Under quantum mechanics understand the physical theory of the dynamic behavior of the forms of radiation and matter. It is 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, developed over 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 solar system had a nucleus at its center, around which the elements revolved. Based on this theory, calculations of the physical and chemical properties of some substances that were built from simple atoms began.
One of the important questions in such a theory as quantum mechanics is the nature of the forces that bound the atom. Thanks to Coulomb's law, E. Rutherford showed that this law is valid on a huge scale. Then it was necessary to determine how the electrons move in their orbit. Helped at this point
In fact, quantum mechanics often conflicts with concepts such as common sense. Along with the fact that our common sense operates and shows only such things that can be taken from everyday experience. And, in turn, everyday experience deals only with the phenomena of the macrocosm and large objects, while material particles at the subatomic and atomic level behave quite differently. For example, in the macrocosm, we can easily 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 object of measurement and the measuring device.
In other words, one 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 ways:
The probable nature of some physical quantities, for example, the speed and position of a microparticle cannot be precisely determined, only the probability of their values can be calculated;
A discrete change, for example, the energy of a microparticle has only certain certain values.
Quantum mechanics is also associated with the notion quantum cryptography, which is a fast-growing technology that can change the world. Quantum cryptography aims to protect communications and the secrecy of information. This cryptography is based on certain phenomena and considers such cases when information can be transferred using the object of quantum mechanics. It is here that with the help of electrons, photons and other physical means, the process of receiving and sending information is determined. Thanks to quantum cryptography it is possible to create and design a communication system that can detect eavesdropping.
To date, there are quite a lot of materials where it is proposed to study such a concept as quantum mechanics fundamentals and directions, as well as activities of quantum cryptography. To gain knowledge in this difficult theory, it is necessary to thoroughly study and delve into this area. After all, quantum mechanics is far from an easy concept that has been studied and proven by the greatest scientists for many years.
Quantum mechanics (QM; also known as quantum physics or quantum theory), including quantum field theory, is a branch of physics that studies the laws of nature at small distances and at low energies of atoms and subatomic particles. Classical physics - physics that existed before quantum mechanics, follows from quantum mechanics as its limiting transition, valid only at large (macroscopic) scales. Quantum mechanics differs from classical physics in that energy, momentum, and other quantities are often limited to discrete values (quantization), objects have characteristics of both particles and waves (wave-particle duality), and there are limits on the precision with which quantities can be determined (uncertainty principle).
Quantum mechanics follows successively from Max Planck's 1900 solution to the black body radiation problem (published in 1859) and Albert Einstein's 1905 work which proposed a quantum theory to explain the photoelectric effect (published in 1887). Early quantum theory was deeply rethought in the mid-1920s.
The rethought theory is formulated in the language of specially developed mathematical formalisms. In one of them, a mathematical function (wave function) provides information about the probability amplitude of the position, momentum, and other physical characteristics of the particle.
Important areas of application of quantum theory are: quantum chemistry, superconducting magnets, light emitting diodes, as well as laser, transistor and semiconductor devices such as the microprocessor, medical and research imaging such as magnetic resonance imaging and electron microscopy, and explanations of many biological and physical phenomena.
The scientific study of the wave nature of light began in the 17th and XVIII centuries when scientists Robert Hook, Christian Huygens and Leonhard Euler proposed a wave theory of light based on experimental observations. In 1803, Thomas Young, an English generalist, performed the famous double slit experiment, which he later described in a paper entitled The Nature of Light and Colors. This experiment played an important role in the general acceptance of the wave theory of light.
In 1838, Michael Faraday discovered cathode rays. These studies were followed by Gustav Kirchhoff's formulation of the blackbody radiation problem in 1859, Ludwig Boltzmann's suggestion in 1877 that the energy states of a physical system could be discrete, and Max Planck's quantum hypothesis in 1900. Planck's hypothesis that energy is emitted and absorbed in discrete "quanta" (or energy packets) corresponds exactly to observable models of blackbody radiation.
In 1896, Wilhelm Wien empirically determined the blackbody radiation distribution law, named after him, Wien's law. Ludwig Boltzmann independently arrived at this result by analyzing Maxwell's equations. However, the law only worked at high frequencies and underestimated the radiation at low frequencies. Planck later corrected this model with a statistical interpretation of Boltzmann's thermodynamics and proposed what is now called Planck's law, leading to the development of quantum mechanics.
After Max Planck's solution in 1900 to the problem of blackbody radiation (published 1859), Albert Einstein proposed a quantum theory to explain the photoelectric effect (1905, published 1887). In the years 1900-1910, the atomic theory and the corpuscular theory of light were first widely accepted as scientific fact. Accordingly, these latter theories can be regarded as quantum theories of matter and electromagnetic radiation.
Among the first to study quantum phenomena in nature were Arthur Compton, C. V. Raman, and Peter Zeeman, after each of whom some quantum effects are named. Robert Andrews Millikan investigated the photoelectric effect experimentally, and Albert Einstein developed a theory for it. At the same time, Ernest Rutherford experimentally discovered the nuclear model of the atom, according to which Niels Bohr developed his theory of the structure of the atom, which was later confirmed by the experiments of Henry Moseley. In 1913, Peter Debye extended Niels Bohr's theory of the structure of the atom by introducing elliptical orbits, a concept also proposed by Arnold Sommerfeld. This stage in the development of physics is known as the old quantum theory.
According to Planck, the energy (E) of a radiation quantum is proportional to the radiation frequency (v):
where h is Planck's constant.
Planck cautiously insisted that this was simply a mathematical expression of the processes of absorption and emission of radiation and had nothing to do with the physical reality of the radiation itself. In fact, he considered his quantum hypothesis a mathematical trick to get the right answer, rather than a major fundamental discovery. However, in 1905, Albert Einstein gave Planck's quantum hypothesis a physical interpretation and used it to explain the photoelectric effect, whereby illuminating certain substances with light can cause electrons to be emitted from the substance. Einstein received the 1921 Nobel Prize in Physics for this work.
Einstein then developed this idea to show that an electromagnetic wave, which is what light is, can also be described as a particle (later called a photon), with a discrete quantum energy that depends on the frequency of the wave.
During the first half of the 20th century, Max Planck, Niels Bohr, Werner Heisenberg, Louis de Broglie, Arthur Compton, Albert Einstein, Erwin Schrödinger, Max Born, John von Neumann, Paul Dirac, Enrico Fermi, Wolfgang Pauli, Max von Laue, Freeman Dyson, David Hilbert, Wilhelm Wien, Shatyendranath Bose, Arnold Sommerfeld and others laid the foundations of quantum mechanics. Niels Bohr's Copenhagen interpretation has received universal acclaim.
In the mid-1920s, developments in quantum mechanics led to it becoming the standard formulation for atomic physics. In the summer of 1925, Bohr and Heisenberg published results that closed the old quantum theory. Out of respect for their particle-like behavior in certain processes and measurements, light quanta came to be called photons (1926). From a simple postulate of Einstein, a flurry of discussions, theoretical constructions and experiments was born. In this way, whole areas of quantum physics emerged, leading to its widespread recognition at the Fifth Solvay Congress in 1927.
It was found that subatomic particles and electromagnetic waves are neither just particles nor waves, but have certain properties of each of them. This is how the concept of wave-particle duality arose.
