated with the work of Max Planck, in which he proposed a solution to the problem of equilibrium thermal radiation. For simplicity, Planck chose as a model of the substance of the cavity walls (see Fig. 1.1.) a system of charged oscillators, that is, particles capable of performing harmonic oscillations around the equilibrium position. If ω is the natural frequency of the oscillator, then it is capable of emitting and absorbing electromagnetic waves of the same frequency. Let the walls of the cavity in Fig. 1.1. contain oscillators with all possible natural frequencies. Then, after thermal equilibrium has been established, the average energy per electromagnetic wave with frequency ω should be equal to the average energy of the oscillator E ω with the same natural frequency of oscillation. Recalling the reasoning given on page 5, let us write the equilibrium spectral radiation density in the following form:

1 In Latin, the word “quantum” literally means “portion” or “piece.”

In turn, the energy quantum is proportional to the frequency of the oscillator:

Some people prefer to use instead of the cyclic frequency ω the so-called linear frequency ν =ω/ 2π, which is equal to the number of oscillations per second. Then expression (1.6) for the energy quantum can be written in the form

The value h = 2π 6, 626176· 10− 34 J· s is also called Planck’s constant1.

Based on the assumption of quantization of the oscillator energy, Planck obtained the following expression for the spectral density of equilibrium radiation2:

In the region of low frequencies (ω k B T ), Planck's formula practically coincides with the Rayleigh-Jeans formula (1.3), and at high frequencies (ω k B T ), the spectral radiation density, in accordance with experiment, quickly tends to zero.

1.3. Einstein's hypothesis about electromagnetic field quanta

Although Planck’s hypothesis about the quantization of oscillator energy “does not fit” into classical mechanics, it could be interpreted in the sense that, apparently, the mechanism of interaction of light with matter is such that radiation energy is absorbed and emitted only in portions, the value of which is given by the formula ( 1.5). In 1900, practically nothing was known about the structure of atoms, so Planck’s hypothesis itself did not yet mean a complete rejection of classical laws. A more radical hypothesis was expressed in 1905 by Albert Einstein. Analyzing the laws of the photoelectric effect, he showed that they are all naturally explained if we accept that light of a certain frequency ω consists of individual particles (photons) with energy

1 Sometimes, to emphasize which Planck constant is meant, it is called “the crossed out Planck constant.”

2 Now this expression is called Planck's formula.

where Aout is the work function, i.e., the energy required to overcome the forces holding the electron in the substance1. The dependence of the photoelectron energy on the frequency of light, described by formula (1.11), was in excellent agreement with the experimental dependence, and the value in this formula turned out to be very close to the value (1.7). Note that by accepting the photon hypothesis, it was also possible to explain the patterns of equilibrium thermal radiation. Indeed, the absorption and emission of electromagnetic field energy by matter occurs in quanta because individual photons having exactly this energy are absorbed and emitted.

1.4. Photon momentum

The introduction of the concept of photons to some extent revived the corpuscular theory of light. The fact that the photon is a “real” particle is confirmed by analysis of the Compton effect. From the point of view of photon theory, the scattering of X-rays can be represented as individual acts of collisions of photons with electrons (see Fig. 1.3.), in which the laws of conservation of energy and momentum must be satisfied.

The law of conservation of energy in this process has the form

By the way, it is immediately clear from here that ω< ω ; это наблюдается и в эксперименте. Чтобы записать закон сохранения импульса в эффекте Комптона, необходимо найти выражение для импульса фотона. Это можно сделать на основе следующих простых рассуждений. Фотон всегда движется со скоростью светаc , но, как известно из теории относительности, частица, движущаяся со скоростью света, должна

have zero mass. So, from the general expression for relativistic

energy E =m 2 c 4 +p 2 c 2 it follows that the energy and momentum of the photon are related by the relation E =pc. Recalling formula (1.10), we obtain

Now the law of conservation of momentum in the Compton effect can be written as

The solution to the system of equations (1.12) and (1.18), which we leave to the reader (see exercise 1.2.), leads to the following formula for changing the wavelength of scattered radiation ∆λ =λ − λ:

is called the Compton wavelength of the particle (mass m) on which radiation is scattered. If m =m e = 0.911· 10− 30 kg is the mass of the electron, then λ C = 0.0243· 10− 10 m. The results of measurements of ∆λ carried out by Compton and then by many other experimenters are completely consistent with the predictions of formula (1.19) , and the value of Planck’s constant, which is included in expression (1.20), coincides with the values ​​obtained from experiments on equilibrium thermal radiation and the photoelectric effect.

After the advent of the photon theory of light and its success in explaining a number of phenomena, a strange situation arose. In fact, let's try to answer the question: what is light? On the one hand, in the photoelectric effect and the Compton effect it behaves as a stream of particles - photons, but, on the other hand, the phenomena of interference and diffraction just as persistently show that light is electromagnetic waves. Based on “macroscopic” experience, we know that a particle is an object that has finite dimensions and moves along a certain trajectory, and a wave fills a region of space, that is, it is a continuous object. How to combine these two mutually exclusive points of view on the same physical reality - electromagnetic radiation? The wave-particle paradox (or, as philosophers prefer to say, wave-particle duality) for light was explained only in quantum mechanics. We will return to it after we get acquainted with the basics of this science.

1 Recall that the modulus of the wave vector is called the wave number.

Exercises

1.1. Using Einstein's formula (1.11), explain the existence of red boundaries of matter. ωmin for photo effect. Expressωmin through the electron work function

1.2. Derive expression (1.19) for the change in radiation wavelength in the Compton effect.

Hint: Dividing equality (1.14) by c and using the relationship between wave number and frequency (k =ω/c), we write

p2 + m2 e c2 = (k − k) + me c.

