QUANTUM MECHANICS
fundamental physical theory of the dynamic behavior of all elementary forms of matter and radiation, as well as their interactions. Quantum mechanics is the theoretical basis on which the modern theory of atoms, atomic nuclei, molecules and physical bodies, as well as the elementary particles that make up all this, is built. Quantum mechanics was created by scientists seeking to understand how the atom works. Atomic processes have been studied for many years by physicists and especially by chemists; in presenting this question, we will, without going into details of the theory, follow the historical course of development of the subject. see also ATOM.
The origin of the theory. When E. Rutherford and N. Bohr proposed the nuclear model of the atom in 1911, it was like a miracle. Indeed, it was built from what has been known for over 200 years. It was, in essence, a Copernican model of the solar system, reproduced on a microscopic scale: in the center is a heavy mass, soon called the nucleus, around which electrons revolve, the number of which determines the chemical properties of the atom. But more than that, behind this illustrative model there was a theory that made it possible to begin calculations of some chemical and physical properties of substances, at least built from the smallest and most simple atoms. The Bohr-Rutherford theory contained a number of provisions that it is useful to recall here, since all of them, in one form or another, have been preserved in modern theory. First, the question of the nature of the forces that bind the atom is important. From the 18th century it was known that electrically charged bodies attract or repel each other with a force inversely proportional to the square of the distance between them. Using alpha particles resulting from radioactive transformations as test bodies, Rutherford showed that the same law of electrical interaction (Coulomb's law) is valid on a scale a million million times smaller than those for which it was originally experimentally established. Secondly, it was necessary to answer the question of how electrons move in orbits under the influence of these forces. Here again, Rutherford's experiments seemed to show (and Bohr accepted this in his theory) that Newton's laws of motion, formulated in his Principia Mathematica, 1687, could be used to describe the motion of particles on these new scales of the microcosm. Third, there was the issue of stability. In the Newtonian-Coulomb atom, as in the solar system, the sizes of the orbits are arbitrary and depend only on how the system was originally set in motion. However, all atoms of one substance are the same and, moreover, stable, which is completely inexplicable from the point of view of old ideas. Bohr suggested that atomic electrons should be considered as moving around the nucleus only in certain orbits, which correspond to certain energy levels, and they should emit a quantum of energy in the form of light, moving from an orbit with a higher energy to an orbit with a lower energy. Such "quantization conditions" did not follow from any experimental data or theories; they were accepted as postulates. On the basis of these conceptual elements, supplemented by the ideas just developed at that time by M. Planck and A. Einstein on the nature of light, Bohr managed to quantitatively explain the entire emission spectrum of hydrogen atoms in a gas discharge tube and give a qualitative explanation of all the basic laws of the periodic system of elements. By 1920, it was time to tackle the problem of the emission spectrum of heavier atoms and calculate the intensity of the chemical forces that bind atoms in compounds. But here the illusion of success faded. For a number of years, Bohr and other researchers unsuccessfully tried to calculate the spectrum of helium, the simplest atom with two electrons following hydrogen. At first nothing worked at all; in the end, several researchers solved this problem in various ways, but the answer turned out to be wrong - it contradicted the experiment. Then it turned out that it is generally impossible to construct any acceptable theory of chemical interaction. By the early 1920s, Bohr's theory had exhausted itself. The time has come to recognize the validity of the prophetic remark that Bohr made back in 1914 in a letter to a friend in his usual intricate style: “I am inclined to believe that the problem is connected with exceptionally great difficulties that can be overcome only by moving much further away from ordinary considerations than required hitherto, and that the success achieved so far has been due solely to the simplicity of the systems considered."
see also
BOR Niels Henrik David;
LIGHT ;
RUTHERFORD Ernest;
SPECTROSCOPY.
First steps. Since Bohr's combination of pre-existing ideas from the field of electricity and mechanics with quantization conditions led to incorrect results, all this had to be completely or partially changed. The main provisions of Bohr's theory were given above, and for the corresponding calculations, not very complicated calculations using ordinary algebra and mathematical analysis were sufficient. In 1925, the young German physicist W. Heisenberg visited Bohr in Copenhagen, where he spent long hours in conversations with him, figuring out what Bohr's theory must necessarily enter into a future theory, and what, in principle, can be abandoned. Bohr and Heisenberg immediately agreed that in a future theory everything directly observable must necessarily be represented, and everything that is not amenable to observation can be changed or excluded from consideration. From the very beginning, Heisenberg believed that atoms should be preserved, but the orbit of an electron in an atom should be considered an abstract idea, since no experiment can determine the electron orbit from measurements in the same way as it can be done for planetary orbits. The reader may notice that there is a certain illogicality here: strictly speaking, the atom is just as unobservable directly as the electron orbits, and in general, in our perception of the surrounding world, there is not a single sensation that would not require explanation. Nowadays, physicists are increasingly quoting the famous aphorism, which was first uttered by Einstein in a conversation with Heisenberg: "What we observe, the theory tells us." Thus the distinction between observable and unobservable quantities is purely practical, having no justification either in strict logic or in psychology, and this distinction, however drawn, must be considered as part of the theory itself. Therefore, the Heisenberg ideal of a theory, cleansed of everything unobservable, is a certain direction of thought, but by no means a consistent scientific approach. Nevertheless, it dominated atomic theory for almost half a century after it was first formulated. We have already mentioned the building blocks of Bohr's early model, such as Coulomb's law for electric forces, Newton's laws of dynamics, and the usual rules of algebra. Through subtle analysis, Heisenberg showed that it was possible to preserve the known laws of electricity and dynamics by finding the proper expression for Newtonian dynamics and then changing the rules of algebra. In particular, Heisenberg suggested that since neither the position q nor the momentum p of an electron are measurable quantities in the sense in which, for example, the position and momentum of a car are, we can, if we wish, preserve them in the theory only by considering as mathematical symbols denoted by letters, but not as numbers. He adopted algebraic rules for p and q, according to which the product pq does not coincide with the product qp. Heisenberg showed that simple calculations of atomic systems give acceptable results, assuming that position q and momentum p hold
Where h is Planck's constant, already known from the quantum theory of radiation and featured in Bohr's theory, a. Planck's constant h is a common number, but very small, approximately 6.6×10-34 J*s. Thus, if p and q are values of the usual scale, then the difference between the products of pq and qp will be extremely small compared to these products themselves, so that p and q can be considered ordinary numbers. Constructed to describe the phenomena of the microcosm, Heisenberg's theory agrees almost completely with Newton's mechanics when applied to macroscopic objects. Already in the earliest works of Heisenberg, it was shown that for all the ambiguity of the physical content of the new theory, it predicts the existence of discrete energy states characteristic of quantum phenomena (for example, for the emission of light by an atom). In later work, carried out jointly with M. Born and P. Jordan in Göttingen, Heisenberg developed the formal mathematical apparatus of the theory. Practical calculations remained, however, extremely complex. After several weeks of hard work, W. Pauli derived a formula for the energy levels of the hydrogen atom, which coincides with the Bohr formula. But before it was possible to simplify the calculations, new and completely unexpected ideas appeared. see also
ALGEBRA, ABSTRACT;
PLANK CONSTANT.