By 1930, quantum mechanics was further unified and formulated in the work of David Hilbert, Paul Dirac, and John von Neumann, which emphasized measurement, the statistical nature of our knowledge of reality, and philosophical reflections on the "observer". It has subsequently penetrated many disciplines, including quantum chemistry, quantum electronics, quantum optics, and quantum information science. Her theoretical contemporary developments include string theory and theories of quantum gravity. It also provides a satisfying explanation for many features of the modern periodic table of elements and describes the behavior of atoms when chemical reactions and the motion of electrons in computer semiconductors, and therefore plays a critical role in many of today's technologies.
Although quantum mechanics was built to describe the microcosm, it is also necessary to explain some macroscopic phenomena such as superconductivity and superfluidity.
The word quantum comes from the Latin "quantum", which means "how much" or "how much". In quantum mechanics, a quantum means a discrete unit attached to certain physical quantities, such as the energy of an atom at rest. The discovery that particles are discrete packets of energy with wave-like properties led to the creation of a branch of physics dealing with atomic and subatomic systems that is now called quantum mechanics. It lays the mathematical foundation for many areas of physics and chemistry, including condensed matter physics, solid state physics, atomic physics, molecular physics, computational physics, computational chemistry, quantum chemistry, particle physics, nuclear chemistry, and nuclear physics. Some fundamental aspects of the theory are still being actively studied.
Quantum mechanics is essential for understanding the behavior of systems at atomic and smaller distance scales. If the physical nature of the atom were described solely by classical mechanics, then the electrons would not have to revolve around the nucleus, since the orbiting electrons should emit radiation (due to circular motion) and eventually collide with the nucleus due to energy loss by radiation. Such a system could not explain the stability of atoms. Instead, the electrons are in indeterminate, non-deterministic, smeared, probabilistic wave-particle orbitals around the nucleus, contrary to the traditional notions of classical mechanics and electromagnetism.
Quantum mechanics was originally developed to better explain and describe the atom, especially the differences in the spectra of light emitted by different isotopes of the same chemical element, as well as descriptions of subatomic particles. In short, the quantum mechanical model of the atom has been remarkably successful in an area where classical mechanics and electromagnetism failed.
Quantum mechanics includes four classes of phenomena that classical physics cannot explain:
In the mathematically rigorous formulation of quantum mechanics, developed by Paul Dirac, David Hilbert, John von Neumann, and Hermann Weyl, the possible states of a quantum mechanical system are symbolized by unit vectors (called state vectors). Formally, they belong to the complex separable Hilbert space - in other words, the state space or the associated Hilbert space of the system, and are defined up to a product by complex number with a unit module (phase factor). In other words, the possible states are points in the projective space of a Hilbert space, commonly referred to as the complex projective space. The exact nature of this Hilbert space depends on the system - for example, the state space of position and momentum is the space of square-integrable functions, while the state space for the spin of a single proton is just the direct product of two complex planes. Each physical quantity is represented by a hypermaximally Hermitian (more precisely: self-adjoint) linear operator acting on the state space. Each eigenstate of a physical quantity corresponds to an eigenvector of the operator, and the associated eigenvalue corresponds to the value of the physical quantity in that eigenstate. If the spectrum of the operator is discrete, the physical quantity can only take on discrete eigenvalues.
In the formalism of quantum mechanics, the state of a system at a given moment is described by a complex wave function, also called a state vector in a complex vector space. This abstract mathematical object allows you to calculate the probabilities of the outcomes of specific experiments. For example, it allows you to calculate the probability of finding an electron in a certain area around the nucleus at a certain time. Unlike classical mechanics, here one can never make simultaneous predictions with arbitrary accuracy for conjugate variables such as position and momentum. For example, electrons can be considered (with some probability) to be somewhere within a given region of space, but their exact location is unknown. You can draw regions of constant probability around the nucleus of an atom, often called "clouds," to represent where an electron might be with most likely. The Heisenberg uncertainty principle quantifies the inability to accurately localize a particle with a given momentum that is conjugate to position.
According to one interpretation, as a result of measurement, the wave function containing information about the probability of the state of the system decays from a given initial state to a certain eigenstate. Possible measurement results are the eigenvalues of the operator representing the physical quantity - which explains the choice of the Hermitian operator, whose eigenvalues are all real numbers. The probability distribution of a physical quantity in a given state can be found by calculating the spectral expansion of the corresponding operator. The Heisenberg uncertainty principle is represented by a formula in which operators corresponding to certain quantities do not commute.
The probabilistic nature of quantum mechanics thus follows from the act of measurement. This is one of the most difficult aspects of quantum systems to understand, and was a central theme in Bohr's famous debate with Einstein, in which both scientists attempted to elucidate these fundamental principles through thought experiments. For decades after the formulation of quantum mechanics, the question of what constitutes a "measurement" was widely studied. New interpretations of quantum mechanics have been formulated to do away with the notion of "wave function collapse". The basic idea is that when a quantum system interacts with a measuring apparatus, their respective wave functions become entangled, so that the original quantum system ceases to exist as an independent entity.
As a rule, quantum mechanics does not assign certain values. Instead, she makes a prediction using a probability distribution; that is, it describes the probability of obtaining possible outcomes from the measurement of a physical quantity. Often these results are warped, like probability density clouds, by many processes. Probability density clouds are an approximation (but better than the Bohr model) in which the position of an electron is given by a probability function, wave functions corresponding to eigenvalues, such that the probability is the square of the modulus of the complex amplitude, or quantum state of nuclear attraction. Naturally, these probabilities will depend on the quantum state at the "moment" of the measurement. Hence, uncertainty is introduced into the measured value. There are, however, some states that are associated with certain values of a particular physical quantity. They are called eigenstates (eigenstates) of a physical quantity ("eigen" can be translated from German as "intrinsic" or "proper").
It is natural and intuitive that everything in Everyday life(all physical quantities) have their own values. Everything seems to have a certain position, a certain moment, a certain energy, and a certain time of the event. However, quantum mechanics does not indicate exact values the position and momentum of the particle (since they are conjugate pairs) or its energy and time (since they are also conjugate pairs); more precisely, it provides only the range of probabilities with which this particle can have a given momentum and momentum probability. Therefore, it is advisable to distinguish between states that have undefined values and states that have definite values (eigenstates). As a rule, we are not interested in a system in which the particle has no eigenvalue of the physical quantity. However, when measuring a physical quantity, the wave function instantly takes on an eigenvalue (or "generalized" eigenvalue) of that quantity. This process is called the collapse of the wave function, a controversial and much discussed process in which the system under study is expanded by adding a measuring device to it. If the corresponding wave function is known immediately before the measurement, then the probability that the wave function will go into each of the possible eigenstates can be calculated. For example, the free particle in the previous example typically has a wave function, which is a wave packet centered around some average position x0 (having no position and momentum eigenstates). When the position of a particle is measured, it is impossible to predict the result with certainty. It is quite probable, but not certain, that it will be near x0, where the amplitude of the wave function is large. After performing the measurement, having obtained some result x, the wave function collapses into an eigenfunction of the position operator centered at x.
The temporal evolution of a quantum state is described by the Schrödinger equation, in which the Hamiltonian (the operator corresponding to the total energy of the system) generates the temporal evolution. The temporal evolution of wave functions is deterministic in the sense that - given what the wave function was at the initial time - one can make a clear prediction of what the wave function will be at any time in the future.
On the other hand, during the measurement, the change from the original wavefunction to another, later wavefunction will not be deterministic, but will be unpredictable (i.e., random). An emulation of time evolution can be seen here.