After squaring both sides, we get

where ϑ is the scattering angle shown in Fig. 1.3. Equating the right-hand sides of (1.21) and (1.22), we arrive at the equality

me c(k − k) = kk(1 −cos ϑ) .

It remains to multiply this equality by 2π, divide by m e ckk and move from wave numbers to wavelengths (2π/k =λ).

2. Quantization of atomic energy. Wave properties of microparticles

2.1. Bohr's atomic theory

Before proceeding directly to the study of quantum mechanics in its modern form, we will briefly discuss the first attempt to apply Planck's idea of ​​quantization to the problem of atomic structure. We will talk about the theory of the atom proposed in 1913 by Niels Bohr. The main goal that Bohr set for himself was to explain the surprisingly simple pattern in the emission spectrum of the hydrogen atom, which was formulated by Ritz in 1908 in the form of the so-called combination principle. According to this principle, the frequencies of all lines in the spectrum of hydrogen can be represented as the differences of certain quantities T (n) (“terms”), the sequence of which is expressed in terms of integers.

A. SHISHLOVA. based on materials from the journals "Advances in Physical Sciences" and "Scientific American".

The quantum mechanical description of the physical phenomena of the microworld is considered the only correct one and most fully consistent with reality. Objects of the macrocosm obey the laws of another, classical mechanics. The boundary between the macro and micro world is blurred, and this causes a number of paradoxes and contradictions. Attempts to eliminate them lead to the emergence of other views on quantum mechanics and the physics of the microworld. Apparently, the American theorist David Joseph Bohm (1917-1992) was able to express them best.

1. A thought experiment on measuring the components of the spin (momentum of motion) of an electron using a certain device - a “black box”.

2. Consecutive measurement of two spin components. The “horizontal” spin of the electron is measured (on the left), then the “vertical” spin (on the right), then again the “horizontal” spin (below).

3A. Electrons with a “right” spin after passing through a “vertical” box move in two directions: up and down.

3B. In the same experiment, we will place a certain absorbing surface on the path of one of the two beams. Further, only half of the electrons participate in the measurements, and at the output, half of them have a “left” spin, and half have a “right” spin.

4. The state of any object in the microworld is described by the so-called wave function.

5. Thought experiment by Erwin Schrödinger.

6. The experiment proposed by D. Bohm and Ya. Aharonov in 1959 was supposed to show that a magnetic field inaccessible to a particle affects its state.

To understand what difficulties modern quantum mechanics is experiencing, we need to remember how it differs from classical, Newtonian mechanics. Newton created a general picture of the world in which mechanics acted as a universal law of motion of material points or particles - small lumps of matter. Any objects could be built from these particles. It seemed that Newtonian mechanics was capable of theoretically explaining all natural phenomena. However, at the end of the last century it became clear that classical mechanics is unable to explain the laws of thermal radiation of heated bodies. This seemingly private question led to the need to revise physical theories and required new ideas.

In 1900, the work of the German physicist Max Planck appeared, in which these new ideas appeared. Planck suggested that radiation occurs in portions, quanta. This idea contradicted classical views, but perfectly explained the results of experiments (in 1918 this work was awarded the Nobel Prize in Physics). Five years later, Albert Einstein showed that not only radiation, but also absorption of energy should occur discretely, in portions, and was able to explain the features of the photoelectric effect (Nobel Prize 1921). According to Einstein, a light quantum - photon, having wave properties, at the same time in many ways resembles a particle (corpuscle). Unlike a wave, for example, it is either completely absorbed or not absorbed at all. This is how the principle of wave-particle duality of electromagnetic radiation arose.

In 1924, the French physicist Louis de Broglie put forward a rather “crazy” idea, suggesting that all particles without exception - electrons, protons and entire atoms - have wave properties. A year later, Einstein said of this work: “Although it seems like it was written by a madman, it was written solidly,” and in 1929 de Broglie received the Nobel Prize for it...

At first glance, everyday experience rejects de Broglie’s hypothesis: there seems to be nothing “wave” in the objects around us. Calculations, however, show that the de Broglie wavelength of an electron accelerated to an energy of 100 electron volts is 10 -8 centimeters. This wave can be easily detected experimentally by passing a stream of electrons through a crystal. Diffraction of their waves will occur on the crystal lattice and a characteristic striped pattern will appear. But for a speck of dust weighing 0.001 grams at the same speed, the de Broglie wavelength will be 10 24 times smaller, and it cannot be detected by any means.

De Broglie waves are unlike mechanical waves - vibrations of matter propagating in space. They characterize the probability of detecting a particle at a given point in space. Any particle appears to be “smeared” in space, and there is a non-zero probability of finding it anywhere. A classic example of a probabilistic description of objects in the microworld is the experiment on electron diffraction by two slits. An electron passing through the slit is recorded on a photographic plate or on a screen in the form of a speck. Each electron can pass through either the right slit or the left slit in a completely random manner. When there are a lot of specks, a diffraction pattern appears on the screen. The blackening of the screen turns out to be proportional to the probability of an electron appearing in a given location.

De Broglie's ideas were deepened and developed by the Austrian physicist Erwin Schrödinger. In 1926, he derived a system of equations - wave functions that describe the behavior of quantum objects in time depending on their energy (Nobel Prize 1933). From the equations it follows that any impact on the particle changes its state. And since the process of measuring the parameters of a particle is inevitably associated with an impact, the question arises: what does the measuring device record, introducing unpredictable disturbances into the state of the measured object?

Thus, the study of elementary particles has made it possible to establish at least three extremely surprising facts regarding the general physical picture of the world.

Firstly, it turned out that the processes occurring in nature are controlled by pure chance. Secondly, it is not always possible in principle to indicate the exact position of a material object in space. And thirdly, what is perhaps most strange, the behavior of such physical objects as a “measuring device” or an “observer” is not described by fundamental laws that are valid for other physical systems.