Particles and waves. By the 1920s, physicists were already quite familiar with the dual nature of light: the results of some experiments with light could be explained by assuming that light was waves, while in others it behaved like a stream of particles. Since it seemed obvious that nothing could be both a wave and a particle at the same time, the situation remained unclear, causing heated debate among specialists. In 1923, the French physicist L. de Broglie, in his published notes, suggested that such paradoxical behavior may not be specific to light, but matter can also behave like particles in some cases, and like waves in others. Based on the theory of relativity, de Broglie showed that if the momentum of a particle is equal to p, then the wave "associated" with this particle must have a wavelength l = h/p. This relation is analogous to the relation E = hn first obtained by Planck and Einstein between the energy of the light quantum E and the frequency n of the corresponding wave. De Broglie also showed that this hypothesis could be easily tested in experiments analogous to the experiment demonstrating the wave nature of light, and he strongly urged such experiments to be carried out. De Broglie's notes attracted the attention of Einstein, and by 1927 K. Davisson and L. Germer in the United States, as well as J. Thomson in England, confirmed for electrons not only de Broglie's basic idea, but also his formula for the wavelength. In 1926, the Austrian physicist E. Schrödinger, then working in Zurich, having heard about de Broglie's work and the preliminary results of experiments confirming it, published four articles in which he presented a new theory, which was a solid mathematical foundation for these ideas. This situation has its analogue in the history of optics. The mere belief that light is a wave of a certain length is not sufficient for a detailed description of the behavior of light. It is also necessary to write and solve the differential equations derived by J. Maxwell, which describe in detail the processes of interaction of light with matter and the propagation of light in space in the form of an electromagnetic field. Schrödinger wrote a differential equation for de Broglie's material waves, similar to Maxwell's equations for light. The Schrödinger equation for one particle has the form
where m is the mass of the particle, E is its total energy, V(x) is the potential energy, and y is the quantity describing the electron wave. In a number of papers, Schrödinger showed how his equation could be used to calculate the energy levels of the hydrogen atom. He also established that there are simple and effective ways to approximate problems that cannot be solved exactly, and that his theory of matter waves is mathematically completely equivalent to Heisenberg's algebraic theory of observable quantities and in all cases leads to the same results. P. Dirac of the University of Cambridge showed that the theories of Heisenberg and Schrödinger are only two of many possible forms of theory. Dirac's theory of transformations, in which relation (1) plays the most important role, provided a clear general formulation of quantum mechanics, covering all its other formulations as special cases. Soon, Dirac achieved an unexpectedly great success by demonstrating how quantum mechanics can be generalized to the region of very high velocities, i.e. takes on a form that satisfies the requirements of the theory of relativity. Gradually it became clear that there are several relativistic wave equations, each of which, in the case of low velocities, can be approximated by the Schrödinger equation, and that these equations describe completely different types of particles. For example, particles can have different "spin"; this is provided for by Dirac's theory. In addition, according to the relativistic theory, each of the particles must correspond to an antiparticle with the opposite sign of the electric charge. At the time when Dirac's work came out, only three elementary particles were known: the photon, the electron, and the proton. In 1932, the antiparticle of the electron, the positron, was discovered. Over the next few decades, many other antiparticles were discovered, most of which turned out to satisfy the Dirac equation or its generalizations. Created in 1925-1928 by the efforts of outstanding physicists, quantum mechanics has not undergone any significant changes in its foundations since then.
see also ANTI-MATTER.
Applications. In all branches of physics, biology, chemistry and engineering, in which the properties of matter on a small scale are essential, quantum mechanics is now systematically addressed. Let's give some examples. The structure of electron orbits, the most remote from the nucleus of atoms, has been comprehensively studied. The methods of quantum mechanics were applied to the problems of the structure of molecules, which led to a revolution in chemistry. The structure of molecules is determined by the chemical bonds of atoms, and today the complex problems arising from the consistent application of quantum mechanics in this area are solved with the help of computers. Much attention has been drawn to the theory of the crystal structure of solids, and especially to the theory of the electrical properties of crystals. The practical results are impressive: examples include the invention of lasers and transistors, as well as significant advances in explaining the phenomenon of superconductivity.
see also
PHYSICS OF THE SOLID STATE;
LASER ;
TRANSISTOR ;
SUPERCONDUCTIVITY. Many problems have not yet been resolved. This concerns the structure of the atomic nucleus and the physics of elementary particles. From time to time, the question is discussed whether the problems of elementary particle physics lie outside quantum mechanics, just as the structure of atoms turned out to be outside the scope of Newtonian dynamics. However, there are still no indications that the principles of quantum mechanics or its generalizations in the field of field dynamics have turned out to be inapplicable somewhere. For more than half a century, quantum mechanics has remained a scientific tool with a unique "explaining ability" and does not require significant changes in its mathematical structure. Therefore, it may seem surprising that there are still heated debates (see below) about the physical meaning of quantum mechanics and its interpretation.
see also
ATOM STRUCTURE;
ATOMIC NUCLEI STRUCTURE;
MOLECULE STRUCTURE;
ELEMENTARY PARTICLES.