Wave functions change over time. The Schrödinger equation describes the change in wave functions with time, and plays a role similar to the role of Newton's second law in classical mechanics. The Schrödinger equation, applied to the free particle example above, predicts that the center of the wave packet will move through space at a constant speed (like a classical particle in the absence of forces acting on it). However, the wave packet will also spread out over time, which means that the position becomes more uncertain over time. This also has the effect of turning the position eigenfunction (which can be thought of as an infinitely sharp wavepacket peak) into an extended wavepacket that no longer represents a (certain) position eigenvalue.
Some wave functions give rise to probability distributions that are constant or independent of time - for example, when in a stationary state with constant energy, time disappears from the modulus of the square of the wave function. Many systems that are considered dynamic in classical mechanics are described in quantum mechanics by such "static" wave functions. For example, one electron in an unexcited atom is classically represented as a particle moving along a circular path around the atomic nucleus, while in quantum mechanics it is described by a static, spherically symmetric wave function surrounding the nucleus (Fig. 1) (note, however, that only the lowest states of orbital angular momentum, denoted as s, are spherically symmetrical).
The Schrödinger equation acts on the entire probability amplitude, not just on its absolute value. While the absolute value of the probability amplitude contains information about the probabilities, its phase contains information about the mutual influence between quantum states. This gives rise to "wave-like" behavior of quantum states. As it turns out, analytic solutions of the Schrödinger equation are possible only for a very small number of Hamiltonians with respect to simple models, such as the quantum harmonic oscillator, the particle in the box, the ion of the hydrogen molecule, and the hydrogen atom are the most important representatives of such models. Even the helium atom, which contains only one electron more than a hydrogen atom, has not succumbed to any attempt at a purely analytical solution.
However, there are several methods for obtaining approximate solutions. An important technique known as perturbation theory takes an analytical result obtained for a simple quantum mechanical model and generates a result for a more complex model that differs from the simpler model (for example) by adding the energy of a weak potential field. Another approach is the "semiclassical approximation" method, which is applied to systems for which quantum mechanics applies only to weak (small) deviations from classical behavior. These deviations can then be calculated based on the classical motion. This approach is especially important in the study of quantum chaos.
There are numerous mathematically equivalent formulations of quantum mechanics. One of the oldest and most used formulations is the "transformation theory" proposed by Paul Dirac, which combines and generalizes the two earliest formulations of quantum mechanics - matrix mechanics (created by Werner Heisenberg) and wave mechanics (created by Erwin Schrödinger).
Given that Werner Heisenberg was awarded the Nobel Prize in Physics in 1932 for the creation of quantum mechanics, Max Born's role in the development of QM was overlooked until the Nobel Prize was awarded to him in 1954. This role is mentioned in Born's 2005 biography, which talks about his role in the matrix formulation of quantum mechanics, as well as the use of probability amplitudes. In 1940, Heisenberg himself admits in a commemorative collection in honor of Max Planck that he learned about matrices from Born. In a matrix formulation, the instantaneous state of a quantum system determines the probabilities of its measurable properties or physical quantities. Example quantities include energy, position, momentum, and orbital momentum. Physical quantities can be either continuous (eg the position of a particle) or discrete (eg the energy of an electron bound to a hydrogen atom). Feynman path integrals – An alternative formulation of quantum mechanics that treats the quantum mechanical amplitude as the sum over all possible classical and non-classical paths between initial and final states. This is the quantum mechanical analogue of the principle of least action in classical mechanics.
The laws of quantum mechanics are fundamental. It is stated that the state space of the system is Hilbert, and the physical quantities of this system are Hermitian operators acting in this space, although it is not said which Hilbert spaces or which operators these are. They can be chosen accordingly to get quantitative characteristic quantum system. An important guideline for making these decisions is the correspondence principle, which states that the predictions of quantum mechanics are reduced to classical mechanics when the system goes into the region of high energies or, what is the same, into the region of large quantum numbers, that is, while a single particle has a certain degree of randomness, in systems containing millions of particles, averaged values prevail and, as we tend to the high-energy limit, the statistical probability of random behavior tends to zero. In other words, classical mechanics is simply the quantum mechanics of large systems. This "high energy" limit is known as the classical or correspondence limit. Thus, the solution can even start with a well-established classical model of a particular system, and then try to guess the underlying quantum model that would give rise to such a classical model when passing to the correspondence limit.
When quantum mechanics was originally formulated, it was applied to models whose limit of fit was nonrelativistic classical mechanics. For example, the well-known model of the quantum harmonic oscillator uses an explicitly non-relativistic expression for the kinetic energy of the oscillator and is thus a quantum version of the classical harmonic oscillator.
Early attempts to combine quantum mechanics with special relativity involved replacing the Schrödinger equation with covariant equations such as the Klein-Gordon equation or the Dirac equation. Although these theories were successful in explaining many experimental results, they had certain unsatisfactory qualities stemming from the fact that they did not take into account the relativistic creation and annihilation of particles. A fully relativistic quantum theory required the development of a quantum field theory that uses a quantization of the field (rather than a fixed set of particles). The first full-fledged quantum field theory, quantum electrodynamics, provides a complete quantum description of the electromagnetic interaction. The full apparatus of quantum field theory is often not required to describe electrodynamic systems. A simpler approach, taken since the inception of quantum mechanics, is to treat charged particles as quantum mechanical objects subjected to a classical electromagnetic field. For example, the elementary quantum model of the hydrogen atom describes the electric field of the hydrogen atom using the classical expression for the Coulomb potential:
E2/(4πε0r)
Such a "quasi-classical" approach does not work if quantum fluctuations electromagnetic field play an important role, for example, in the emission of photons by charged particles.
Quantum field theories have also been developed for strong and weak nuclear forces. Quantum field theory for strong nuclear interactions is called quantum chromodynamics and describes the interaction of subnuclear particles such as quarks and gluons. The weak nuclear and electromagnetic forces were unified in their quantized forms into a unified quantum field theory (known as the electroweak theory) by the physicists Abdus Salam, Sheldon Glashow and Steven Weinberg. For this work, all three received the Nobel Prize in Physics in 1979.
It turned out to be difficult to build quantum models for the fourth remaining fundamental force - gravity. Semiclassical approximations are made that lead to predictions such as Hawking radiation. However, the formulation of a complete theory of quantum gravity is hampered by apparent inconsistencies between general relativity (which is the most accurate theory of gravity currently known) and some of the fundamental tenets of quantum theory. Resolving these incompatibilities is an area of active research and theories such as string theory, one of the possible candidates for a future theory of quantum gravity.
Classical mechanics was also expanded into the complex realm, with complex classical mechanics beginning to behave like quantum mechanics.
The predictions of quantum mechanics have been confirmed experimentally to a very high degree of accuracy. According to the principle of correspondence between classical and quantum mechanics, all objects obey the laws of quantum mechanics, and classical mechanics is only an approximation for large systems of objects (or statistical quantum mechanics for a large set of particles). Thus, the laws of classical mechanics follow from the laws of quantum mechanics as a statistical average as the number of elements of the system or the values of quantum numbers tend to a very large limit. However, chaotic systems lack good quantum numbers, and quantum chaos studies the relationship between the classical and quantum descriptions of these systems.