For the first time, the founders of quantum theory themselves - Niels Bohr, Werner Heisenberg, Wolfgang Pauli - came to such conclusions. Later, this point of view, called the Copenhagen interpretation of quantum mechanics, was accepted in theoretical physics as official, which was reflected in all standard textbooks.

It is quite possible, however, that such conclusions were made too hastily. In 1952, the American theoretical physicist David D. Bohm created a deeply developed quantum theory, different from the generally accepted one, which also well explains all the currently known features of the behavior of subatomic particles. It represents a unified set of physical laws that allows us to avoid any randomness in describing the behavior of physical objects, as well as the uncertainty of their position in space. Despite this, Bohm's theory was almost completely ignored until very recently.

To better imagine the complexity of describing quantum phenomena, let’s conduct several thought experiments to measure the spin (intrinsic angular momentum) of an electron. Mental because no one has yet succeeded in creating a measuring device that allows accurately measuring both components of spin. Equally unsuccessful are attempts to predict which electrons will change their spin during the experiment described and which will not.

These experiments include the measurement of two spin components, which we will conventionally call “vertical” and “horizontal” spins. Each of the components, in turn, can take one of the values, which we will also conventionally call “upper” and “lower”, “right” and “left” spins, respectively. The measurement is based on the spatial separation of particles with different spins. Devices that carry out separation can be imagined as some kind of “black boxes” of two types - “horizontal” and “vertical” (Fig. 1). It is known that different components of the spin of a free particle are completely independent (physicists say they do not correlate with each other). However, during the measurement of one component, the value of another may change, and in a completely uncontrollable manner (2).

Trying to explain the results obtained, traditional quantum theory came to the conclusion that it is necessary to completely abandon the deterministic, that is, completely determining state

object, description of microworld phenomena. The behavior of electrons is subject to the uncertainty principle, according to which the spin components cannot be accurately measured simultaneously.

Let's continue our thought experiments. Now we will not only split electron beams, but also make them reflect from certain surfaces, intersect and reconnect into one beam in a special “black box” (3).

The results of these experiments contradict conventional logic. Indeed, let us consider the behavior of any electron in the case when there is no absorbing wall (3 A). Where will he go? Let's say it's down. Then, if the electron initially had a “right-handed” spin, it will remain right-handed until the end of the experiment. However, applying the results of another experiment (3 B) to this electron, we will see that its “horizontal” spin at the output should be “right” in half the cases, and “left” in half the cases. An obvious contradiction. Could the electron go up? No, for the same reason. Perhaps he was moving not down, not up, but in some other way? But by blocking the upper and lower routes with absorbing walls, we will get nothing at all at the exit. It remains to assume that the electron can move in two directions at once. Then, having the opportunity to fix its position at different times, in half the cases we would find it on the way up, and in half - on the way down. The situation is quite paradoxical: a material particle can neither bifurcate nor “jump” from one trajectory to another.

What does traditional quantum theory say in this case? It simply declares all the situations considered impossible, and the very formulation of the question about a certain direction of motion of the electron (and, accordingly, the direction of its spin) is incorrect. The manifestation of the quantum nature of the electron lies in the fact that in principle there is no answer to this question. The electron state is a superposition, that is, the sum of two states, each of which has a certain value of “vertical” spin. The concept of superposition is one of the fundamental principles of quantum mechanics, with the help of which for more than seventy years it has been possible to successfully explain and predict the behavior of all known quantum systems.

To mathematically describe the states of quantum objects, a wave function is used, which in the case of a single particle simply determines its coordinates. The square of the wave function is equal to the probability of detecting a particle at a given point in space. Thus, if a particle is located in a certain region A, its wave function is zero everywhere except for this region. Similarly, a particle localized in region B has a wave function that is nonzero only in B. If the state of the particle turns out to be a superposition of its presence in A and B, then the wave function describing such a state is nonzero in both regions of space and is equal to zero everywhere outside of them. However, if we set up an experiment to determine the position of such a particle, each measurement will give us only one value: in half the cases we will find the particle in region A, and in half - in B (4). This means that when a particle interacts with its environment, when only one of the states of the particle is fixed, its wave function seems to collapse, “collapse” into a point.

One of the fundamental claims of quantum mechanics is that physical objects are completely described by their wave functions. Thus, the whole point of the laws of physics comes down to predicting changes in wave functions over time. These laws fall into two categories depending on whether the system is left to itself or whether it is directly observed and measured.

In the first case, we are dealing with linear differential “equations of motion”, deterministic equations that completely describe the state of microparticles. Therefore, knowing the wave function of a particle at some point in time, one can accurately predict the behavior of the particle at any subsequent moment. However, when trying to predict the results of measurements of any properties of the same particle, we will have to deal with completely different laws - purely probabilistic ones.

A natural question arises: how to distinguish the conditions of applicability of one or another group of laws? The creators of quantum mechanics point to the need for a clear division of all physical processes into “measurements” and “physical processes themselves,” that is, into “observers” and “observeds,” or, in philosophical terminology, into subject and object. However, the difference between these categories is not fundamental, but purely relative. Thus, according to many physicists and philosophers, quantum theory in such an interpretation becomes ambiguous and loses its objectivity and fundamentality. The "measurement problem" has become a major stumbling block in quantum mechanics. The situation is somewhat reminiscent of Zeno's famous aporia "Heap". One grain is clearly not a heap, but a thousand (or, if you prefer, a million) is a heap. Two grains are also not a heap, but 999 (or 999999) are a heap. This chain of reasoning leads to a certain number of grains at which the concepts of “heap - not heap” become vague. They will depend on the subjective assessment of the observer, that is, on the method of measurement, even by eye.