The question of the physical meaning. Wave-particle duality, so evident in experiment, creates one of the most difficult problems in the physical interpretation of the mathematical formalism of quantum mechanics. Consider, for example, a wave function that describes a particle freely moving in space. The traditional idea of a particle, among other things, assumes that it moves along a certain trajectory with a certain momentum p. The wave function is assigned the de Broglie wavelength l = h/p, but this is a characteristic of such a wave, which is infinite in space, and therefore does not carry information about the location of the particle. The wave function that localizes a particle in a certain region of space with a length Dx can be constructed as a superposition (packet) of waves with the corresponding set of momenta, and if the required range of momenta is Dp, then it is quite simple to show that the relation DxDp і h/4p. This relation, first obtained in 1927 by Heisenberg, expresses the well-known uncertainty principle: the more precisely one of the two variables x and p is given, the less accuracy with which the theory allows one to determine the other.
The Heisenberg relation could be regarded simply as a flaw in the theory, but, as shown by Heisenberg and Bohr, it corresponds to a deep and previously unnoticed law of nature: even in principle, no experiment will allow one to determine the x and p values of a real particle more accurately than the Heisenberg relation allows . Heisenberg and Bohr differed in the interpretation of this conclusion. Heisenberg viewed it as a reminder that all our knowledge is experimental in origin and that experiment inevitably perturbs the system under study, while Bohr viewed it as a limitation on the accuracy with which the very concept of wave and particle is applicable to the world of the atom. The range of opinions about the nature of the statistical uncertainty itself turns out to be much wider. These uncertainties are nothing new; they are inherent in almost every measurement, but they are usually considered to be due to the shortcomings of the instruments or methods used: the exact value exists, but it is very difficult to find it in practice, and therefore we consider the results obtained as probable values with their inherent statistical uncertainty. One of the schools of physical and philosophical thought, headed at one time by Einstein, believes that the same is true for the microworld, and that quantum mechanics with its statistical results gives only average values that would be obtained by repeatedly repeating the experiment in question with small differences due to the imperfection of our control. With this view, an exact theory of each individual case exists in principle, it just has not yet been found. Another school, historically associated with the name of Bohr, is that indeterminism is inherent in the very nature of things and that quantum mechanics is the theory that best describes each individual case, and the accuracy with which this quantity can be determined and determined is reflected in the uncertainty of a physical quantity. be used. The opinion of most physicists leaned in favor of Bohr. In 1964, J. Bell, who was then working at CERN (Geneva), showed that, in principle, this problem could be solved experimentally. Bell's result was perhaps the most important shift since the 1920s in the search for the physical meaning of quantum mechanics. Bell's theorem, as this result is now called, states that some predictions made on the basis of quantum mechanics cannot be reproduced by computing on the basis of any exact, deterministic theory and then averaging the results. Since two such calculation methods should give different results, the possibility of experimental verification appears. Measurements made in the 1970s convincingly confirmed the adequacy of quantum mechanics. Still, it would be premature to say that the experiment ended the debate between Bohr and Einstein, since such problems often arise as if anew, in a different linguistic guise, every time when, it would seem, all the answers have already been found. Be that as it may, other puzzles remain, reminding us that physical theories are not only equations, but also verbal explanations, connecting the crystalline realm of mathematics with the nebulous realms of language and sensory experience, and that this is often the most difficult.
LITERATURE
Vihman E. Quantum physics. M., 1977 Jammer M. Evolution of the concepts of quantum mechanics. M., 1985 Migdal A.B. Quantum physics for big and small. M., 1989 Volkova E.L. and others. Quantum mechanics on a personal computer. M., 1995
Collier Encyclopedia. - Open society. 2000 .
The word "quantum" comes from the Latin quantum(“how much, how much”) and English quantum("quantity, portion, quantum"). "Mechanics" has long been called the science of the movement of matter. Accordingly, the term "quantum mechanics" means the science of the movement of matter in portions (or, in modern scientific language, the science of the movement quantized matter). The term "quantum" was introduced by the German physicist Max Planck ( cm. Planck's constant) to describe the interaction of light with atoms.
Quantum mechanics often contradicts our notions of common sense. And all because common sense tells us things that are taken from everyday experience, and in our everyday experience we have to deal only with large objects and phenomena of the macrocosm, and at the atomic and subatomic level, material particles behave quite differently. The Heisenberg Uncertainty Principle is precisely the meaning of these differences. In the macrocosm, we can reliably and unambiguously determine the location (spatial coordinates) of any object (for example, this book). It doesn't matter if we use a ruler, radar, sonar, photometry or any other measurement method, the measurement results will be objective and independent of the position of the book (of course, provided that you are careful in the measurement process). That is, some uncertainty and inaccuracy are possible - but only due to the limited capabilities of measuring instruments and observation errors. To get more accurate and reliable results, we just need to take a more accurate measuring device and try to use it without errors.
Now, if instead of the coordinates of a book, we need to measure the coordinates of a microparticle, such as an electron, then we can no longer neglect the interactions between the measuring device and the object of measurement. The force of the action of a ruler or other measuring device on the book is negligible and does not affect the measurement results, but in order to measure the spatial coordinates of an electron, we need to launch a photon, another electron or another elementary particle of energies comparable to the measured electron in its direction and measure its deviation. But at the same time, the electron itself, which is the object of measurement, will change its position in space as a result of interaction with this particle. Thus, the very act of measurement leads to a change in the position of the object being measured, and the inaccuracy of the measurement is due to the very fact of the measurement, and not the degree of accuracy of the measuring device used. This is the situation we have to put up with in the microworld. Measurement is impossible without interaction, and interaction without impact on the measured object and, as a result, distortion of the measurement results.
Only one thing can be said about the results of this interaction:
spatial coordinate uncertainty × particle velocity uncertainty > h/m,
or, in mathematical terms:
Δ x × Δ v > h/m
where ∆ x and Δ v - the uncertainty of the spatial position and velocity of the particle, respectively, h- Planck's constant, and m - particle mass.
