Physico-chemical microstructure of matter - atoms and atomic nuclei

1. Nuclear and radiation physics
1.0. Physics - fundamental natural science
1.1. Atoms and atomic nuclei
1.2. Radioactivity
1.3. Nuclear reactions and nuclear energy
1.4. Radionuclides
1.5. Elementary particles and accelerators
1.6. Ionizing radiation

1.1. Atoms and atomic nuclei

Substance, Fields, Particles, Interactions
In the introduction to our treatise on atoms, atomic nuclei, and the physics of the microworld, we make a few preliminary remarks about the basic building blocks of matter and the nature of the forces that govern their behavior. All these findings, which are only outlined here, will always be substantially expanded and specified in the appropriate places during the interpretation .
  In the physical study of nature, we divide the whole material world into two basic forms of matter :

• Substance ,
  formed material particles (particles with nonzero rest mass - see below) - elementary particles, atoms, molecules and more complex sets and structures created from them .
• Field ,
  mediating reciprocal force action (interaction) between the particles of the substance - electromagnetic fields, gravitational, strong and weak interactions (discussed in more detail below) .

Modern physics shows that this division is to some extent conventional - the two forms change with each other; particles of matter can be interpreted as quantum states of specific fields (unitary field and particle theory ) and physical fields can be described using quantum - particles (see "Quantum field theory" below).

Discrete particle and continuous field model of matter
We model the structure and behavior of matter in physics by the two different ways mentioned above (forms of matter) :
-> Discrete particle model , according to which all bodies and material environments consist of a large number of small spatially localized objects - particles. According to classical ideas, particles have a certain non-zero mass or own energy, have a certain position in space and time, a certain speed, momentum and kinetic energy. The motion of particles is governed by the universal laws of mechanics (idealized "material point") - classical mechanics (Newton's laws of motion) or relativistic kinematics and dynamics. Now we know that for a detailed analysis of particles in the microworld, classical mechanics is not sufficient, but we must use its generalization - quantum mechanics.
-> Continuous field model , describing the structure and behavior of matter by quantities continuously distributed in space. In modern physics, this field description is used for forces - interactions - between particles of matter. It is perfectly elaborated especially for electromagnetic phenomena between charged bodies and particles - Faraday-Maxwell electrodynamics (see below "Electromagnetic field and radiation"). Here, too, for a detailed analysis of phenomena in the microworld, quantum field theory must be used, not only electromagnetic, but also the field of strong and weak interactions (see below "Quantum field theory", "Strong nuclear interactions" and "Beta radioactivity. Weak interactions").
  However, the field description can be used (and especially often used earlier) in continuum physics for the study of liquids, gases and partly also solids. The movement of liquids and gases is internally caused by the movements of their atoms and molecules - a perfectly proven kinetic theory . However, for the macroscopic description of the behavior of gases and liquids, the motion of individual atoms or molecules is not investigated, but the collective motions of a set of particles. "Averaged" quantities are used, which are continuously distributed throughout the volume of gas or liquid. Individual data on the position of individual atoms are replaced by the average spatial distribution of their number - the density distribution. In the macroscopic description , the speed of the disordered motion of individual atoms is replaced by temperature, and the ordered motion by flow rate (momentum transfer) in various places of the material environment. Collisions and forces between atoms or molecules inside gases and liquids are expressed by the distribution of pressure. Important equations of state apply to the interdependencies between these quantities . The relationship between the mechanical characteristics of particles (atoms and molecules) and field state quantities in continuum physics is derived by the methods of statistical physics. All properties of material environments and the events observed in them are an integral manifestation of a number of chaotic or coordinated simpler movements of the building blocks of the respective substance.

Basic building particles of matter
With ever deeper penetration into the microworld of building matter, physics discovers that atoms (previously considered indivisible) are composed of particles that can no longer be decomposed into simpler objects capable of independent existence. These smallest particles, which are no longer indivisible, are called elementary particles and can be considered as the basic "building blocks" of matter. However, these elementary particles are not static and immutable, but can undergo mutual changes and some of them may have a certain internal structure. In the study of the structure of atoms, we encounter mainly the three most important particles - the electron, the proton and the neutron *). In the study of excitations and radiation of atoms and atomic nuclei then with a photon - a quantum of electromagnetic radiation, with radioactivity also with a neutrino and a positron (antiparticle to an electron) - §1.2, part "Radioactivity beta". The properties of these and many other particles are more fully discussed in §1.5 "Elementary particle and accelerators" dedicated elementary particle physics, where it is administered also systematics of elementary particles (neutrinos is moreover closely discussed in §1.2, section "Neutrinos - "ghosts" between particles").
*) The reason why the observed matter is composed only of electrons, protons and neutrons is that all other material particles are very unstable.