All macroscopic bodies surrounding us are assumed to be point (or extended) objects with fixed coordinates, which obey the laws of classical mechanics. But this means that the classical description can be continued down to the smallest particles. On the other hand, coming from the microcosm side, one should include in the wave description objects of increasingly larger sizes up to the Universe as a whole. The boundary between the macro- and microworld is not defined, and attempts to define it lead to a paradox. The most clear example of this is the so-called “Schrödinger's cat problem,” a thought experiment proposed by Erwin Schrödinger in 1935 (5).

A cat is sitting in a closed box. There is also a bottle of poison, a radiation source and a charged particle counter connected to a device that breaks the bottle at the moment the particle is detected. If the poison spills, the cat will die. Whether the counter has registered a particle or not, we cannot know in principle: the laws of quantum mechanics are subject to the laws of probability. And from this point of view, until the counter has made measurements, it is in a superposition of two states - “registration - non-registration”. But then at this moment the cat finds itself in a superposition of states of life and death.

In reality, of course, there can be no real paradox here. Registration of a particle is an irreversible process. It is accompanied by a collapse of the wave function, followed by a mechanism that breaks the bottle. However, orthodox quantum mechanics does not consider irreversible phenomena. The paradox that arises in full agreement with its laws clearly shows that between the quantum microworld and the classical macroworld there is a certain intermediate region in which quantum mechanics does not work.

So, despite the undoubted success of quantum mechanics in explaining experimental facts, at the moment it can hardly claim to be a complete and universal description of physical phenomena. One of the most daring alternatives to quantum mechanics was the theory proposed by David Bohm.

Having set out to build a theory free from the uncertainty principle, Bohm proposed to consider a microparticle as a material point capable of occupying an exact position in space. Its wave function receives the status not of a characteristic of probability, but of a very real physical object, a kind of quantum mechanical field that exerts an instantaneous force effect. In the light of this interpretation, for example, the “Einstein-Podolsky-Rosen paradox” (see “Science and Life” No. 5, 1998) ceases to be a paradox. All laws governing physical processes become strictly deterministic and take the form of linear differential equations. One group of equations describes the change in wave functions over time, the other - their effect on the corresponding particles. The laws apply to all physical objects without exception - both “observers” and “observed”.

Thus, if at some moment the position of all particles in the Universe and the complete wave function of each are known, then in principle it is possible to accurately calculate the position of the particles and their wave functions at any subsequent moment in time. Consequently, there can be no talk of any randomness in physical processes. Another thing is that we will never be able to have all the information necessary for accurate calculations, and the calculations themselves turn out to be insurmountably complex. Fundamental ignorance of many system parameters leads to the fact that in practice we always operate with certain averaged values. It is this “ignorance,” according to Bohm, that forces us to resort to probabilistic laws when describing phenomena in the microcosm (a similar situation arises in classical statistical mechanics, for example, in thermodynamics, which deals with a huge number of molecules). Bohm's theory provides certain rules for averaging unknown parameters and calculating probabilities.

Let us return to the experiments with electrons shown in Fig. 3 A and B. Bohm's theory gives them the following explanation. The direction of motion of the electron at the exit from the “vertical box” is completely determined by the initial conditions - the initial position of the electron and its wave function. While the electron moves either up or down, its wave function, as follows from the differential equations of motion, will split and begin to propagate in two directions at once. Thus, one part of the wave function will be “empty”, that is, it will propagate separately from the electron. Having been reflected from the walls, both parts of the wave function will reunite in the “black box”, and at the same time the electron will receive information about that part of the path where it was not. The content of this information, for example about an obstacle in the path of the “empty” wave function, can have a significant impact on the properties of the electron. This removes the logical contradiction between the results of the experiments shown in the figure. It is necessary to note one curious property of “empty” wave functions: being real, they nevertheless do not affect foreign objects in any way and cannot be recorded by measuring instruments. And the “empty” wave function exerts a force on “its” electron regardless of the distance, and this influence is transmitted instantly.

Many researchers have made attempts to “correct” quantum mechanics or explain the contradictions that arise in it. For example, de Broglie tried to build a deterministic theory of the microworld, who agreed with Einstein that “God does not play dice.” And the prominent Russian theorist D.I. Blokhintsev believed that the features of quantum mechanics stem from the impossibility of isolating a particle from the surrounding world. At any temperature above absolute zero, bodies emit and absorb electromagnetic waves. From the standpoint of quantum mechanics, this means that their position is continuously “measured”, causing the collapse of wave functions. “From this point of view, there are no isolated, left to themselves “free” particles,” wrote Blokhintsev. “It is possible that in this connection between particles and the environment the nature of the impossibility of isolating a particle, which manifests itself in the apparatus of quantum mechanics, is hidden.”

And yet, why has the interpretation of quantum mechanics proposed by Bohm still not received due recognition in the scientific world? And how to explain the almost universal dominance of traditional theory, despite all its paradoxes and “dark places”?

For a long time, they did not want to consider the new theory seriously on the grounds that in predicting the outcome of specific experiments it completely coincides with quantum mechanics, without leading to significantly new results. Werner Heisenberg, for example, believed that “for any experiment of his (Bohm’s) the results coincide with the Copenhagen interpretation. Hence the first consequence: Bohm’s interpretation cannot be refuted by experiment...” Some consider the theory to be erroneous, since it gives a predominant role to the position of the particle in space. In their opinion, this contradicts physical reality, since phenomena in the quantum world cannot in principle be described by deterministic laws. There are many other, no less controversial arguments against Bohm's theory, which themselves require serious evidence. In any case, no one has really been able to completely refute it yet. Moreover, many researchers, including domestic ones, continue to work on its improvement.

Mechanics is a science that describes the movement of bodies and correlates physical quantities, such as energy or momentum. It provides accurate and reliable results for many phenomena. This applies to both microscopic-scale phenomena (here classical mechanics is not able to explain even the existence of a stable atom) and some macroscopic phenomena, such as superconductivity, superfluidity or black-body radiation. During the century that quantum mechanics has been around, its predictions have never been challenged by experiment. Quantum mechanics explains at least three types of phenomena that classical mechanics and classical electrodynamics cannot describe:

1) quantization of some physical quantities;

2) wave-particle duality;

3) the existence of mixed quantum states.