Accordingly, uncertainty arises when determining the spatial coordinates of not only an electron, but also any subatomic particle, and not only coordinates, but also other properties of particles, such as speed. The measurement error of any such pair of mutually related particle characteristics is determined in a similar way (an example of another pair is the energy emitted by an electron and the length of time during which it is emitted). That is, if we, for example, managed to measure the spatial position of an electron with high accuracy, then we at the same moment in time we have only the vaguest idea of its speed, and vice versa. Naturally, with real measurements, these two extremes do not reach, and the situation is always somewhere in the middle. That is, if we managed, for example, to measure the position of an electron with an accuracy of 10 -6 m, then we can simultaneously measure its speed, at best, with an accuracy of 650 m/s.
Due to the uncertainty principle, the description of the objects of the quantum microworld is of a different nature than the usual description of the objects of the Newtonian macrocosm. Instead of spatial coordinates and speed, which we used to describe the mechanical movement of, for example, a ball on a billiard table, in quantum mechanics, objects are described by the so-called wave function. The crest of the "wave" corresponds to the maximum probability of finding a particle in space at the moment of measurement. The motion of such a wave is described by the Schrödinger equation, which tells us how the state of a quantum system changes over time.
The picture of quantum events in the microcosm, drawn by the Schrödinger equation, is such that the particles are likened to individual tidal waves propagating over the surface of the ocean-space. Over time, the wave crest (corresponding to the peak of the probability of finding a particle, such as an electron, in space) moves in space in accordance with the wave function, which is the solution of this differential equation. Accordingly, what is traditionally represented to us as a particle, at the quantum level, exhibits a number of characteristics inherent in waves.
Coordination of wave and corpuscular properties of microworld objects ( cm. The de Broglie relation) became possible after physicists agreed to consider the objects of the quantum world not as particles or waves, but as something intermediate and having both wave and corpuscular properties; there are no analogues to such objects in Newtonian mechanics. Although even with such a solution, there are still enough paradoxes in quantum mechanics ( cm. Bell's theorem), no one has yet proposed the best model for describing the processes occurring in the microworld.
MINISTRY OF EDUCATION OF THE RUSSIAN FEDERATION
MOSCOW STATE INSTITUTE OF RADIO ENGINEERING, ELECTRONICS AND AUTOMATION (TECHNICAL UNIVERSITY)
A.A. BERZIN, V.G. MOROZOV
BASICS OF QUANTUM MECHANICS
Tutorial
Moscow - 2004
Introduction
Quantum mechanics appeared a hundred years ago and took shape in 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 invention of the transistor 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 great practical potential and a serious acquaintance with this science is necessary not only for professional physicists, but also for representatives of other specialties - chemists, engineers and even biologists.
Since quantum mechanics has increasingly begun to acquire the features of not only fundamental, but also applied science, the problem of teaching its fundamentals to students of non-physical specialties has arisen. Some quantum ideas are first introduced to a student in a course in general physics, but as a rule, this acquaintance is limited to nothing more than random facts and their highly simplified explanations. On the other hand, the full course of quantum mechanics taught at the physics departments of universities 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” the course of quantum mechanics to the needs of teaching students of applied specialties was noticed long ago and has not been completely overcome, despite numerous attempts to create “transitional” courses focused on practical applications of quantum laws. This is due to the specifics of quantum mechanics itself. First, to understand quantum mechanics, a student needs a thorough knowledge of classical physics: Newtonian mechanics, classical theory of electromagnetism, special relativity, optics, etc. Secondly, in quantum mechanics, for a correct description of phenomena in the microcosm, one has to sacrifice visibility. Classical physics operates with more or less visual concepts; their connection with experiment is relatively simple. Another position 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, the 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, in order to learn how to solve quantum mechanical problems, one must know differential equations, handle complex numbers fairly freely, and be able to do many other things. All this, however, does not go beyond the mathematical training of a student of a modern technical university. The real difficulty of quantum mechanics is connected not only and even not so much with mathematics. The fact is that the conclusions of quantum mechanics, like any physical theory, must predict and explain real experiments, so you need to learn how to associate abstract mathematical constructions with measured physical quantities and observed phenomena. This skill is developed by each person individually, mainly by independently solving problems and understanding the results. Newton also remarked: “in the study of sciences, examples are often more important than rules.” With respect to quantum mechanics, these words contain a great deal of truth.
The manual offered to the reader is based on the long-term practice of reading 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 to 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 course "Physics 4" 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 fields of application of quantum mechanics, have been selected. However, in contrast to courses in general physics and applied technical disciplines, we tried to present these sections within the framework of a single and fairly modern approach, taking into account the ability of students to master it. The volume of the manual exceeds the content of lectures and practical classes, since the course "Physics 4" provides for students to complete term papers or individual assignments that require independent study of issues not included in the lecture plan. The presentation of these questions in textbooks on quantum mechanics, aimed at students of physical faculties of universities, often exceeds the level of preparation of a student of a technical university. Thus, this manual can be used as a source of material for term papers and individual assignments.
An important part of the manual are exercises. Some of them are given directly in the text, the rest are placed at the end of each paragraph. Many of the exercises are provided with directions for the reader. In connection with the “unusualness” of the concepts and methods of quantum mechanics noted above, the execution of 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 led, in the end, to the emergence of quantum theory.
Equilibrium radiation spectrum of a black body. Recall that in physics
a black body (often called “absolutely black body”) is a body that completely absorbs electromagnetic radiation of any frequency incident on it.
A 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 opening, 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 kept constant, then eventually thermal equilibrium will be established between the wall material
Rice. 1.1. and electromagnetic radiation in the cavity. One of the problems that physicists actively discussed at the end of the 19th century was the following: how is the energy of equilibrium radiation distributed over
Rice. 1.2.
frequencies? Quantitatively, this distribution is described by the spectral density of radiation energy u ω . The product u ω dω is the energy of electromagnetic waves per unit volume with frequencies in the range from ω to ω +dω . The spectral energy density can be measured by analyzing the emission spectrum from the opening of the cavity shown in Fig. 1.1. The experimental dependence u ω for two temperatures is shown in Fig. . 1.2. As the temperature increases, 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 purple.