Four basic physical interactions
The interaction between particles of matter can be explained by four basic physical interactions. At the level of atomic nuclei and elementary particles, two short-range interactions are dominant :
¨ Strong interaction , important especially by holding atomic nuclei together (see "Atomic nucleus" below). It primarily combines quarks into protons and neutrons, mesons and other hadrons . The inherent strong interaction between quarks, mediated by gluons, has a long range, but the nuclear strong interaction, as its "residual manifestation", is short-range (see "Strong interaction" below).
¨ Weak interaction , which is applied in mutual transformations of neutrons and protons with the participation of neutrinos, in practice mainly in radioactivity b (§1.2, part "Radioactivity beta", passage "Mechanism of weak interactions"). It is also short-range.
¨ Certain types of particles, which we call electrically charged, show a force interaction described by an electromagnetic interaction. When these electrically charged particles are at rest, an attractive or repulsive electric force acts between them according to Coulomb's law, when they are in motion, there is also a magnetic force, in the case of uneven movements of charges, then there is also the emission of electromagnetic waves - photon radiation (see the section "Electromagnetic fields and radiation" below). The electromagnetic interaction has a long range (more precisely, the range is infinite).
¨ The fourth interaction, also of long range, is the gravitational interaction, which acts universally between all particles, is attractive and has a significant effect on high-mass bodies. Its force manifestations are described in classical physics by Newton's law of gravitation, in relativistic physics by Einstein's equations of the gravitational field - see the book "Gravity, black holes and space-time physics", §1.2 "Newton's law of gravitation" and §2.5 "Einstein's equations of the gravitational field".
  The biggest and most difficult task of contemporary theoretical physics is to find the so-called unitary field theory, which would unify the 4 basic interactions and explain them as special cases of a single general interaction - see the section "Unitary theory of fields and elementary particles" in §1.5, more details "Unitary field theory and quantum gravity" of the above-mentioned monograph "Gravity, black holes and space-time physics".
  The magnitudes of the force effect of these basic interactions are diametrically different and decisively depend on the distances of the interacting particles. For distances of the order of 10^-13 cm corresponding to the dimensions of atomic nuclei, the relative ratio (or rather, "disproportion") of force effect strong, electromagnetic, weak and gravitational interaction is about 1 : 10^-(2-3) : 10^-15 : 10^-40. At distances of the order of 10^-8 cm, corresponding to the dimensions of the atomic shell, short-range strong and weak interactions are practically no longer act, and the electromagnetic interaction has a decisive influence.
  In our treatise on nuclear and radiation physics, we will not deal with the gravitational interaction, which is more pronounced in macroscopic bodies and acquires a dominant character in bodies of cosmic dimensions and masses. The strong and weak interaction will be discussed in more detail below in the relevant passages on the atomic nucleus ("Atomic Nucleus") and in §1.5 on elementary particles (section "Four types of interactions"). We will say some basic information about the electromagnetic interaction here (below "Electromagnetic fields and radiation"), because we will need it first to explain it - already in the science of atoms.

Classical and quantum models in the microworld
In atomic and nuclear physics, we study objects and processes whose behavior is beyond our imagination based on experience from the macroscopic world - the behavior of objects composed of a large set of atoms. Even in the microworld, controlled by quantum laws (see below), we can sometimes help out by use illustrative mechanical comparisons to macroscopic systems known to us. For example we imagine electrons in atoms as light negatively charged "globules" orbiting a heavy positively charged small "sphere" - the nucleus of an atom. Or other times we imagine the particles as waves or a wave pack. However, we must always keep in mind that these are just models, expressing only some selected properties of these microsystems, not their actual material structure in the usual sense! They are all just our human models, how to at least roughly understand the phenomena, that are very foreign to our daily experience. Importantly, it works in a theory-experiment relationship; and we believe that it will also help us to understand the internal mechanisms..?..
  An important difference compared to classical physics is the stochastic (probabilistic) character of quantum phenomena in the microworld. For individual processes, we cannot determine exactly when they will occur, but only their probability. The individual causality of particle behavior is lost, but a new kind of stochastic regularity emerges. Chaotic randomness (apparent or principled?) in the behavior of individual particles results in a regularity for the statistical set of these particles as a whole (not for its individual elements). These aspects of quantum physics will be briefly discussed below ("The Quantum Nature of the Microworld").
  From the philosophical-scientific point of view, the relations between the macroworld, the microworld and the megaworld are discussed in §1.0 "Physics - fundamental natural science".

