What is quantum mechanics?
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 the characteristics of both particles and waves (wave-particle duality), and there are limits to 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.
History of quantum mechanics
Scientific research wave nature of light began in the XVII 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, Schatien Ranatom Bose, Arnold Sommerfeld and others laid the foundations of quantum mechanics. Niels Bohr's Copenhagen interpretation has received universal acclaim.
In the mid-1920s, the development of 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 works of David Hilbert, Paul Dirac and John von Neumann, who focused on great attention 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 of the features of the modern periodic table of the elements and describes the behavior of atoms in chemical reactions and the movement of electrons in computer semiconductors, and therefore plays a critical role in many 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.
What does the word quantum mean?
The word quantum comes from the Latin "quantum", which means "how much" or "how much". In quantum mechanics, a quantum means a discrete unit 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.
Significance of quantum mechanics
Quantum mechanics has importance to understand 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, and to describe 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:
- quantization of individual physical properties
- quantum entanglement
- uncertainty principle
- wave-particle duality
Mathematical foundations of quantum mechanics
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 - otherwise, the state space or the associated Hilbert space of the system, and are defined up to a product by a complex number with a unit modulus (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 areas of constant probability, often called "clouds," around the nucleus of an atom to represent where an electron is most likely to be. 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.
Measurement in quantum mechanics
The probabilistic nature of quantum mechanics thus follows from the act of measurement. This is one of the most difficult aspects of quantum systems to understand, and 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.
The probabilistic nature of the predictions of quantum mechanics
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 specify the exact position and momentum of a 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.
Schrödinger equation in quantum mechanics
The temporal evolution of a quantum state is described by the Schrödinger equation, in which the Hamiltonian (the operator corresponding to the total energy of the system) generates the temporal evolution. The temporal evolution of wave functions is deterministic in the sense that - given what the wave function was at the initial 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, a single electron in an unexcited atom is classically represented as a particle moving in 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 symmetric).
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, analytical solutions to the Schrödinger equation are only possible for a very small number of relatively simple Hamiltonians, such as the quantum harmonic oscillator, the particle in a box, the hydrogen molecule ion, and the hydrogen atom - these 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.
Mathematically equivalent formulations of quantum mechanics
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.
Laws of quantum 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 appropriately to quantify the quantum system. An important guideline for making these decisions is the correspondence principle, which states that the predictions of quantum mechanics are reduced to classical mechanics when the system goes into the region of high energies or, what is the same, into the region of large quantum numbers, that is, while an individual particle has a certain degree of randomness, in systems containing millions of particles, average values prevail and, as the high-energy limit tends, 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.
Interaction with other scientific theories
Early attempts to combine quantum mechanics with special relativity involved replacing the Schrödinger equation with covariant equations such as the Klein-Gordon equation or the Dirac equation. Although these theories were successful in explaining many experimental results, they had certain unsatisfactory qualities 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 the quantum fluctuations of the electromagnetic field play an important role, for example, when charged particles emit photons.
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.
Relationship between quantum mechanics and classical 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 of a classical system are a direct consequence of the quantum behavior of its parts. For example, the stability of the main part of matter (consisting of atoms and molecules, which would quickly collapse under the action of electrical forces alone), the rigidity of solids, as well as the mechanical, thermal, chemical, optical and magnetic properties of matter are the result of the interaction of electric charges in accordance with the rules of quantum mechanics.
While the seemingly "exotic" behavior of matter postulated by quantum mechanics and 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 at 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 smaller than the speed of light.
What is the difference between quantum mechanics and classical mechanics?
Classical and quantum mechanics are very different in that they use very different kinematic descriptions.
According to the well-established opinion of Niels Bohr, 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 term). 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.
General theory of relativity
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 coherent, unified 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).
Basic theories of quantum mechanics
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, successfully unifies the weak nuclear forces into the electroweak force, and work is underway to unify the electroweak and strong forces into the electrostrong force. 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 not easily measurable or examined.