Quantum mechanics can be formulated as a relativistic or non-relativistic theory. Although relativistic quantum mechanics is one of the most fundamental theories, non-relativistic quantum mechanics is also often used for convenience.

Theoretical basis of quantum mechanics

Various formulations of quantum mechanics

One of the first formulations of quantum mechanics is “wave mechanics”, proposed by Erwin Schrödinger. In this concept, the state of the system under study is determined by a “wave function” that reflects the probability distribution of all measured physical quantities of the system. Such as energy, coordinates, momentum or angular momentum. The wave function (from a mathematical point of view) is a complex quadratically integrable function of the coordinates and time of the system.

In quantum mechanics, physical quantities are not associated with specific numerical values. On the other hand, assumptions are made about the probability distribution of the values ​​of the measured parameter. As a rule, these probabilities will depend on the type of state vector at the time of the measurement. Although, to be more precise, each specific value of the measured quantity corresponds to a certain state vector, known as the “eigenstate” of the measured quantity.

Let's take a specific example. Let's imagine a free particle. Its state vector is arbitrary. Our task is to determine the coordinate of the particle. The eigenstate of a particle's coordinate in space is a state vector; the norm at a certain point x is quite large, while at any other place in space it is zero. If we now make measurements, then with one hundred percent probability we will get the very value of x.

Sometimes the system we are interested in is not in its own state or the physical quantity we are measuring. However, if we try to make measurements, the wave function instantly becomes an eigenstate of the quantity being measured. This process is called wave function collapse. If we know the wave function at the moment before the measurement, we are able to calculate the probability of collapse into each of the possible eigenstates. For example, a free particle in our previous example for measurement will have a wave function, is a wave packet centered at some point x0, and is not an eigenstate of the coordinate. When we start measuring the coordinates of a particle, it is impossible to predict the result we will get. It is likely, but not certain, that it will be close to x0, where the amplitude of the wave function is large. After the measurement, when we get some result x, the wave function collapses into a position with an eigenstate concentrated precisely at x.

State vectors are functions of time. ψ = ψ (t) The Schrödinger equation determines the change in the state vector over time.

Some state vectors result in probability distributions that are constant over time. Many systems that are considered dynamic in classical mechanics are in fact described by such “static” functions. For example, an electron in an unexcited atom in classical physics is depicted as a particle that moves in a circular path around the nucleus of the atom, while in quantum mechanics it is static, a spherically symmetric probabilistic cloud around the nucleus.

The evolution of a state vector over time is deterministic in the sense that, given a certain state vector at the initial time, one can make an accurate prediction of what it will be at any other moment. During the measurement process, the change in the configuration of the state vector is probabilistic and not deterministic. The probabilistic nature of quantum mechanics thus manifests itself precisely in the process of performing measurements.

There are several interpretations of quantum mechanics that introduce a new concept into the very act of measurement in quantum mechanics. The main interpretation of quantum mechanics, which is generally accepted today, is the probabilistic interpretation.

Physical foundations of quantum mechanics

The uncertainty principle, which states that there are fundamental obstacles to accurately measuring two or more parameters of a system simultaneously with arbitrary uncertainty. In the example with a free particle, this means that it is fundamentally impossible to find a wave function that would be an eigenstate of both momentum and coordinates at the same time. From this it follows that the coordinate and momentum cannot be simultaneously determined with an arbitrary error. As the accuracy of coordinate measurement increases, the maximum accuracy of impulse measurement decreases and vice versa. Those parameters for which such a statement is true are called canonically conjugate in classical physics.

Experimental basis of quantum mechanics

There are experiments that cannot be explained without the use of quantum mechanics. The first type of quantum effects is the quantization of certain physical quantities. If we localize a free particle from the example considered above in a rectangular potential well - a proto-area of ​​size L, bounded on both sides by an infinitely high potential barrier, then it turns out that the momentum of the particle can only have certain discrete values, Where h is Planck’s constant, and n is an arbitrary natural number. Parameters that can only acquire discrete values ​​are said to be quantized. Examples of quantized parameters are also angular momentum, the total energy of a spatially limited system, as well as the energy of electromagnetic radiation of a certain frequency.

Another quantum effect is wave-particle duality. It can be shown that under certain experimental conditions, microscopic objects, such as atoms or electrons, acquire the properties of particles (that is, they can be localized in a certain region of space). Under other conditions, the same objects acquire the properties of waves and exhibit effects such as interference.

The next quantum effect is the effect of entangled quantum states. In some cases, the state vector of a system of many particles cannot be represented as the sum of the individual wave functions corresponding to each of the particles. In this case, they say that the states of the particles are confused. And then, measurements that were carried out for only one particle will result in the collapse of the overall wave function of the system, i.e. such a measurement will have an instantaneous effect on the wave functions of other particles in the system, even if some of them are located at a considerable distance. (This does not contradict the special theory of relativity, since the transfer of information over a distance is impossible in this way.)

Mathematical apparatus of quantum mechanics

In the rigorous mathematics of quantum mechanics, which was developed by Paul Dirac and John von Neumann, the possible states of a quantum mechanical system are represented by state vectors in a complex separable Hilbert space. The evolution of a quantum state is described by the Schrödinger equation, in which the Hamilton operator, or Hamiltonian, corresponding to the total energy of the system, determines its evolution in time.