While we talked about experimental data. Interest in the spectrum of black body radiation was due to the fact that the function u ω can be accurately calculated by 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 uniformly 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 tok B T , wherek 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 density of radiation has the form
u ω= | k B T ω2 | |||
π2 c3 |
||||
This formula is the famous Rayleigh-Jeans formula. In classical physics, it is exact and, at the same time, absurd. Indeed, according to it, in thermal equilibrium at any temperature there are electromagnetic waves of arbitrarily high frequencies (i.e., ultraviolet radiation, X-ray radiation, and even gamma radiation that is fatal to humans), and the higher the radiation frequency, the more energy falls on him. The obvious contradiction between the classical theory of equilibrium radiation and experiment has received an emotional name in the physical literature - ultraviolet
disaster. Note that the well-known 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.
Photoelectric effect. Another "weak point" of classical physics turned out to be the photoelectric effect - knocking out electrons from matter under the action 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
in light wave and is equal to the average energy flux incident on the substance. On the other hand, the energy of the emitted electrons essentially depends on the frequency of the light and increases linearly with increasing frequency. It's also impossible to explain
in 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 its 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 the knockout of electrons begins. If ω< 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 the incident and scattered radiation must be exactly equal. To be convinced of the above, complex mathematics is not needed. Suffice it to recall the classical mechanism of electromagnetic wave scattering by charged particles. Scheme
reasoning is like this. Variable electric field E (t) \u003d E 0 sinωt |
|||||
of the incident wave acts on each electron with the force F (t) = −eE (t), where −e - |
|||||
(me | |||||
electron charge | The electron acquires acceleration a (t) \u003d F (t) / m e |
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electron), which changes with time with the same frequency ω as the field in the incident wave. According to classical electrodynamics, an accelerating charge radiates electromagnetic waves. This is the scattered radiation. If the 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 the incident radiation clearly contradicts classical electrodynamics.
Atomic Stability. In 1912, a very important event for the entire further development of the natural sciences occurred - the structure of the atom was elucidated. The English physicist E. Rutherford, conducting experiments on the scattering of α-particles in matter, found 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 is similar to the solar system: light electrons move in orbits around a massive nucleus, just as the planets move around the Sun. The force holding electrons in their orbits is the force of the Coulomb attraction of the nucleus. At first glance, such a “planetary model” seems very
1 The symbol e everywhere denotes a positive elementary charge e = 1.602 10− 19 C.
attractive: it is illustrative, 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 point 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 an orbit with acceleration. Even if the magnitude of the electron's velocity does not change, there is still an acceleration directed towards the nucleus (normal or "centripetal" acceleration). But, as noted above, a charge moving with acceleration must radiate electromagnetic waves. These waves carry away energy, so the energy of the electron decreases. The radius of its orbit decreases and in the end the electron must fall into the nucleus. Simple calculations, which we will not present here, show that the characteristic “lifetime” of an electron in orbit is about 10−8 seconds. Thus, classical physics is unable to explain the stability of atoms.
The examples given do not exhaust all the difficulties that classical physics encountered at the turn of the 19th and 20th centuries. Other phenomena, where its conclusions contradict experiment, we will consider later, when the apparatus of quantum mechanics is developed and we can immediately give a correct explanation. Gradually accumulating, the 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 conjecture about the quantization of the energy of an oscillator
December 2000 marks one hundred years 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 the establishment of thermal equilibrium, 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 oscillation frequency. Recalling the reasoning given on page 5, we write the equilibrium spectral density of radiation 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 as
ε = hv. |
The value h = 2π 6.626176 10− 34 J s is also called Planck's constant1.
Based on the assumption of oscillator energy quantization, Planck derived the following expression for the spectral density of equilibrium radiation2:
π2 c3 | e ω/kB T | − 1 | |||||
At low frequencies (ω k B T ) the Planck formula practically coincides with the Rayleigh-Jeans formula (1.3), and at high frequencies (ω k B T ) the spectral density of radiation, in accordance with the experiment, rapidly tends to zero.
1.3. Einstein's hypothesis about the quanta of the electromagnetic field
Although Planck's hypothesis about the quantization of the 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 the 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 in itself did not yet mean a complete rejection of classical laws. A more radical hypothesis was proposed in 1905 by Albert Einstein. Analyzing the patterns of the photoelectric effect, he showed that all of them can be explained in a natural way if we assume 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 A out is the work function, i.e., the energy necessary to overcome the forces that hold 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 regularities of equilibrium thermal radiation. Indeed, the absorption and emission of the energy of the electromagnetic field by matter occurs by quanta ω because individual photons are absorbed and emitted, having just such an energy.
1.4. photon momentum
The introduction of the idea of photons to some extent revived the corpuscular theory of light. The fact that the photon is a “real” particle is confirmed by the 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 fulfilled.
The law of conservation of energy in this process has the form
commensurate with the speed of light, so | ||||||||
the expression for the energy of an electron is needed | ||||||||
take in relativistic form, i.e. | ||||||||
Eel \u003d me c2, | E email= | |||||||
m e 2c 4+ p 2c 2 | ||||||||
where p is the momentum of the electron after the collision with the photon, am | ||||||||
electron. The law of conservation of energy in the Compton effect looks like this: |
||||||||
ω + me c2 = ω+ | ||||||||
m e 2c 4+ p 2c 2 | ||||||||
Incidentally, 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 in this way, from the general expression for the relativistic
energy E \u003d m 2 c 4 + p 2 c 2 it follows that the energy and momentum of a photon are related by the relation E \u003d pc. Recalling formula (1.10), we obtain
Now the law of conservation of momentum in the Compton effect can be written as
The solution of 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 the scattered radiation ∆λ =λ − λ :
is called the Compton wavelength of the particle (of mass m) on which radiation is scattered. If m \u003d m e \u003d 0.911 10− 30 kg is the electron mass, then λ C \u003d 0. 0243 10− 10 m. The results of measurements of ∆λ carried out by Compton, and then by many other experimenters, are fully consistent with the predictions of formula (1.19) , and the value of Planck's constant, which enters 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 like a stream of particles - photons, but, on the other hand, the phenomena of interference and diffraction just as stubbornly show that light is electromagnetic waves. On the basis of “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 the red the boundaries of matter. ωmin for photoelectric effect. expressωmin through the work function of an electron
1.2. Derive expression (1.19) for changing the radiation wavelength in the Compton effect.
Hint: Dividing equation (1.14) by c and using the relation between wavenumber 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 go from wavenumbers to wavelengths (2π/k =λ ).