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.
Philosophical aspects of quantum mechanics
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 an experimental measurement setup. He believed that natural phenomena occur according to their own laws, regardless of whether they are observed and how. In this regard, it is supported by the currently accepted definition of a quantum state, which remains invariant 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.
Quantum physics in electronics
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 mode of operation with negative resistance can only be explained by quantum mechanics: as the bound-carrier state energy 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.
quantum cryptography
Researchers are currently looking for reliable methods for directly manipulating quantum states. Efforts are being made to fully develop quantum cryptography, which theoretically will guarantee the secure transmission of information.
quantum computing
A more distant goal is to develop quantum computers that are expected to perform certain computational tasks exponentially faster than classical computers. Instead of classical bits, quantum computers use qubits, which can be in a superposition of states. Another active research topic is quantum teleportation, which deals with methods for transmitting quantum information over arbitrary distances.
quantum effects
While quantum mechanics is primarily applied to atomic systems with less matter and energy, some systems exhibit quantum mechanical effects on a large scale. Superfluidity, the ability to move fluid without friction at temperatures near absolute zero, is one well-known example of such effects. Closely related to this phenomenon is the phenomenon of superconductivity - a flow of electron gas (electric current) moving without resistance in a conducting material at sufficiently low temperatures. The fractional quantum Hall effect is a topologically ordered state that corresponds to long-range 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
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.
Free particle motion
For example, consider a free particle. In quantum mechanics, wave-particle duality is observed, so that the properties of a particle can be described as properties of a wave. Thus, a quantum state can be represented as a wave of arbitrary shape and extending through 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.
Rectangular potential barrier
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.
Particle in a one-dimensional potential box
A particle in a one-dimensional potential box is the simplest mathematical example in which spatial constraints lead to quantization of energy levels. A box is defined as having zero potential energy everywhere within a certain area, and infinite potential energy everywhere outside that area.
Ultimate potential well
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.
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M. G. Ivanov
How to understand quantum mechanics
Moscow Izhevsk
UDC 530.145.6 LBC 22.314
Ivanov M. G.
How to understand quantum mechanics. - M.–Izhevsk: Research Center "Regular and Chaotic Dynamics", 2012. - 516 p.
This book is devoted to a discussion of issues that, from the point of view of the author, contribute to the understanding of quantum mechanics and the development of quantum intuition. The purpose of the book is not just to give a summary of the basic formulas, but also to teach the reader to understand what these formulas mean. Special attention devoted to a discussion of the place of quantum mechanics in modern scientific picture world, its meaning (physical, mathematical, philosophical) and interpretations.
The book fully includes the material of the first semester of the standard annual course in quantum mechanics and can be used by students as an introduction to the subject. For the novice reader, discussions of the physical and mathematical meaning of the introduced concepts should be useful, however, many subtleties of the theory and its¨ interpretations may turn out to be unnecessary and even confusing, and therefore should be omitted at the first reading.