Each variable parameter of the system is represented by Hermitian operators in the state space. Each eigenstate of the measured parameter corresponds to an eigenvector of the operator, and the corresponding eigenvalue is equal to the value of the measured parameter in that eigenstate. During the measurement process, the probability of the system transitioning to one of its eigenstates is determined as the square of the scalar product of the eigenstate vector and the state vector before the measurement. The possible results of the measurement are the eigenvalues ​​of the operator, explains the choice of Hermitian operators for which all eigenvalues ​​are real numbers. The probability distribution of the measured parameter can be obtained by calculating the spectral decomposition of the corresponding operator (here the spectrum of the operator is the sum of all possible values ​​of the corresponding physical quantity). The Heisenberg uncertainty principle corresponds to the fact that the operators of the corresponding physical quantities do not commute with each other. The details of the mathematical apparatus are presented in a special article, Mathematical apparatus of quantum mechanics.

An analytical solution to the Schrödinger equation exists for a small number of Hamiltonians, for example, for a harmonic oscillator, a model of the hydrogen atom. Even the helium atom, which differs from the hydrogen atom by one electron, does not have a completely analytical solution to the Schrödinger equation. However, there are certain methods for solving these equations approximately. For example, perturbation theory methods, where the analytical result of solving a simple quantum mechanical model is used to obtain solutions for more complex systems by adding a certain “perturbation” in the form of, for example, potential energy. Another method, "Squasi-classical equations of motion" is applied to systems for which quantum mechanics produces only weak deviations from classical behavior. Such deviations can be calculated using classical physics methods. This approach is important in the theory of quantum chaos, which has been rapidly developing recently.

Interaction with other theories

The fundamental principles of quantum mechanics are quite abstract. They claim that the state space of the system is Hilbert, and the physical quantities correspond to Hermitian operators acting in this space, but do not specifically indicate what kind of Hilbert space it is and what kind of operators they are. They must be chosen appropriately to obtain a quantitative description of the quantum system. An important guide here is the correspondence principle, which states that quantum mechanical effects cease to be significant, and the system acquires classical features as its size increases. This “large system” limit is also called a classic or compliance limit. Alternatively, one can start by considering a classical model of a system, and then try to understand which quantum model corresponds to which classical one is outside the matching limit.

When quantum mechanics was first formulated, it was applied to models that corresponded to classical models of nonrelativistic mechanics. For example, the well-known harmonic oscillator model uses a frankly non-relativistic description of the oscillator's kinetic energy, just like the corresponding quantum model.

The first attempts to connect quantum mechanics with the special theory of relativity led to the replacement of the Schrödinger equation with the Dirac equations. These theories were successful in explaining many experimental results, but ignored facts such as relativistic creation and the annihilation of elementary particles. A fully relativistic quantum theory requires the development of a quantum field theory that would apply the concept of quantization to a field rather than to a fixed list of particles. The first completed quantum field theory, quantum electrodynamics, provides a fully quantum description of the processes of electromagnetic interaction.

The full apparatus of quantum field theory is often excessive for describing electromagnetic systems. A simple approach, taken from quantum mechanics, suggests that charged particles are quantum mechanical objects in a classical electromagnetic field. For example, the elementary quantum model of the hydrogen atom describes the electromagnetic field of the atom using the classical Coulomb potential (i.e., inversely proportional to distance). This “pseudoclassical” approach does not work if quantum fluctuations of the electromagnetic field, such as the emission of photons by charged particles, begin to play a significant role.

Quantum field theories for strong and weak nuclear interactions have also been developed. Quantum field theory for strong interactions is called quantum chromodynamics and describes the interaction of subnuclear particles - quarks and gluons. The weak nuclear and electromagnetic forces have been combined in their quantum form into a single quantum field theory called electroweak force theory.

It has not yet been possible to build a quantum model of gravity, the last of the fundamental forces. Pseudoclassical approximations work, and even provide for some effects such as Hawking radiation. But formulating a complete theory of quantum gravity is complicated by existing contradictions between general relativity, the most accurate theory of gravity known today, and some fundamental principles of quantum theory. The intersection of these contradictions is an area of ​​active scientific research, and theories such as string theory are possible candidates for a future theory of quantum gravity.

Application of quantum mechanics

Quantum mechanics has had great success in explaining many environmental phenomena. The behavior of microscopic particles that form all forms of matter - electrons, protons, neutrons, etc. - can often be satisfactorily explained only by the methods of quantum mechanics.

Quantum mechanics is important in understanding how individual atoms combine to form chemical elements and compounds. The application of quantum mechanics to chemical processes is known as quantum chemistry. Quantum mechanics can further provide a qualitatively new understanding of the processes of formation of chemical compounds, showing which molecules are energetically more favorable than others, and by how much. Most of the calculations done in computational chemistry are based on quantum mechanical principles.

Modern technology has already reached the scale where quantum effects become important. Examples are lasers, transistors, electron microscopes, magnetic resonance imaging. The development of semiconductors led to the invention of the diode and transistor, which are indispensable in modern electronics.

Researchers today are searching for reliable methods for directly manipulating quantum states. Successful attempts have been made to create the foundations of quantum cryptography, which will allow guaranteed secret transfer of information. A more distant goal is the development of quantum computers, which are expected to be able to implement certain algorithms much more efficiently than classical computers. Another topic of active research is quantum teleportation, which deals with technologies for transmitting quantum states over significant distances.

Philosophical aspect of quantum mechanics

From the very moment of the creation of quantum mechanics, its conclusions contradicted the traditional understanding of the world order, resulting in active philosophical discussion and the emergence of many interpretations. Even such fundamental principles as the rules of probability amplitudes and probability distributions formulated by Max Born took decades to be accepted by the scientific community.

Another problem with quantum mechanics is that the nature of the object it studies is unknown. In the sense that the coordinates of an object, or the spatial distribution of the probability of its presence, can be determined only if it has certain properties (charge, for example) and environmental conditions (the presence of electric potential).