2. Quantization of atomic energy. Wave properties of microparticles
2.1. Bohr's theory of the atom
Before proceeding directly to the study of quantum mechanics in its modern form, we briefly discuss the first attempt to apply Planck's idea of quantization to the problem of the structure of the atom. We will talk about the theory of the atom, proposed in 1913 by Niels Bohr. Bohr's main goal was to explain a surprisingly simple pattern in the emission spectrum of the hydrogen atom, which Ritz formulated 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 differences of some quantities T (n) (“terms”), the sequence of which is expressed in terms of integers.
Surely you have heard many times about the inexplicable mysteries of quantum physics and quantum mechanics. Its laws fascinate with mysticism, and even the 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 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, so many inquisitive people type in the search line: “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 post 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 (a 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 macro- and microworlds: in our macrocosm, if you put a ball into one of the 2 boxes, then one of them will be empty, and the other - a ball. But in the microcosm (if instead of a ball - an atom), an atom can be simultaneously in two boxes. This has been repeatedly confirmed experimentally. Isn't it hard to put it in your head? But you can't argue with the facts.
One more example. You photographed a fast racing red sports car and in the photo you saw a blurry horizontal strip, as if the car at the time of the photo was from several points in space. Despite what you see in the photo, you are still sure that the car was at the moment when you photographed it. in one specific place in space. Not so in the micro world. An electron that revolves around the nucleus of an atom does not actually revolve, but 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 small digression into history. For the first time, scientists thought 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 quantum. Before that, 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 only 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 certain sense, to abstract from the laws of classical physics familiar to us. And imagine that you dived, like Alice, down the rabbit hole, into Wonderland.
And here is a cartoon for children and adults. Talks about the fundamental experiment of quantum mechanics with 2 slits and an observer. Lasts only 5 minutes. Watch it before we delve 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. 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 slots. 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 "draw" on the screen only 2 vertical stripes.
Light interference- This is the "wave" behavior of light, when a lot of alternating bright and dark vertical stripes are displayed on the screen. And those vertical stripes called an 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! 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 consider the 2nd experiment. This is the famous Stern-Gerlach experiment (which was carried out in the 20s of the last century).
In the installation described in the cartoon, they did not shine with light, but “shot” with electrons (as separate 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 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, they 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
It is today that you and I are so smart and understand that the 2 experiments described above - firing electrons and illuminating slots with light - are one and the same. Because we're firing quantum particles at the slits. Now we know that both light and electrons are of 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 issue.
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 to see how each of the electrons flies through the slit.
Presumably, one electron flies to the left slit, the other to the right. But then 2 vertical stripes should appear on the screen directly opposite the slots. Why is an interference pattern obtained? 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 follow this?
We will throw electrons not in a beam, but one at a time. Drop it, 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 on the screen each electron after the throw. One or two, of course, will not “paint” a clear picture for us. But when one by one we send a lot of them into the slots, we will notice ... oh horror - they again “drawn” an interference wave pattern!
We start 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 it were, through 2 slits at the same time and interfered with itself. Fiction! We will return to the explanation of 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 (let's call them photons for simplicity from now on).
As a result of the experiment, when we threw 1 photon into 2 slits, we realized that it flies as if 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: 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 have arrived at the "path integral". This is Richard Feynman's formulation of quantum mechanics.
By the way, exactly Richard Feynman belongs to the 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 we know that a photon can behave both as a particle and as a wave. That he can, in some way incomprehensible to 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 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. This is the point.
It remains for us to simply accept 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 photon parameter - 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 magnet vector (spin) is either directed up or down. But the electron or 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 "wavefunction collapse"?
It remains for us a little - to understand what is "measurement" and what is "collapse of the wave function".
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 indeterminate 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 will deviate in the other direction. Photons can also be sent to a polarizing filter. If the spin (polarization) of a photon is +1, it passes through the filter, and if it is -1, then it does not.
Stop! This is where the question inevitably arises: before the measurement, after all, the electron did not have any particular spin direction, right? Was he in all states at the same time?
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 its own angular momentum vector - spin). But at the moment when you measured his state, he seems to be deciding which spin vector to take.
This quantum object is so cool - it makes a decision 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 decides 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 " wave function collapse". The wave function before the measurement was indefinite, i.e. the electron spin vector was simultaneously in all directions, after the measurement, the electron fixed a certain direction of its spin vector.
Attention! An excellent example-association from our macrocosm for understanding:
Spin a coin on the table like a top. While the coin is spinning, it has no specific meaning - heads or tails. But as soon as you decide to "measure" this value and slam the coin with your hand, this is where you get the specific state of the coin - heads or tails. Now imagine that this coin 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 passed through the slits, they behaved like a wave and showed an interference pattern on the screen. And when the scientists wanted to fix (measure) the moment when photons passed through the slit and put an “observer” behind the screen, the photons began to behave not like waves, but like particles. And “drawn” 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.
And now, do you want the information to be put on the shelves? Watch a documentary produced by the Canadian Institute for Theoretical Physics. In 20 minutes, it will tell you very briefly and in chronological order about all the discoveries of quantum physics, starting with the discovery of Planck in 1900. And then they will tell you what practical developments are currently being carried out on the basis of knowledge of quantum physics: from the most accurate atomic clocks to super-fast calculations 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 you interested in?
P.S.3 Subscribe to the blog - the subscription form under the article.