ISBN 978-5-93972-944-4 |
c M. G. Ivanov, 2012
c Research Center "Regular and Chaotic Dynamics", 2012
1. Thanks. . . . . . . . . . . . . . . . . . . . . . . . . . . xvii
2. About the distribution of this book. . . . . . . . . . . . . . . .xviii
1.1.2. How interactions work. . . . . . . . . . . . . . 3
1.1.3. Statistical physics and quantum theory. . . . . . . 5
1.1.4. Fundamental fermions. . . . . . . . . . . . . . . 5
1.1.8. The Higgs field and the Higgs boson (*) . . . . . . . . . . . . . 15
1.1.9. Vacuum (*) . . . . . . . . . . . . . . . . . . . . . . . . . 18
1.2. Where did quantum theory come from? . . . . . . . . . . . . . . . . 20
1.3. Quantum mechanics and complex systems. . . . . . . . . . . . 21
1.3.1. Phenomenology and quantum theory. . . . . . . . . . . 21
2.3.1. When the observer turned away. . . . . . . . . . . . . . . thirty
2.3.2. Before our eyes. . . . . . . . . . . . . . . . . . . . . . . 31
2.4. Correspondence principle (f). . . . . . . . . . . . . . . . . . . . 33
2.5. A few words about classical mechanics (f). . . . . . . . . . 34
2.5.1. Probabilistic nature of classical mechanics (f) . . 35
ABOUT HEAD |
2.5.2. The Heresy of Analytic Determinism and Perturbation Theory (f) . . . . . . . . . . . . . . . . . . . . . . . . . . 36
Theoretical mechanics classical and quantum (f) . . . . |
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A few words about optics (f). . . . . . . . . . . . . . . . . . |
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Mechanics and optics geometric and wave (f) . . |
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2.7.2. Complex amplitude in optics and the number of photons (φ*) |
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Fourier transform and relations undefined¨- |
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2.7.4. The Heisenberg microscope and the relation is indeterminate¨- |
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news. . . . . . . . . . . . . . . . . . . . . . . . . . . . |
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CHAPTER 3. Conceptual foundations of quantum theory. . . . . . . . . 47
3.1. Probabilities and probability amplitudes. . . . . . . . . . . . . 47
3.1.1. Addition of probabilities and amplitudes. . . . . . . . . . . 49
3.1.2. Multiplication of probabilities and amplitudes. . . . . . . . . . 51
3.1.3. Association of independent subsystems. . . . . . . . . . 51
3.1.4. Probability distributions and wave functions in measurement. . . . . . . . . . . . . . . . . . . . . . . 52
3.1.5. Amplitude at measurement and scalar product. 56
3.2. Everything is possible¨ that can happen (f*). . . . . . . . . . . . 58
3.2.1. Big in small (f*). . . . . . . . . . . . . . . . . . . 63
CHAPTER 4. Mathematical concepts of quantum theory . . . . . . 66 4.1. The space of wave functions. . . . . . . . . . . . . . . . 66
4.1.1. What variable is the wave function a function of? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
4.1.2. Wave function as a state vector. . . . . . . . 69
4.2. Matrices (l) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
4.3. Dirac notation. . . . . . . . . . . . . . . . . . . . . 75
4.3.1. Basic "building blocks" of Dirac notation. . . . . . . . . . . . . . . . . . . . . . . . . . . 76
4.3.2. Combinations of the main blocks and their meaning. . . . . . 77
4.3.3. Hermitian conjugation. . . . . . . . . . . . . . . . . . . 79
4.4. Multiplication on the right, on the left, . . . top, bottom and obliquely**. . 80
4.4.1. Diagram notation* . . . . . . . . . . . . . . . 81
4.4.2. Tensor Notation in Quantum Mechanics* . . . . 82
4.4.3. Dirac notation for complex systems* . . . . 83
4.4.4. Comparison of different designations * . . . . . . . . . . . . . 84
4.5. The meaning of the scalar product. . . . . . . . . . . . . . . . . 86
4.5.1. Normalization of wave functions to unity. . . . . . 86
ABOUT HEAD |
4.5.2. The physical meaning of the scalar square. Probability normalization. . . . . . . . . . . . . . . . . . . . . . . 87
4.5.3. The physical meaning of the scalar product. . . . . . 89