Quantum coherence is an essential difference between classical and quantum theories, exemplified by the Einstein-Podolsky-Rosen (EPR) paradox, it has become an attack on the well-known philosophical interpretation of quantum mechanics by resorting to local realism. Quantum interference involves the addition of probability amplitudes, while classical "waves" involve the addition of intensities. For microscopic bodies, the extent of the system is much smaller than the coherence length, which leads to entanglement at large distances and other non-local phenomena characteristic of quantum systems. Quantum coherence does not usually show up on macroscopic scales, although an exception to this rule can occur at extremely low temperatures (i.e., approaching absolute zero), at which quantum behavior can show up on a macroscopic scale. This is in line with the following observations:
Many macroscopic properties classical system are a direct consequence of the quantum behavior of its parts. For example, the stability of the main part of matter (consisting of atoms and molecules, which would quickly collapse under the action of electrical forces alone), rigidity solids, as well as mechanical, thermal, chemical, optical and magnetic properties of matter are the result of the interaction electric charges according to the rules of quantum mechanics.
While the seemingly "exotic" behavior of matter postulated by quantum mechanics and relativity becomes more apparent when dealing with very small particles or moving at speeds approaching the speed of light, the laws of classical, often referred to as "Newtonian", physics remain accurate in predicting the behavior of the vast majority of "large" objects (on the order of the size of large molecules or even larger) and at speeds much lower than the speed of light.
Classical and quantum mechanics are very different in that they use very different kinematic descriptions.
According to the well-established opinion of Niels Bohr, experiments are required to study quantum mechanical phenomena, with a complete description of all the devices of the system, preparatory, intermediate and final measurements. Descriptions are presented in macroscopic terms, expressed in ordinary language, supplemented by the concepts of classical mechanics. The initial conditions and the final state of the system are respectively described by a position in the configuration space, for example, in coordinate space, or some equivalent space, such as momentum space. Quantum mechanics does not allow for a completely accurate description, both in terms of position and momentum, of an accurate deterministic and causal prediction of an end state from initial conditions or "states" (in the classical sense of the word). In this sense, promoted by Bohr in his mature writings, a quantum phenomenon is a process of transition from an initial to a final state, and not an instantaneous "state" in the classical sense of the word. Thus, there are two types of processes in quantum mechanics: stationary and transitional. For stationary processes, the start and end positions are the same. For transitional - they are different. It is obvious by definition that if only the initial condition is given, then the process is not defined. Given the initial conditions, the prediction of the final state is possible, but only at a probabilistic level, since the Schrödinger equation is determined for the evolution of the wave function, and the wave function describes the system only in a probabilistic sense.
In many experiments it is possible to take the initial and final state of the system as a particle. In some cases, it turns out that there are potentially several spatially distinguishable paths or trajectories along which the particle can pass from the initial to the final state. An important feature of the quantum kinematic description is that it does not allow one to unambiguously determine which of these paths the transition between states takes place. Only the initial and final conditions are defined, and, as indicated in the previous paragraph, they are defined only to the extent that the description of the spatial configuration or its equivalent permits. In every case for which a quantum kinematic description is needed, there is always a good reason for such a limitation on kinematic accuracy. The reason is that in order to experimentally find a particle in a certain position, it must be stationary; to experimentally find a particle with a certain momentum, it must be in free motion; these two requirements are logically incompatible.
Initially, classical kinematics does not require an experimental description of its phenomena. This makes it possible to completely accurately describe the instantaneous state of the system by a position (point) in the phase space - the Cartesian product of the configuration and momentum spaces. This description simply assumes or imagines the state as a physical entity without worrying about its experimental measurability. Such a description of the initial state, together with Newton's laws of motion, makes it possible to accurately make a deterministic and causal prediction of the final state, along with a certain trajectory of the system's evolution. For this, the Hamiltonian dynamics can be used. Classical kinematics also makes it possible to describe the process, similar to the description of the initial and final states used by quantum mechanics. Lagrangian mechanics allows you to do this. For processes in which it is necessary to take into account the magnitude of the action of the order of several Planck constants, classical kinematics is not suitable; here it is required to use quantum mechanics.
Even though the defining postulates of general relativity and Einstein's quantum theory are unequivocally supported by rigorous and repetitive empirical evidence, and although they do not contradict each other theoretically (at least in regard to their primary claims), they have proven extremely difficult to integrate into one consistent , a single model.
Gravity can be neglected in many areas of particle physics, so the unification between general relativity and quantum mechanics is not a pressing issue in these particular applications. However, the lack of a correct theory of quantum gravity is an important issue in physical cosmology and physicists' search for an elegant "Theory of Everything" (TV). Therefore, resolving all inconsistencies between both theories is one of the main goals for 20th and 21st century physics. Many prominent physicists, including Stephen Hawking, have labored over the years in an attempt to discover the theory behind everything. This TV will combine not only different models of subatomic physics, but also derive the four fundamental forces of nature - strong interaction, electromagnetism, weak interaction and gravity - from one force or phenomenon. While Stephen Hawking initially believed in TV, after considering Gödel's incompleteness theorem, he concluded that such a theory was not feasible and stated this publicly in his lecture Gödel and the End of Physics (2002).
The quest to unify the fundamental forces through quantum mechanics is still ongoing. Quantum electrodynamics (or "quantum electromagnetism"), which is currently (at least in the perturbative mode) the most accurate proven physical theory in competition with general relativity, has successfully unified the weak nuclear forces into the electroweak force and is currently being worked on. on the unification of the electroweak and strong interactions into the electrostrong interaction. Current predictions state that around 1014 GeV, the above three forces merge into a single unified field. In addition to this "grand unification", it is assumed that gravity can be unified with the other three gauge symmetries, which is expected to happen at about 1019 GeV. However - and while special relativity is carefully incorporated into quantum electrodynamics - extended general relativity, currently the best theory to describe the forces of gravity, is not fully incorporated into quantum theory. One of those who develops a consistent theory of everything, Edward Witten, a theoretical physicist, formulated M-theory, which is an attempt to explain supersymmetry on the basis of superstring theory. M-theory suggests that our apparent 4-dimensional space is actually an 11-dimensional space-time continuum containing ten space dimensions and one time dimension, although the 7 space dimensions at low energies are completely "condensed" (or infinitely curved) and are not easy to measure or study.
Another popular theory is Loop quantum gravity (LQG), a theory pioneered by Carlo Rovelli that describes the quantum properties of gravity. It is also a theory of quantum space and quantum time, since in general relativity the geometric properties of space-time are a manifestation of gravity. LQG is an attempt to unify and adapt standard quantum mechanics and standard general relativity. The main result of the theory is a physical picture in which space is granular. Graininess is a direct consequence of quantization. It has the same graininess of photons in the quantum theory of electromagnetism or discrete energy levels of atoms. But here space itself is discrete. More precisely, space can be viewed as an extremely thin fabric or network "woven" from finite loops. These loop networks are called spin networks. The evolution of a spin network over time is called spin foam. The predicted size of this structure is the Planck length, which is approximately 1.616 × 10-35 m. According to the theory, there is no point in a shorter length than this. Therefore, LQG predicts that not only matter, but space itself, has an atomic structure.
Since its inception, many of the paradoxical aspects and results of quantum mechanics have given rise to heated philosophical debates and many interpretations. Even fundamental questions, such as Max Born's basic rules about probability amplitude and probability distribution, took decades to be appreciated by the public and by many leading scientists. Richard Feynman once said, “I think I can safely say that no one understands quantum mechanics. In the words of Steven Weinberg, “At present, in my opinion, there is no absolutely satisfactory interpretation of quantum mechanics.
The Copenhagen interpretation - largely thanks to Niels Bohr and Werner Heisenberg - has remained the most accepted among physicists for 75 years after its announcement. According to this interpretation, the probabilistic nature of quantum mechanics is not a temporary feature that will eventually be replaced by a deterministic theory, but should be seen as a final rejection of the classical idea of "causation". In addition, it is believed that any well-defined applications of the quantum mechanical formalism in it must always make reference to the design of the experiment, due to the conjugate nature of the evidence obtained in different experimental situations.