PLAN
INTRODUCTION 2
1. HISTORY OF THE CREATION OF QUANTUM MECHANICS 5
2. THE PLACE OF QUANTUM MECHANICS AMONG OTHER SCIENCES OF MOTION. fourteen
CONCLUSION 17
LITERATURE 18
Introduction
Quantum mechanics is a theory that establishes the method of describing and the laws of motion of microparticles (elementary particles, atoms, molecules, atomic nuclei) and their systems (for example, crystals), as well as the relationship of quantities characterizing particles and systems with physical quantities directly measured in macroscopic experiments . The laws of quantum mechanics (hereinafter referred to as quantum mechanics) form the foundation for studying the structure of matter. They made it possible to elucidate the structure of atoms, establish the nature of the chemical bond, explain the periodic system of elements, understand the structure of atomic nuclei, and study the properties of elementary particles.
Since the properties of macroscopic bodies are determined by the motion and interaction of the particles of which they are composed, the laws of quantum mechanics underlie the understanding of most macroscopic phenomena. The quantum mechanics made it possible, for example, to explain the temperature dependence and to calculate the heat capacity of gases and solids, to determine the structure and understand many properties of solids (metals, dielectrics, and semiconductors). Only on the basis of quantum mechanics was it possible to consistently explain such phenomena as ferromagnetism, superfluidity, and superconductivity, to understand the nature of such astrophysical objects as white dwarfs and neutron stars, and to elucidate the mechanism of thermonuclear reactions in the Sun and stars. There are also phenomena (for example, the Josephson effect) in which the laws of quantum mechanics are directly manifested in the behavior of macroscopic objects.
Thus, quantum mechanical laws underlie the operation of nuclear reactors, determine the possibility of carrying out thermonuclear reactions under terrestrial conditions, manifest themselves in a number of phenomena in metals and semiconductors used in the latest technology, and so on. The foundation of such a rapidly developing field of physics as quantum electronics is the quantum mechanical theory of radiation. The laws of quantum mechanics are used in the purposeful search for and creation of new materials (especially magnetic, semiconductor, and superconducting materials). Quantum mechanics is becoming largely an "engineering" science, the knowledge of which is necessary not only for research physicists, but also for engineers.
1. The history of the creation of quantum mechanics
At the beginning of the 20th century two (seemingly unrelated) groups of phenomena were discovered, indicating the inapplicability of the usual classical theory of the electromagnetic field (classical electrodynamics) to the processes of interaction of light with matter and to the processes occurring in the atom. The first group of phenomena was associated with the establishment by experience of the dual nature of light (dualism of light); the second - with the impossibility of explaining on the basis of classical concepts the stable existence of the atom, as well as the spectral patterns discovered in the study of the emission of light by atoms. The establishment of a connection between these groups of phenomena and attempts to explain them on the basis of a new theory ultimately led to the discovery of the laws of quantum mechanics.
For the first time, quantum representations (including the quantum constant h) were introduced into physics in the work of M. Planck (1900), devoted to the theory of thermal radiation.
The theory of thermal radiation that existed by that time, built on the basis of classical electrodynamics and statistical physics, led to a meaningless result, which consisted in the fact that thermal (thermodynamic) equilibrium between radiation and matter cannot be achieved, because all energy must sooner or later turn into radiation. Planck resolved this contradiction and obtained results in perfect agreement with experiment, on the basis of an extremely bold hypothesis. In contrast to the classical theory of radiation, which considers the emission of electromagnetic waves as a continuous process, Planck suggested that light is emitted in certain portions of energy - quanta. The value of such an energy quantum depends on the light frequency n and is equal to E=h n. From this work of Planck, two interrelated lines of development can be traced, culminating in the final formulation of K. m. in its two forms (1927).
The first one begins with the work of Einstein (1905), in which the theory of the photoelectric effect was given - the phenomenon of pulling electrons out of matter by light.
In developing Planck's idea, Einstein suggested that light is not only emitted and absorbed in discrete portions - radiation quanta, but light propagation occurs in such quanta, i.e. that discreteness is inherent in light itself - that light itself consists of separate portions - light quanta (which were later called photons). Photon energy E is related to the oscillation frequency n of the wave by the Planck relation E= hn.
Further proof of the corpuscular nature of light was obtained in 1922 by A. Compton, who showed experimentally that the scattering of light by free electrons occurs according to the laws of elastic collision of two particles - a photon and an electron. The kinematics of such a collision is determined by the laws of conservation of energy and momentum, and the photon, along with the energy E= hn momentum must be assigned p = h / l = h n / c, where l- the length of the light wave.
The energy and momentum of a photon are related by E = cp , valid in relativistic mechanics for a particle with zero mass. Thus, it was experimentally proved that, along with the known wave properties (manifested, for example, in the diffraction of light), light also has corpuscular properties: it consists, as it were, of particles - photons. This manifests the dualism of light, its complex corpuscular-wave nature.
Dualism is already contained in the formula E= hn, which does not allow choosing any one of the two concepts: on the left side of the equality, the energy E refers to the particle, and on the right, the frequency n is the characteristic of the wave. A formal logical contradiction arose: to explain some phenomena, it was necessary to assume that light has a wave nature, and to explain others - corpuscular. In essence, the resolution of this contradiction led to the creation of the physical foundations of quantum mechanics.
In 1924, L. de Broglie, trying to find an explanation for the conditions for quantization of atomic orbits postulated in 1913 by N. Bohr, put forward a hypothesis about the universality of wave-particle duality. According to de Broglie, each particle, regardless of its nature, should be associated with a wave whose length L related to the momentum of the particle R ratio. According to this hypothesis, not only photons, but also all “ordinary particles” (electrons, protons, etc.) have wave properties, which, in particular, should manifest themselves in the phenomenon of diffraction.
In 1927, K. Davisson and L. Germer first observed electron diffraction. Later, wave properties were discovered in other particles, and the validity of the de Broglie formula was confirmed experimentally
In 1926, E. Schrödinger proposed an equation describing the behavior of such "waves" in external force fields. This is how wave mechanics was born. The Schrödinger wave equation is the basic equation of nonrelativistic quantum mechanics.
In 1928, P. Dirac formulated a relativistic equation describing the motion of an electron in an external force field; The Dirac equation has become one of the fundamental equations of relativistic quantum mechanics.