4.6. Bases in the state space. . . . . . . . . . . . . . . . 90
4.6.1. Expansion in a basis in the state space, normal
adjustment of basis vectors. . . . . . . . . . . . . . . |
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The nature of the states of the continuous spectrum* . . . . . . |
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Change of basis. . . . . . . . . . . . . . . . . . . . . . . |
4.7. Operators. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
4.7.1. Operator kernel* . . . . . . . . . . . . . . . . . . . . . . 99
4.7.2. Matrix element of the operator. . . . . . . . . . . . . . 100
4.7.3. Basis of eigenstates. . . . . . . . . . . . . . 101
4.7.4. Vectors and their components** . . . . . . . . . . . . . . . 101
4.7.5. Average from the operator. . . . . . . . . . . . . . . . . . . 102
4.7.6. Expansion of the operator in terms of the basis. . . . . . . . . . . . . 103
4.7.7. Domains of definition of operators in infinity* 104
4.7.8. Operator trace* . . . . . . . . . . . . . . . . . . . . . . 106
4.8.2. Density matrix for subsystem* . . . . . . . . . . 111
4.9. Observed* . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
4.9.1. Quantum Observables* . . . . . . . . . . . . . . . . 114
4.9.2. Classic observables** . . . . . . . . . . . . . . 115
4.9.3. Reality of observables*** . . . . . . . . . . . . 116
4.10. Position and momentum operators. . . . . . . . . . . . . . . 119
4.11. variational principle. . . . . . . . . . . . . . . . . . . . . . 121
4.11.1. Variational principle and Schrödinger equations**¨ . 121
4.11.2. Variational principle and ground state. . . . . 123
4.11.3. Variational principle and excited¨ states*. 124
CHAPTER 5. Principles of quantum mechanics. . |
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5.1. Quantum mechanics of a closed system |
5.1.1. Unitary evolution and conservation of probability. . . . 125
5.1.2. Unitary evolution of the density matrix* . . . . . . . 128
5.1.3. (Non)unitary evolution***** . . . . . . . . . . . . . . 128
5.1.4. The Schrödinger equation¨ and the Hamiltonian. . . . . . . . . 130
5.2.4. Functions from operators in different representations. . . 136
5.2.5. Hamiltonian in the Heisenberg representation. . . . . . 137
5.2.6. Heisenberg equation. . . . . . . . . . . . . . . . . . 137
5.2.7. Poisson bracket and commutator* . . . . . . . . . . . . . 141
5.2.8. Pure and mixed states in theoretical mechanics*. . . . . . . . . . . . . . . . . . . . . . . . . . . 143
5.2.9. The representations of Hamilton and Liouville in the theoretical
what mechanics** . . . . . . . . . . . . . . . . . . . . . |
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5.2.10. Equations in Interaction View* . . . . |
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5.3. Measurement. . . . . . . . . . . . . . . . . . . . . . . . . . . . |
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projection postulate. . . . . . . . . . . . . . . . |
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Selective and non-selective measurement* . . . . . . |
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State preparation. . . . . . . . . . . . . . . . |
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CHAPTER 6. One-dimensional quantum systems. . . . . . . . . . . . |
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6.1. Spectrum structure. . . . . . . . . . . . . . . . . . . . . . . . . 157
6.1.1. Where does spectrum come from? . . . . . . . . . . . . . . . . . . 157
6.1.2. Reality of eigenfunctions. . . . . . . . . 158
6.1.3. The structure of the spectrum and the asymptotics of the potential. . . . . 158
6.2. Oscillatory theorem. . . . . . . . . . . . . . . . . . . . . . 169
6.2.3. Wronskian (l*) . . . . . . . . . . . . . . . . . . . . . . . 172
6.2.4. Growth in the number of zeros with the level number* . . . . . . . . . . 173
6.3.1. Formulation of the problem. . . . . . . . . . . . . . . . . . . . . 176
6.3.2. Example: scattering on a step. . . . . . . . . . . . . 178
7.1.2. The meaning of the probability space*. . . . . . . . . . 195