Albert Einstein, being one of the founders of quantum theory, did not himself accept some of the more philosophical or metaphysical interpretations of quantum mechanics, such as the rejection of determinism and causation. His most quoted famous response to this approach is: "God does not play dice." He rejected the concept that the state of a physical system depends on the experimental measuring setup. He believed that natural phenomena occur according to their own laws, regardless of whether they are observed and how. In this regard, it is supported by the currently accepted definition of a quantum state, which remains invariant for an arbitrary choice of the configuration space for its representation, that is, the method of observation. He also believed that quantum mechanics should be based on a theory that carefully and directly expresses the rule that rejects the principle of long-range action; in other words, he insisted on the principle of locality. He considered, but theoretically justifiably dismissed, the private notion of latent variables in order to avoid uncertainty or lack of causality in quantum mechanical measurements. He believed that quantum mechanics was at that time the valid, but not the final and unshakable theory of quantum phenomena. He believed that its future replacement would require deep conceptual advances, and that it would not happen so quickly and easily. The Bohr-Einstein discussions provide a vivid critique of the Copenhagen interpretation from an epistemological point of view.
John Bell showed that this "EPR" paradox led to experimentally verifiable differences between quantum mechanics and theories that rely on the addition of hidden variables. Experiments have been carried out confirming the accuracy of quantum mechanics, thereby demonstrating that quantum mechanics cannot be improved by adding hidden variables. Alain Aspect's initial experiments in 1982, and many subsequent experiments since then, have definitively confirmed quantum entanglement.
Entanglement, as Bell's experiments showed, does not violate causality, since no information is transmitted. Quantum entanglement forms the basis of quantum cryptography, which is proposed for use in highly secure commercial applications in banking and government.
Everett's many-worlds interpretation, formulated in 1956, assumes that all the possibilities described by quantum theory occur simultaneously in a multiverse consisting mainly of independent parallel universes. This is not achieved by introducing some "new axiom" into quantum mechanics, but, on the contrary, is achieved by removing the axiom of wave packet decay. All possible successive states of the measured system and the measuring device (including the observer) are present in a real physical - and not just in a formal mathematical, as in other interpretations - quantum superposition. Such a superposition of successive combinations of states of different systems is called an entangled state. While the multiverse is deterministic, we perceive non-deterministic behavior, random in nature, since we can only observe the universe (i.e., the contribution of the compatible state to the aforementioned superposition) in which we, as observers, inhabit. Everett's interpretation fits in perfectly with John Bell's experiments and makes them intuitive. However, according to the theory of quantum decoherence, these "parallel universes" will never be available to us. Inaccessibility can be understood as follows: once a measurement is made, the system being measured becomes entangled both with the physicist who measured it and with a huge number of other particles, some of which are photons flying away at the speed of light to the other end of the universe. In order to prove that the wave function has not decayed, it is necessary to return all these particles back and measure them again along with the system that was originally measured. Not only is this completely impractical, but even if theoretically it could be done, any evidence that the original measurement took place would have to be destroyed (including the physicist's memory). In light of these Bell experiments, Cramer formulated his transactional interpretation in 1986. In the late 1990s, relational quantum mechanics emerged as a modern derivative of the Copenhagen interpretation.
Quantum mechanics has been a huge success in explaining many features of our universe. Quantum mechanics is often the only tool available that can reveal the individual behavior of the subatomic particles that make up all forms of matter (electrons, protons, neutrons, photons, etc.). Quantum mechanics has strongly influenced string theory - a contender for the theory of everything (a Theory of Everything).
Quantum mechanics is also critical to understanding how individual atoms create covalent bonds to form molecules. The application of quantum mechanics to chemistry is called quantum chemistry. Relativistic quantum mechanics can, in principle, mathematically describe most of chemistry. Quantum mechanics can also give a quantitative idea of the processes of ionic and covalent bonding, explicitly showing which molecules are energetically suitable for other molecules and at what energies. In addition, most calculations in modern computational chemistry rely on quantum mechanics.
In many industries, modern technologies operate at scales where quantum effects are significant.
Many modern electronic devices are designed using quantum mechanics. For example, the laser, the transistor (and thus the microchip), the electron microscope, and magnetic resonance imaging (MRI). The study of semiconductors led to the invention of the diode and transistor, which are indispensable components of modern electronic systems, computer and telecommunication devices. Another application is the light emitting diode, which is a highly efficient light source.
Many electronic devices operate under the influence of quantum tunneling. It is even present in a simple switch. The switch wouldn't work if the electrons couldn't quantum tunnel through the oxide layer on the metal contact surfaces. Flash memory chips, the heart of USB drives, use quantum tunneling to erase the information in their cells. Some negative differential resistance devices, such as the resonant tunnel diode, also use the quantum tunnel effect. Unlike classical diodes, the current in it flows under the action of resonant tunneling through two potential barriers. Its negative resistance mode of operation can only be explained by quantum mechanics: as the energy of the bound carrier state approaches the Fermi level, the tunneling current increases. As you move away from the Fermi level, the current decreases. Quantum mechanics is vital to understanding and designing these types of electronic devices.
Researchers are currently looking for reliable methods for directly manipulating quantum states. Efforts are being made to fully develop quantum cryptography, which theoretically will guarantee the secure transmission of information.
A more distant goal is to develop quantum computers that are expected to perform certain computational tasks exponentially faster than classical computers. Instead of classical bits, quantum computers use qubits, which can be in a superposition of states. Another active research topic is quantum teleportation, which deals with methods for transmitting quantum information over arbitrary distances.
While quantum mechanics is primarily applied to atomic systems with less matter and energy, some systems exhibit quantum mechanical effects on a large scale. Superfluidity - the ability to move a fluid flow without friction at a temperature near absolute zero, is one famous example such effects. Closely related to this phenomenon is the phenomenon of superconductivity - the flow of electron gas ( electricity), moving without resistance in a conducting material at sufficiently low temperatures. The fractional quantum Hall effect is a topologically ordered state that corresponds to long-range models of quantum entanglement. States with a different topological order (or a different configuration of far-range entanglement) cannot change the states into each other without phase transformations.
Quantum theory also contains accurate descriptions of many previously unexplained phenomena, such as blackbody radiation and the stability of orbital electrons in atoms. It also gave insight into how many different biological systems work, including olfactory receptors and protein structures. A recent study of photosynthesis has shown that quantum correlations play an important role in this fundamental process in plants and many other organisms. However, classical physics can often provide good approximations to the results obtained by quantum physics, usually under conditions of large numbers of particles or large quantum numbers. Since classical formulas are much simpler and easier to calculate than quantum formulas, the use of classical approximations is preferred when the system is large enough to make the effects of quantum mechanics negligible.
For example, consider a free particle. In quantum mechanics, wave-particle duality is observed, so that the properties of a particle can be described as properties of a wave. Thus, quantum state can be represented as a wave of arbitrary shape and extending in space as a wave function. The position and momentum of a particle are physical quantities. The uncertainty principle states that position and momentum cannot be measured exactly at the same time. However, it is possible to measure the position (without measuring momentum) of a moving free particle by creating an eigenstate of position with a wave function (Dirac delta function) that is very large at a certain position x, and zero at other positions. If you make a position measurement with such a wave function, then the result x will be obtained with a probability of 100% (that is, with full confidence, or with full accuracy). This is called the eigenvalue (state) of the position or, in mathematical terms, the eigenvalue of the generalized coordinate (eigendistribution). If a particle is in an eigenstate of position, then its momentum is absolutely undeterminable. On the other hand, if the particle is in an eigenstate of momentum, then its position is completely unknown. In an eigenstate of an impulse whose eigenfunction is in the form of a plane wave, one can show that the wavelength is h/p, where h is Planck's constant and p is the eigenstate momentum.