The second line of development begins with the work of Einstein (1907) on the theory of the heat capacity of solids (it is also a generalization of Planck's hypothesis). Electromagnetic radiation, which is a set of electromagnetic waves of different frequencies, is dynamically equivalent to a certain set of oscillators (oscillatory systems). The emission or absorption of waves is equivalent to the excitation or damping of the corresponding oscillators. The fact that the emission and absorption of electromagnetic radiation by matter occur in energy quanta h n. Einstein generalized this idea of quantizing the energy of an electromagnetic field oscillator to an oscillator of an arbitrary nature. Since the thermal motion of solids is reduced to vibrations of atoms, then a solid body is dynamically equivalent to a set of oscillators. The energy of such oscillators is also quantized, i.e., the difference between neighboring energy levels (the energies that an oscillator can have) should be equal to h n, where n is the frequency of vibrations of atoms.
Einstein's theory, refined by P. Debye, M. Born, and T. Karman, played an outstanding role in the development of the theory of solids.
In 1913, N. Bohr applied the idea of energy quantization to the theory of the structure of the atom, whose planetary model followed from the results of E. Rutherford's experiments (1911). According to this model, in the center of the atom there is a positively charged nucleus, in which almost the entire mass of the atom is concentrated; Negatively charged electrons revolve around the nucleus.
Consideration of such a motion on the basis of classical concepts led to a paradoxical result - the impossibility of a stable existence of atoms: according to classical electrodynamics, an electron cannot stably move in orbit, since a rotating electric charge must radiate electromagnetic waves and, therefore, lose energy. The radius of its orbit should decrease and in a time of about 10 -8 sec the electron should fall on the nucleus. This meant that the laws of classical physics are not applicable to the motion of electrons in an atom, since atoms exist and are extremely stable.
To explain the stability of atoms, Bohr suggested that of all the orbits allowed by Newtonian mechanics for the motion of an electron in the electric field of an atomic nucleus, only those that satisfy certain quantization conditions are actually realized. That is, discrete energy levels exist in the atom (as in an oscillator).
These levels follow a certain pattern, deduced by Bohr based on a combination of the laws of Newtonian mechanics with quantization conditions requiring that the magnitude of the action for the classical orbit be an integer multiple of Planck's constant.
Bohr postulated that, being at a certain energy level (i.e., performing the orbital motion allowed by the conditions of quantization), the electron does not emit light waves.
Radiation occurs only when an electron moves from one orbit to another, i.e., from one energy level E i , to another with less energy E k , in this case, a light quantum is born with an energy equal to the difference in the energies of the levels between which the transition is carried out:
h n= E i- E k . (one)
This is how the line spectrum arises - the main feature of atomic spectra, Bohr received the correct formula for the frequencies of the spectral lines of the hydrogen atom (and hydrogen-like atoms), covering a set of previously discovered empirical formulas.
The existence of energy levels in atoms was directly confirmed by Frank-Hertz experiments (1913-14). It was found that electrons bombarding a gas lose only certain portions of energy when they collide with atoms, equal to the difference in the energy levels of the atom.
N. Bohr, using the quantum constant h, reflecting the dualism of light, showed that this quantity also determines the motion of electrons in an atom (and that the laws of this motion differ significantly from the laws of classical mechanics). This fact was later explained on the basis of the universality of the wave-particle duality contained in the de Broglie hypothesis. The success of Bohr's theory, like the previous successes of quantum theory, was achieved by violating the logical integrity of the theory: on the one hand, Newtonian mechanics was used, on the other hand, artificial quantization rules alien to it were involved, which, moreover, contradicted classical electrodynamics. In addition, Bohr's theory was unable to explain the movement of electrons in complex atoms, the emergence of molecular bonds.
Bohr's "semi-classical" theory could also not answer the question of how an electron moves during the transition from one energy level to another.
Further intense development of questions of the theory of the atom led to the conviction that, while maintaining the classical picture of the motion of an electron in orbit, it is impossible to construct a logically coherent theory.
The realization of the fact that the movement of electrons in an atom is not described in terms (concepts) of classical mechanics (as movement along a certain trajectory), led to the idea that the question of the movement of an electron between levels is incompatible with the nature of the laws that determine the behavior of electrons in an atom, and that a new theory is needed, which would include only quantities related to the initial and final stationary states of the atom.
In 1925, W. Heisenberg succeeded in constructing such a formal scheme in which, instead of the coordinates and velocities of an electron, some abstract algebraic quantities - matrices - appeared; the relationship of matrices with observable quantities (energy levels and intensities of quantum transitions) was given by simple consistent rules. Heisenberg's work was developed by M. Born and P. Jordan. This is how matrix mechanics arose. Shortly after the appearance of the Schrödinger equation, the mathematical equivalence of wave (based on the Schrödinger equation) and matrix mechanics was shown. In 1926 M. Born gave a probabilistic interpretation of de Broglie waves (see below).
An important role in the creation of quantum mechanics was played by Dirac's works dating back to the same time. The final formation of quantum mechanics as a consistent physical theory with clear foundations and a coherent mathematical apparatus occurred after the work of Heisenberg (1927), in which the uncertainty relation was formulated - the most important relation that illuminates the physical meaning of the equations of quantum mechanics, its connection with classical mechanics, and other questions of principle as well as qualitative results of quantum mechanics. This work was continued and summarized in the writings of Bohr and Heisenberg.
A detailed analysis of the spectra of atoms led to the representation (introduced for the first time by J. Yu. Uhlenbeck and S. Goudsmit and developed by W. Pauli) that the electron, in addition to charge and mass, must be assigned one more internal characteristic (quantum number) - spin.
An important role was played by the so-called exclusion principle discovered by W. Pauli (1925), which is of fundamental importance in the theory of the atom, molecule, nucleus, and solid state.
Within a short time, quantum mechanics was successfully applied to a wide range of phenomena. Theories of atomic spectra, the structure of molecules, chemical bonding, the periodic system of D. I. Mendeleev, metallic conductivity and ferromagnetism were created. These and many other phenomena have become (at least qualitatively) understandable.