7.1.3. Averaging (integration) over measure* . . . . . . . . . 196
7.1.4. Probability spaces in quantum mechanics (φ*)196
7.2. Uncertainty relations¨. . . . . . . . . . . . . . . . 197
7.2.1. Uncertainty relations¨ and (anti)commutators 197
7.2.2. So what did we count? (f) . . . . . . . . . . . . . . 199
7.2.3. coherent states. . . . . . . . . . . . . . . . . . 200
7.2.4. Uncertainty Relations¨ time is energy. . . . 202
7.3. Measurement without interaction* . . . . . . . . . . . . . . . . . 207
7.3.1. Penrose experiment with bombs (f *) . . . . . . . . . 209
7.4. The quantum Zeno effect (the paradox of a non-boiling teapot)
7.5. Quantum (non)locality. . . . . . . . . . . . . . . . . . . . 218
7.5.1. Entangled states (f*) . . . . . . . . . . . . . . . . 218
7.5.2. Entangled states in selective measurement (φ*) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219
7.5.3. Entangled states in a non-selective measurement
7.5.5. Relative states (f*) . . . . . . . . . . . . . . 224
7.5.6. Bell's inequality and its violation (f**) . . . . . . . 226
7.6. Theorem on the impossibility of cloning a quantum state** . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233
7.6.1. The meaning of the impossibility of cloning (f *) . . . . . . . 235
8.1. The structure of quantum theory (f) . . . . . . . . . . . . . . . . . 243
8.1.1. The concept of classical selective measurement (f) . . 243
8.1.2. Quantum theory in large blocks. . . . . . . . . . 244
8.1.3. Quantum locality (f) . . . . . . . . . . . . . . . . 245
8.1.4. Questions about the self-consistency of quantum theory (f) 245
8.2. Measuring instrument simulation* . . . . . . . . . . . 246
8.2.1. Measuring device according to von Neumann** . . . . . . . 246
8.3. Is another theory of measurements possible? (ff) . . . . . . . . . . . 250
8.3.2. "Rigidity"¨ formulas for probabilities (ff) . . . . . 253
8.3.3. Theorem of quantum telepathy (ff *) . . . . . . . . . . 254
8.3.4. "Softness" of the projection postulate (ff). . . . . . . 256
8.4. Decoherence (ff) . . . . . . . . . . . . . . . . . . . . . . . . . 257
CHAPTER 9. On the verge of physics and philosophy (ff *) . . . . . . . . . . 259
9.1. Riddles and paradoxes of quantum mechanics (f *) . . . . . . . . . 259
9.1.1. Einstein's mouse (f *) . . . . . . . . . . . . . . . . . . 260
9.1.2. Schrödinger's cat¨ (f *) . . . . . . . . . . . . . . . . . . . 261
9.1.3. Friend of Wigner (f *) . . . . . . . . . . . . . . . . . . . . . 265
9.2. What is the misunderstanding of quantum mechanics? (ff) . . . . 267
9.3.2. Copenhagen interpretation. Reasonable self-restraint (f). . . . . . . . . . . . . . . . . . . . . . . . . 276
9.3.3. Quantum Theories with Hidden Parameters (ff). . 278
9.3.6. "Abstract Self" von Neumann (ff). . . . . . . . . . . 284
9.3.7. Everett's Many Worlds Interpretation (ff). . . . . . 285
9.3.8. Consciousness and Quantum Theory (ff). . . . . . . . . . . . 289
9.3.9. Active consciousness (ff *) . . . . . . . . . . . . . . . . . 292
CHAPTER 10 Quantum informatics**. . . . . . . . . . . . . . . 294 10.1. Quantum Cryptography** . . . . . . . . . . . . . . . . . . . . 294
10.4. The concept of a universal quantum computer. . . . . . . 298
10.5. Quantum parallelism. . . . . . . . . . . . . . . . . . . . . . 299
10.6. Logic and calculations. . . . . . . . . . . . . . . . . . . . . . . 300
ABOUT HEAD |
10.6.3. Reversible classical computations. . . . . . . . . . 302
10.6.4. Reversible calculations. . . . . . . . . . . . . . . . . . 302
10.6.5. Gates are purely quantum. . . . . . . . . . . . . . . . 303
10.6.6. Reversibility and cleaning of "garbage". . . . . . . . . . . . . 304
CHAPTER 11. Symmetries-1 (Noether's theorem)¨. . . . . . . . . . . . . . 306 11.1. What is symmetry in quantum mechanics. . . . . . . . . . 306 11.2. Operator transformations "together" and "instead of". . . . . . . 308
11.2.1. Continuous transformations of operators and commutators. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309
11.3. Continuous symmetries and conservation laws. . . . . . . . 309