This is a model of the quantum tunneling effect, which plays an important role in the production of modern technological devices such as flash memory and scanning tunneling microscope. Quantum tunneling is the central physical process occurring in superlattices.
A particle in a one-dimensional potential box is the simplest mathematical example in which spatial constraints lead to quantization of energy levels. A box is defined as having zero potential energy everywhere within a certain area, and infinite potential energy everywhere outside that area.
A finite potential well is a generalization of the problem of an infinite potential well with a finite depth.
The problem of a finite potential well is mathematically more complex than the problem of a particle in an infinite potential box, since the wave function does not vanish on the walls of the well. Instead, the wave function must satisfy more complex mathematical boundary conditions, since it is non-zero in the region outside the potential well.
If you suddenly realized that you have forgotten the basics and postulates of quantum mechanics or do not know what kind of mechanics it is, then it's time to refresh this information in your memory. After all, no one knows when quantum mechanics can come in handy in life.
In vain you grin and sneer, thinking that you will never have to deal with this subject in your life at all. After all, quantum mechanics can be useful to almost every person, even those who are infinitely far from it. For example, you have insomnia. For quantum mechanics, this is not a problem! Read a textbook before going to bed - and you sleep soundly on the third page already. Or you can name your cool rock band that way. Why not?
Joking aside, let's start a serious quantum conversation.
Where to begin? Of course, from what a quantum is.
A quantum (from the Latin quantum - “how much”) is an indivisible portion of some physical quantity. For example, they say - a quantum of light, a quantum of energy or a field quantum.
What does it mean? This means that it simply cannot be less. When they say that some value is quantized, they understand that this value takes on a number of specific, discrete values. So, the energy of an electron in an atom is quantized, light propagates in "portions", that is, quanta.
The term "quantum" itself has many uses. A quantum of light (electromagnetic field) is a photon. By analogy, particles or quasi-particles corresponding to other fields of interaction are called quanta. Here we can recall the famous Higgs boson, which is a quantum of the Higgs field. But we do not climb into these jungles yet.
Quantum mechanics for dummies As you have already noticed, in our conversation we have mentioned particles many times. Perhaps you are used to the fact that light is a wave that simply propagates at a speed With . But if you look at everything from the point of view of the quantum world, that is, the world of particles, everything changes beyond recognition.
Quantum mechanics is a branch of theoretical physics, a component of quantum theory that describes physical phenomena at the most elementary level - the level of particles.
The effect of such phenomena is comparable in magnitude to Planck's constant, and Newton's classical mechanics and electrodynamics turned out to be completely unsuitable for their description. For example, according to the classical theory, an electron, rotating at high speed around the nucleus, must radiate energy and eventually fall onto the nucleus. This, as you know, does not happen. That is why they came up with quantum mechanics - the discovered phenomena needed to be explained somehow, and it turned out to be exactly the theory in which the explanation was the most acceptable, and all the experimental data "converged".
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The birth of quantum theory took place in 1900, when Max Planck spoke at a meeting of the German Physical Society. What did Planck say then? And the fact that the radiation of atoms is discrete, and the smallest portion of the energy of this radiation is equal to
Where h is Planck's constant, nu is the frequency.
Then Albert Einstein, introducing the concept of “light quantum”, used Planck's hypothesis to explain the photoelectric effect. Niels Bohr postulated the existence of stationary energy levels in an atom, and Louis de Broglie developed the idea of wave-particle duality, that is, that a particle (corpuscle) also has wave properties. Schrödinger and Heisenberg joined the cause, and so, in 1925, the first formulation of quantum mechanics was published. Actually, quantum mechanics is far from a complete theory; it is actively developing at the present time. It should also be recognized that quantum mechanics, with its assumptions, is unable to explain all the questions it faces. It is quite possible that a more perfect theory will come to replace it.
In the transition from the quantum world to the world of familiar things, the laws of quantum mechanics are naturally transformed into the laws of classical mechanics. We can say that classical mechanics is a special case of quantum mechanics, when the action takes place in our familiar and familiar macrocosm. Here, the bodies move quietly in non-inertial frames of reference at a speed much lower than the speed of light, and in general - everything around is calm and understandable. If you want to know the position of the body in the coordinate system - no problem, if you want to measure the momentum - you are always welcome.
Quantum mechanics has a completely different approach to the question. In it, the results of measurements of physical quantities are of a probabilistic nature. This means that when a value changes, several outcomes are possible, each of which corresponds to a certain probability. Let's give an example: a coin is spinning on a table. While it is spinning, it is not in any particular state (heads-tails), but only has the probability of being in one of these states.
Here we are slowly approaching Schrödinger equation And Heisenberg's uncertainty principle.
According to legend, Erwin Schrödinger, speaking at a scientific seminar in 1926 with a report on wave-particle duality, was criticized by a certain senior scientist. Refusing to listen to the elders, after this incident, Schrödinger actively engaged in the development of the wave equation for describing particles in the framework of quantum mechanics. And he did brilliantly! The Schrödinger equation (the basic equation of quantum mechanics) has the form:
This type equations - the one-dimensional stationary Schrödinger equation is the simplest.
Here x is the distance or coordinate of the particle, m is the mass of the particle, E and U are its total and potential energies, respectively. The solution to this equation is the wave function (psi)
The wave function is another fundamental concept in quantum mechanics. So, any quantum system that is in some state has a wave function that describes this state.
For example, when solving the one-dimensional stationary Schrödinger equation, the wave function describes the position of the particle in space. More precisely, the probability of finding a particle at a certain point in space. In other words, Schrödinger showed that probability can be described by a wave equation! Agree, this should have been thought of!
But why? Why do we have to deal with these incomprehensible probabilities and wave functions, when, it would seem, there is nothing easier than just taking and measuring the distance to a particle or its speed.
Everything is very simple! Indeed, in the macrocosm this is true - we measure the distance with a tape measure with a certain accuracy, and the measurement error is determined by the characteristics of the device. On the other hand, we can almost accurately determine the distance to an object, for example, to a table, by eye. In any case, we accurately differentiate its position in the room relative to us and other objects. In the world of particles, the situation is fundamentally different - we simply do not physically have measurement tools to measure the required quantities with accuracy. After all, the measurement tool comes into direct contact with the measured object, and in our case both the object and the tool are particles. It is this imperfection, the fundamental impossibility to take into account all the factors acting on a particle, as well as the very fact of a change in the state of the system under the influence of measurement, that underlie the Heisenberg uncertainty principle.
Let us present its simplest formulation. Imagine that there is some particle, and we want to know its speed and coordinate.
In this context, the Heisenberg Uncertainty Principle states that it is impossible to accurately measure the position and velocity of a particle at the same time. . Mathematically, this is written like this:
Here delta x is the error in determining the coordinate, delta v is the error in determining the speed. We emphasize that this principle says that the more accurately we determine the coordinate, the less accurately we will know the speed. And if we define the speed, we will not have the slightest idea about where the particle is.