The Copenhagen interpretation, thanks primarily to Niels Bohr, has been the basic interpretation of quantum mechanics from its formulation to the present day. She argued that the probabilistic nature of quantum mechanical predictions could not be explained in terms of other deterministic theories and placed limits on our knowledge of the environment. Quantum mechanics therefore provides only probabilistic results; the very nature of the Universe is probabilistic, albeit deterministic in the new quantum sense.

Albert Einstein, himself one of the founders of quantum theory, was uncomfortable with the fact that in this theory there was a departure from classical determinism in determining the values ​​of physical quantities of objects. He believed that the existing theory was incomplete and there should have been some additional theory. Therefore, he put forward a series of comments on quantum theory, the most famous of which was the so-called EPR paradox. John Bell showed that this paradox could lead to discrepancies in quantum theory that could be measured. But experiments have shown that quantum mechanics is correct. However, some "inconsistencies" in these experiments leave questions that are still unanswered.

Everett's multiple worlds interpretation, formulated in 1956, proposes a model of the world in which all the possibilities for physical quantities to take on certain values ​​in quantum theory simultaneously occur in reality, in a “multiverse” assembled from mostly independent parallel universes. The multiverse is deterministic, but we get the probabilistic behavior of the universe only because we cannot observe all the universes at the same time.

Story

The foundation of quantum mechanics was laid in the first half of the 20th century by Max Planck, Albert Einstein, Werner Heisenberg, Erwin Schrödinger, Max Born, Paul Dirac, Richard Feynman and others. Some fundamental aspects of the theory still need to be studied. In 1900, Max Planck proposed the concept of energy quantization in order to obtain the correct formula for the energy of blackbody radiation. In 1905, Einstein explained the nature of the photoelectric effect, postulating that light energy is absorbed not continuously, but in portions, which he called quanta. In 1913, Bohr explained the configuration of the spectral lines of the hydrogen atom, again using quantization. In 1924, Louis de Broglie proposed the hypothesis of wave-corpuscular duality.

These theories, although successful, were too fragmentary and together constitute the so-called old quantum theory.

Modern quantum mechanics was born in 1925, when Heisenberg developed matrix mechanics and Schrödinger proposed wave mechanics and his equation. Subsequently, Janos von Neumann proved that both approaches are equivalent.

The next step came when Heisenberg formulated the uncertainty principle in 1927, and around then the probabilistic interpretation began to take shape. In 1927, Paul Dirac combined quantum mechanics with special relativity. He was also the first to use operator theory, including the popular bracket notation. In 1932, John von Neumann formulated the mathematical basis of quantum mechanics based on operator theory.

The era of quantum chemistry was started by Walter Heitler and Fritz London, who published the theory of the formation of covalent bonds in the hydrogen molecule in 1927. Quantum chemistry was subsequently developed by a large community of scientists around the world.

Beginning in 1927, attempts began to apply quantum mechanics to multiparticle systems, resulting in the emergence of quantum field theory. Work in this direction was carried out by Dirac, Pauli, Weiskopf, and Jordan. This line of research culminated in quantum electrodynamics, formulated by Feynman, Dyson, Schwinger and Tomonaga during the 1940s. Quantum electrodynamics is the quantum theory of electrons, positrons and the electromagnetic field.

The theory of quantum chromodynamics was formulated in the early 1960s. This theory, as we know it now, was proposed by Polizter, Gross and Wilczek in 1975. Building on the work of Schwinger, Higgs, Goldston and others, Glashow, Weinberg and Salam independently showed that weak nuclear forces and quantum electrodynamics can be unified and be considered as a single electroweak force.

Quantization

In quantum mechanics, the quantization term is used in several close but different meanings.

Quantization is the discerization of the values ​​of a physical quantity, which in classical physics is continuous. For example, electrons in atoms can only be in certain orbitals with certain energy values. Another example is that the orbital momentum of a quantum mechanical particle can only have very specific values. Discretization of the energy levels of a physical system as its dimensions decrease is called size quantization.
Quantization is also called the transition from a classical description of a physical system to a quantum one. In particular, the procedure for decomposing classical fields (for example, the electromagnetic field) into normal modes and representing them in the form of field quanta (for the electromagnetic field, these are photons) is called secondary quantization.

You've probably heard it many times about the inexplicable 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 interesting to understand these laws, but on the other hand, there is no time to read multi-volume and complex books on physics. I understand you very much, 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, many curious people type in the search bar: “quantum physics for dummies, quantum mechanics for dummies, quantum physics for beginners, quantum mechanics for beginners, basics of quantum physics, basics of quantum mechanics, quantum physics for children, what is quantum mechanics". This publication is exactly 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 a 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 happens to electrons and photons in the microcosm.

An example of the difference between the laws of the 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 will have a ball. But in the microcosm (if there is an atom instead of a ball), an atom can be in two boxes at the same time. This has been confirmed experimentally many times. Isn't it hard to wrap your head around this? But you can't argue with the facts.

Another example. You took a photograph of a fast racing red sports car and in the photo you saw a blurry horizontal stripe, as if the car was located at several points in space at the time of the photo. Despite what you see in the photo, you are still sure that the car was in one specific place in space. In the micro world, everything is different. An electron that rotates around the nucleus of an atom does not actually rotate, but is located simultaneously at all points of the sphere around the nucleus of an atom. Like a loosely wound ball of fluffy wool. This concept in physics is called "electronic cloud" .

A short excursion into history. Scientists first thought about the quantum world when, in 1900, German physicist Max Planck tried to figure out why metals change color when heated. It was he who introduced the concept of quantum. Until then, scientists thought that light traveled continuously. The first person to take Planck's discovery seriously was the then unknown Albert Einstein. He realized that light is not just a wave. Sometimes he 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) .