11.3.1. Saving a single operator. . . . . . . . . . . . 311
11.3.2. Generalized¨ momentum. . . . . . . . . . . . . . . . . . . 311
11.3.3. Momentum as a generalized¨ coordinate*. . . . . . . . . 314
11.4. Conservation laws for previously discrete symmetries. . . . . 316
11.4.1. Mirror symmetry and more. . . . . . . . . . . . 317
11.4.2. Parity*¨ . . . . . . . . . . . . . . . . . . . . . . . . . . 319
11.4.3. Quasi-momentum* . . . . . . . . . . . . . . . . . . . . . . . 320
11.5. Shifts in phase space** . . . . . . . . . . . . . . . . 322
11.5.1. Group shift switch* . . . . . . . . . . . . . 322
11.5.2. Classical and quantum observables**. . . . . . . 324
11.5.3. Curvature of the phase space**** . . . . . . . . . . 326
CHAPTER 12 Harmonic oscillator. . . . . . . . . . . . . . . 328
12.2.1. ladder operators. . . . . . . . . . . . . . . . . . 330
12.2.2. Basis of eigenfunctions. . . . . . . . . . . . . . . 335
12.3. Transition to coordinate representation. . . . . . . . . . . 337
12.4. Calculation example¨ in filling numbers representation* . . . . . 342
12.5. Symmetries of a harmonic oscillator. . . . . . . . . . . . 343
12.5.1. Mirror symmetry. . . . . . . . . . . . . . . . . . . 343
12.5.2. Fourier symmetry and the transition from the coordinate
ABOUT HEAD |
12.7.2. Coherent states in the representation of occupation numbers** . . . . . . . . . . . . . . . . . . . . . . . . . . . 351
12.8. Expansion in terms of coherent states** . . . . . . . . . . . 353
12.9. Compressed States** . . . . . . . . . . . . . . . . . . . . . . . . 356
13.1. De Broglie waves. Phase and group velocity. . . . . . . 363 13.2. What is a function from operators? . . . . . . . . . . . . . . . . 365 13.2.1. Power series and polynomials of commuting arguments
cops. . . . . . . . . . . . . . . . . . . . . . . . . . . 366
13.2.2. Functions of simultaneously diagonalizable operators. 366
13.2.3. Functions of noncommuting arguments. . . . . . . . 367
13.2.4. Derivative with respect to operator argument. . . . . . . . 368
13.5. semiclassical approximation. . . . . . . . . . . . . . . . . 375
13.5.1. How to guess and remember the semiclassical wave function. . . . . . . . . . . . . . . . . . . . . . . . 375
13.5.2. How to derive a semiclassical wave function. 377
13.5.3. Semiclassical wave function near the turning point 379
13.5.4. Semiclassical quantization. . . . . . . . . . . . . 383
13.5.5. Spectral density of the semiclassical spectrum. 384
13.5.6. Quasi-stationary states in quasi-classics. . . . 386
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.
Quantum
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 How can mechanics be quantum?
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 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".
By the way! For our readers there is now a 10% discount on
A bit of history
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 special case 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 of equation, 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!
Welcome to the blog! I am very glad to you!
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, the basics of quantum physics, the 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 interference?
- What is spin and superposition?
- What is "measurement" or "wavefunction collapse"?
- What is quantum entanglement (or quantum teleportation for dummies)? (see article)
- What's happened thought experiment"Shroedinger `s cat"? (see article)
What is quantum physics and quantum mechanics?
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. Fantastic! 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 fly through 2 slots at the same time in some way that is incomprehensible to us. 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 in 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.
Fantastic! Is not it?
But that is 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's happened ?
- 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!
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