There are many jokes and anecdotes about the uncertainty principle. Here is one of them:
A policeman stops a quantum physicist.
- Sir, do you know how fast you were moving?
- No, but I know exactly where I am.
And, of course, we remind you! If suddenly, for some reason, the solution of the Schrödinger equation for a particle in a potential well does not let you fall asleep, contact - professionals who were brought up with quantum mechanics on their lips!
Quantum mechanics is a branch of theoretical physics that describes physical phenomena in which the action is comparable in magnitude to Planck's constant. The predictions of quantum mechanics can differ significantly from those of classical mechanics. Since Planck's constant is an extremely small quantity compared to the action of objects in macroscopic motion, quantum effects mostly appear on microscopic scales. If the physical action of the system is much greater than Planck's constant, quantum mechanics goes organically into classical mechanics. In turn, quantum mechanics is a non-relativistic approximation (that is, an approximation of small energies compared to the rest energy of the massive particles of the system) of quantum field theory.
Classical mechanics, which describes macroscopic systems well, is not able to describe all phenomena at the level of molecules, atoms, electrons and photons. Quantum mechanics adequately describes the basic properties and behavior of atoms, ions, molecules, condensed matter, and other systems with an electron-nuclear structure. Quantum mechanics is also able to describe: the behavior of electrons, photons, and other elementary particles, however, a more accurate relativistically invariant description of the transformations of elementary particles is built within the framework of quantum field theory. Experiments confirm the results obtained with the help of quantum mechanics.
The basic concepts of quantum kinematics are the concepts of observable and state.
The basic equations of quantum dynamics are the Schrödinger equation, the von Neumann equation, the Lindblad equation, the Heisenberg equation, and the Pauli equation.
The equations of quantum mechanics are closely related to many branches of mathematics, among which are: operator theory, probability theory, functional analysis, operator algebras, group theory.
At a meeting of the German Physical Society, Max Planck read out his historic paper "On the theory of radiation energy distribution in the normal spectrum", in which he introduced the universal constant h (\displaystyle h). It is the date of this event, December 14, 1900, that is often considered the birthday of quantum theory.
To explain the structure of the atom, Niels Bohr proposed in 1913 the existence of stationary states of the electron, in which the energy can only take on discrete values. This approach, developed by Arnold Sommerfeld and other physicists, is often referred to as the old quantum theory (1900-1924). A distinctive feature of the old quantum theory is the combination of classical theory with additional assumptions that contradict it.
where through ⟨ψ | ϕ ⟩ (\displaystyle \langle \psi |\phi \rangle ) denoted by the scalar product of vectors | ψ ⟩ (\displaystyle |\psi \rangle ) And | ϕ ⟩ (\displaystyle |\phi \rangle ).
Where H ^ (\displaystyle (\hat(H))) is the Hamiltonian.
The main consequences of these provisions are:
These provisions make it possible to create a mathematical apparatus suitable for describing a wide range of problems in quantum mechanics of Hamiltonian systems in pure states. Not all states of quantum mechanical systems, however, are pure. In the general case, the state of the system is mixed and is described by the density matrix , for which the generalization of the Schrödinger equation - the von Neumann equation (for Hamiltonian systems) is valid. Further generalization of quantum mechanics to the dynamics of open, non-Hamiltonian, and dissipative quantum systems leads to the Lindblad equation.
Let the amplitude of the probability of finding a particle at a point M. The stationary Schrödinger equation allows us to determine it.
Function ψ (r →) (\displaystyle \psi ((\vec (r)))) satisfies the equation:
Where ∇ 2 (\displaystyle (\nabla )^(\,2)) is the Laplace operator, and U = U (r →) (\displaystyle U=U((\vec (r)))) is the potential energy of the particle as a function of .
The solution of this equation is the main problem of quantum mechanics. It is noteworthy that the exact solution of the stationary Schrödinger equation can be obtained only for a few relatively simple systems. Among such systems one can single out the quantum harmonic oscillator and the hydrogen atom. For most real systems, various approximate methods such as perturbation theory can be used to obtain solutions.
Solution of the stationary equation
Let E and U be two constants independent of r → (\displaystyle (\vec (r))).
By writing the stationary equation as:
The uncertainty relation arises between any quantum observables defined by non-commuting operators.
Let be the standard deviation of the particle coordinate M (\displaystyle M) moving along the axis x (\displaystyle x), and - standard deviation of its momentum . Quantities ∆ x (\displaystyle \Delta x) And ∆ p (\displaystyle \Delta p) are related by the following inequality:
Δ x Δ p ⩾ ℏ 2 (\displaystyle \Delta x\Delta p\geqslant (\frac (\hbar )(2)))Where h (\displaystyle h) is Planck's constant, and ℏ = h 2 π . (\displaystyle \hbar =(\frac (h)(2\pi )).)
According to the uncertainty relation, it is impossible to absolutely accurately determine both the coordinates and momentum of a particle. With an increase in the accuracy of measuring the coordinates, maximum accuracy momentum measurement decreases and vice versa. Those parameters for which such a statement is true are called canonically conjugate.
This centering on the dimension, coming from N. Bohr, is very popular. However, the uncertainty relation is derived theoretically from the postulates of Schrödinger and Born and concerns not the measurement, but the states of the object: it states that for any possible state, the corresponding uncertainty relations hold. Naturally, it will be carried out for measurements as well. Those. instead of "with increasing accuracy of measuring the coordinate, the maximum accuracy of measuring the momentum decreases" one should say: "in states where the uncertainty of the coordinate is less, the uncertainty of the momentum is greater."
Let ∆ E (\displaystyle \Delta E) is the root-mean-square deviation when measuring the energy of a certain state of a quantum system, and Δt (\displaystyle \Delta t) is the lifetime of this state. Then the following inequality holds,
Δ E Δ t ⩾ ℏ 2 . (\displaystyle \Delta E\Delta t\geqslant (\frac (\hbar )(2)).)In other words, the state of living a short time, cannot have a well-defined energy.
At the same time, although the form of these two uncertainty relations is similar, their nature (physics) is completely different.
A. SHISHLOV. based on the materials of the journals "Uspekhi fizicheskikh nauk" and "Scientific american".
The quantum-mechanical description of the physical phenomena of the microworld is considered the only true and most fully consistent with reality. Objects of the macrocosm obey the laws of another, classical mechanics. The boundary between the macro- and microworld 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) managed to express them in the best way.
1. Mental experiment to measure the components of the spin (proper momentum) of an electron using a device - a "black box".
2. Sequential measurement of two spin components. The "horizontal" spin of an electron is measured (left), then the "vertical" spin (right), then the "horizontal" spin again (bottom).
3A. Electrons with "right" spin after passing through the "vertical" box move in two directions: up and down.
3B. In the same experiment, we put some 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 microworld object is described by the so-called wave function.
5. Thought experiment by Erwin Schrödinger.
6. An experiment proposed by D. Bohm and J. 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 big picture world, in which mechanics acted as a universal law of motion of material points or particles - small lumps of matter. From these particles it was possible to build any objects. It seemed that Newtonian mechanics could theoretically explain everything. 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 reconsider physical theories and demanded 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. Such an 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 the absorption of energy must occur discretely, in portions, and managed to explain the features of the photoelectric effect (Nobel Prize in 1921). A light quantum - a photon, according to Einstein, 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 corpuscular-wave dualism 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 whole atoms - have wave properties. A year later, Einstein commented on this work: “Although it seems that 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 length of the de Broglie wave of an electron accelerated to an energy of 100 electron volts is 10 -8 cm. This wave is easy to detect experimentally by passing a stream of electrons through the crystal. Diffraction of their waves will occur on the crystal lattice and a characteristic striped pattern will appear. And for a dust particle weighing 0.001 grams at the same speed, the de Broglie wavelength will be 1024 times smaller, and it cannot be detected by any means.