To make it easier to understand the laws of quantum physicists And mechanics (Wikipedia), we must, in a sense, abstract from the laws of classical physics that are familiar to us. And imagine that you dived, like Alice, into the rabbit hole, into 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. Watch it before we dive into the fundamental questions and concepts of quantum physics.

Quantum physics for dummies video. In the cartoon, pay attention to the “eye” of the observer. It has become a serious mystery 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 slits and land on the screen directly opposite the slits. And they “draw” only 2 vertical stripes on the screen.

Interference of light- This is the “wave” behavior of light, when the screen displays many alternating bright and dark vertical stripes. Also these vertical stripes called interference pattern.

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

So, it's not all that complicated! It is now 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 20s of the last century).

The installation described in the cartoon was not shined with light, but “shot” with electrons (as individual particles). Then, at the beginning of the last century, physicists around 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 you “throw” them into 2 slits, like pebbles, then on the screen behind the slits we should see 2 vertical stripes.

But... The result was stunning. Scientists saw an interference pattern - many vertical stripes. That is, electrons, like light, can also have a wave nature and can interfere. On the other hand, it became clear that light is not only a wave, but also a bit of 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).

Maybe you remember, at school we were told in physics about "wave-particle duality"? It means that when we are talking about very small particles (atoms, electrons) of the microcosm, then They are both waves and particles

Today you and I are so smart and we understand that the 2 experiments described above - shooting with electrons and illuminating slits with light - are the same thing. Because we shoot quantum particles at the slits. We now know that both light and electrons are of a quantum nature, that 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 issue.

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

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

We will throw electrons not in a beam, but one at a time. Let's throw it, wait, let's throw the next one. Now that the electron is flying alone, it will no longer be able to interact with other electrons on the screen. 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 slits one at a time, we will notice... oh horror - they again “drew” an interference wave pattern!

We are slowly starting to 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 it were, through 2 slits at the same time and interfered with itself. Fantastic! Let's return to explaining this phenomenon in the next section.

What is spin and superposition?

We now know what interference is. This is the wave behavior of micro particles - photons, electrons, other micro particles (for simplicity, let's call them photons 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. Otherwise, how can we explain the interference pattern on the screen?

But how can we imagine a photon flying 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 two, but 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 there is a well-known expression that We can confidently say that no one understands quantum mechanics

But this expression of his 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 way incomprehensible to us, fly through 2 slits at the same time. Therefore, it will be easy for us to understand the following important statement of quantum mechanics:

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

Objects of the macroworld can only be in one specific place and in one specific state. But a quantum particle exists according to its own laws. And she doesn’t care that we don’t understand them. That's 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, in 2 or more points at the same time

The same applies to another photon parameter – spin (its own angular momentum). Spin is a vector. A quantum object can be thought of as a microscopic magnet. We are accustomed to the fact that the magnet vector (spin) is either directed up or down. But the electron or photon again tells us: “Guys, we don’t care what you’re 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 "wavefunction collapse"?

There is little left for us to understand what “measurement” is and what “wave function collapse” 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 using a magnetic field: electrons whose spin was directed in the direction of the field will deviate in one direction, and electrons whose spin is directed against the field - in the other. More photons can be directed into a polarizing filter. If the spin (polarization) of the photon is +1, it passes through the filter, but if it is -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, wasn't he?

This is the trick and sensation of quantum mechanics. As long as you do not 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 making a decision which spin vector to accept.

This quantum object is so cool - it makes decisions about its state. And we cannot predict in advance what decision it will make when it flies into the magnetic field in which we measure it. The probability that he will decide to have a spin vector “up” or “down” is 50 to 50%. But as soon as he decides, he is in a certain state with a specific spin direction. The reason for his decision is our “dimension”!

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

Attention! An excellent example for understanding is an association from our macrocosm:

Spin a coin on the table like a spinning top. While the coin is spinning, it does not have a specific meaning - heads or tails. But as soon as you decide to “measure” this value and slam the coin with your hand, that’s when you get the specific state of the coin - heads or tails. Now imagine that this coin decides which value to “show” you - heads or tails. The electron behaves in approximately the same way.

Now remember the experiment shown at the end of the cartoon. When photons were passed through the slits, they behaved like a wave and showed an interference pattern on the screen. And when scientists wanted to record (measure) the moment of photons flying through the slit and placed an “observer” behind the screen, the photons began to behave not like waves, but like particles. And they “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.

Fantastic! Isn't it true?

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

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

• What's happened ?
• What is a thought experiment?

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

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I wish everyone inspiration for all their plans and projects!

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

Such a concept as quantum mechanics, has been brewing for many years. In 1911, scientists N. Bohr proposed a nuclear model of the atom, which resembled the model of Copernicus with his solar system. After all, the solar system had a core at its center around which the elements revolved. Based on this theory, calculations of the physical and chemical properties of some substances, which were built from simple atoms, began.

One of the important issues in such a theory is quantum mechanics- this 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

In fact, quantum mechanics often contradicts concepts such as common sense. Along with the fact that our common sense acts and shows only such things that can be taken from everyday experience. And, in turn, everyday experience deals only with the phenomena of the macroworld and large objects, while material particles at the subatomic and atomic level behave completely differently. 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, which describes the movement of microparticles, quantum mechanics differs in two respects:

The probable nature of some physical quantities, for example, the speed and position of a microparticle cannot be determined exactly; 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 also associated with such a concept as quantum cryptography, which is a fast-growing technology with the potential to change the world. Quantum cryptography aims to protect communications and information privacy. This cryptography is based on certain phenomena and considers such cases when information can be transferred using an object of quantum mechanics. It is here that the process of receiving and sending information is determined with the help of electrons, photons and other physical means. Thanks to quantum cryptography, it is possible to create and design a communication system that can detect eavesdropping.

At the moment, there are quite a lot of materials that offer the study of such a concept as quantum mechanics basics 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, which has been studied and proven by the greatest scientists for many years.