De Broglie waves are unlike mechanical waves - fluctuations of matter propagating in space. They characterize the probability of finding a particle at a given point in space. Any particle appears as if "smeared" in space, and there is a non-zero probability of finding it anywhere. A classic example of a probabilistic description of microworld objects is an experiment on electron diffraction by two slits. An electron passing through a slit is registered on a photographic plate or on a screen in the form of a spot. Each electron can pass through either the right slot or the left slot in a completely random way. When there are a lot of spots, a diffraction pattern appears on the screen. The blackening of the screen is proportional to the probability of an electron appearing at 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 in 1933). It follows from the equations that any impact on a 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 registers measuring device, introducing unpredictable perturbations into the state of the measured object?
Thus, the study of elementary particles made it possible to establish at least three extremely surprising facts concerning the general physical picture of the world.
First, 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, and perhaps most strangely, the behavior of such physical objects as a "measuring device" or "observer" is not described by fundamental laws that are valid for other physical systems.
For the first time such conclusions were reached by the founders of quantum theory themselves - Niels Bohr, Werner Heisenberg, Wolfgang Pauli. Later, this point of view, called the Copenhagen Interpretation of Quantum Mechanics, was adopted in theoretical physics as the official one, which was reflected in all standard textbooks.
It is quite possible, however, that such conclusions were drawn too hastily. In 1952, the American theoretical physicist David D. Bohm created a deeply developed quantum theory, different from the generally accepted one, which just as well explains all the currently known features of the behavior of subatomic particles. It is a single set of physical laws that allows avoiding any randomness in the description of 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 a few thought experiments to measure the spin (intrinsic angular momentum) of an electron. Thoughtful because so far no one has been able to create a measuring device that allows you to accurately measure both components of the spin. Equally unsuccessful are attempts to predict which electrons will change their spin in the course of the described experiment, 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 conditionally call "upper" and "lower", "right" and "left" spins, respectively. The measurement is based on the spatial separation of particles with different spins. Separation devices can be imagined as some kind of "black boxes" of two types - "horizontal" and "vertical" (Fig. 1). It is known that the 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 the other may change, and in a completely uncontrolled way (2).
Trying to explain the results obtained, the traditional quantum theory came to the conclusion that it is necessary to completely abandon the deterministic, that is, completely determining state
object, description of the phenomena of the microworld. 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 recombine into one beam in a special "black box" (3).
The results of these experiments contradict conventional logic. Indeed, let us consider the behavior of some electron in the case when there is no absorbing wall (3 A). Where will he move? Let's say down. Then, if initially the electron had a "right" spin, it will remain right 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 the other half. An obvious contradiction. Could an electron go up? No, for the same reason. Perhaps he did not move down, not up, but in some other way? But, having blocked the upper and lower routes with absorbing walls, we will not get anything at all at the output. It remains to assume that the electron can move in two directions at once. Then, being able to fix its position at different moments of time, in half of 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 split into two nor "jump" from one trajectory to another.
What does the 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 electron motion (and, accordingly, about the direction of its spin) is incorrect. The manifestation of the quantum nature of the electron lies in the fact that there is no answer to this question in principle. The state of an electron is a superposition, that is, the sum of two states, each of which has a certain value of the "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.
For the mathematical description of the states of quantum objects, the 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 finding a particle at a given point in space. Thus, if a particle is located in some region A, its wave function is equal to zero everywhere except for this region. Similarly, a particle localized in region B has a wave function that is non-zero only in B. If the state of the particle turns out to be a superposition of its being in A and B, then the wave function describing such a state is non-zero 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 a 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 collapses, as it were, "collapses" into a point.
One of the main claims of quantum mechanics is that physical objects are completely described by their wave functions. Thus, the whole point of the laws of physics is to predict 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 for the 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 "actually physical processes", that is, into "observers" and "observed", 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, the quantum theory in such an interpretation becomes ambiguous, loses its objectivity and fundamentality. The "problem of measurement" 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 like, a million) is a heap. Two grains is also not a heap, but 999 (or 999999) is a heap. This chain of reasoning leads to a certain number of grains, at which the concepts of "heap - not a heap" become indefinite. They will depend on the subjective assessment of the observer, that is, on the method of measurement, even if only 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 extended down to the smallest particles. On the other hand, going from the side of the microcosm, one should include in the wave description objects of ever larger size up to the Universe as a whole. The boundary between the macro- and microcosm is not defined, and attempts to designate it lead to a paradox. The so-called Schrödinger's cat problem, a thought experiment proposed by Erwin Schrödinger in 1935, points to it most clearly (5).
A cat is sitting in a closed box. There is also a vial of poison, a radiation source, and a counter of charged particles connected to a device that breaks the vial 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 obey 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 also finds itself in a superposition of states of life and death.
In reality, of course, there can be no real paradox here. Particle registration is an irreversible process. It is accompanied by the collapse of the wave function, followed by a mechanism that breaks the vial. However, orthodox quantum mechanics does not consider irreversible phenomena. The paradox that arises in full accordance with its laws clearly shows that between the quantum microcosm and the classical macrocosm there is some intermediate region in which quantum mechanics does not work.
So, despite the undoubted successes of quantum mechanics in explaining experimental facts, at the present moment it can hardly lay claim to the completeness and universality of the description of physical phenomena. One of the most daring alternatives to quantum mechanics was the theory proposed by David Bohm.
Having set himself the goal of constructing 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 probability characteristic, but of a very real physical object, a kind of quantum mechanical field that has 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 in time, the other describes their effect on the corresponding particles. The laws are applicable to all physical objects without exception - both to "observers" and to "observed".
Thus, if at some point the position of all particles in the Universe and the total 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 point in time. Therefore, no accident in physical processes out of the question. 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 parameters of the system leads to the fact that in practice we always operate with some average values. It is this "ignorance", according to Bohm, that makes us resort to probabilistic laws when describing phenomena in the microcosm (a similar situation occurs 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 electron movement 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 reflected from the walls, both parts of the wave function will reunite in the "black box", and in this case the electron will receive information about the 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 interesting property of "empty" wave functions: being real, they nevertheless have no effect on extraneous objects and cannot be registered by measuring instruments. And the "empty" wave function has a force effect on its "own" electron, regardless of the distance, and this effect is transmitted instantly.
Attempts to "correct" quantum mechanics or to explain the contradictions that arise in it have been made by many researchers. 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 the wave functions. "From this point of view, there are no isolated, 'free' particles left to themselves," wrote Blokhintsev. "It is possible that in this connection between particles and the medium, the nature of the impossibility of isolating a particle, which manifests itself in the apparatus of quantum mechanics, is hidden."
And yet - why is 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, no one wanted 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 essentially new results. Werner Heisenberg, for example, believed that "for any experiment, his (Bohm's) results coincide with the Copenhagen interpretation. Hence the first consequence: Bohm's interpretation cannot be refuted by experiment..." space. In their opinion, this contradicts physical reality, because phenomena in the quantum world cannot in principle be described by deterministic laws. There are many other equally controversial arguments against Bohm's theory, which themselves require serious evidence. In any case, so far no one has been able to completely refute it. Moreover, many researchers, including domestic ones, continue to work on its improvement.