Vacuum - emptiness - nothingness ?
In fundamental physics, phenomena occurring with bodies, particles and fields are mostly studied in a vacuum. Vacuum in classical physics means empty space (lat. Vacuus = empty), approximately achieved in terrestrial conditions in closed vessels by exhausting air so that the gas pressure is significantly lower than at normal atmospheric pressure. An ideal or perfect vacuum is a state of space in which no particles of matter (such as electrons, protons, etc.) nor radiation (photons) are present. Creating such a perfect vacuum is very difficult, even impossible in practice (it is impossible to get rid of, for example, the ubiquitous neutrinos or weakly interacting massive WIMP particles forming hidden matter in space - §1.5). Even if it succeeds, it will not be an empty space, where there is nothing and nothing happens - they can reach a physical fields such as electromagnetic and gravitational (gravitational fields cannot be shielded). Any vacuum is not actually empty - according to quantum field theory, there are many processes of quantum fluctuations, virtual pairs of particles and antiparticles are constantly formed (see "Quantum field theory" below).
  And in any case no means can a vacuum (even the "perfect") be considered as "nothingness"! Nothingness means the absence of anything - matter, energy, even space and time; it is therefore a synonym for "non-existence" - a fictitious philosophical concept without physical content.
From a philosophical point of view, a physical vacuum is not a state of pure nothingness, but contains potentiality of all forms of the world of particles (cf. "Anthropic principle or cosmic God"). Vacuum is a "living void" pulsating in the infinite rhythm of formation and extinction of structures, virtual and real particles ...
Vacuum energy
In classical (non-quantum) physics is the energy density of itself vacuum (without fields) zero. A completely marginal exception here is (non-quantum) relativistic cosmology, some models of which introduce the so - called cosmological constant, which generates a certain immanent fundamental density of vacuum energy in space (§5.2, part "Cosmological constant" in the above - mentioned book "Gravity, black holes and space - time physics").
  According to quantum field theory, however, countless processes of spontaneous quantum fluctuations take place everywhere and constantly in a vacuum - virtual pairs of particles and antiparticles are constantly being created and destroyed. The duration of these fluctuations is too short for us to directly detect these particles, so they are called virtual. Quantum field fluctuations have different intensities and spatial dimensions and interfere with each other. The result of this wave interference is averaged over time. If the contributions of individual field fluctuations are canceled on average, the mean energy of the vacuum will be zero - this is the so-called "true vacuum". However, if such a canceled does not occur, the mean energy of the vacuum will be non-zero - such a state is called "false vacuum".
  According to current cosmology, a "strongly false" high - energy vacuum could have been the driving force behind the rapid inflationary expansion of the very early universe (§5.5 "Microphysics and cosmology. Inflationary universe." in the book "Gravity, the black hole ..."). The present value of the vacuum energy is very close to zero, less than about 10^-9 J/m³, which corresponds to the mass density of approximately 10^-26 kg/m³. Attempts have been made explain the vacuum energy through quantum field theory - as a consequence of quantum fluctuations of vacuum. A straightforward computation (resp. dimensional estimation), encompassing all vibrational modes of energy with a wavelength greater than the Planck length (10^-35 meters), can be in and yet incredibly high density vacuum energy, corresponding to a mass density of about 10⁹⁶ kg/m³..!.. In order for the vacuum to look like an empty space, far-reaching compensations must be applied between the vacuum fluctuations of the different fields, which cancel out the vast majority of the fluctuations. This "scandalous discrepancy" of the 120 orders has not yet been satisfactorily explained; perhaps the unitary field theories promise some hope (§B-.6 "Unification of fundamental interactions. Supergravity. Superstrings." in the above-mentioned book "Gravity, Black Holes ...") .

The movement of microparticles in mass-forming ensembles. Thermals, thermodynamics.
Before we begin to deal with the properties and composition of individual microparticles (atoms, molecules, electrons, atomic nuclei, protons, neutrons, ...), it will be useful to talk briefly about the general aspects of the movement of these particles in sets of their large number, forming macroscopic matter. Each substance and the system or body formed from it consists of particles - molecules, atoms, ions - which are composed of smaller "elementary" particles of electrons, protons, neutrons. These molecules, atoms, or ions, are in constant disordered (chaotic) movement in different directions and at different speeds - "thermal" movement. As the temperature (discussed briefly below) increases, the speed of particle movement increases. The disordered thermal movement of atoms and molecules causes several effects in substances :
-> Diffusion is the process of spontaneous dispersion of particles into space and penetration of particles of one substance between particles of another substance. It takes place willingly mainly in gases and liquids, during the dissolution of solid substances in liquids (e.g. salt or sugar in water), to a lesser extent also between solid substances (observed at the interface of plates of different metals pressed together). It goes faster at higher temperatures.
-> The pressure of the gas on the walls of the container is caused by the impact of atoms or molecules hitting the walls of the container. As the temperature increases, the gas pressure increases - the particles have a higher speed and thus a higher kinetic energy.
-> Thermal expansion of solids and liquids. At a higher temperature, due to the higher speed of the particles, their mutual distances increase, which leads to an increase in the volume of the substance.
-> Changes in electrical conductivity of metals, electrolytes and semiconductors. In metals, at a higher temperature, the intensity of collisions of electrons with the atoms of the crystal lattice increases, so the electrical resistance of the conductor increases slightly with increasing temperature. The exception is the area of ??very low temperatures of the °K unit, when in some materials the resistance drops to zero, to superconductivity. In electrolytes, on the other hand, the dissociation of molecules into cations and anions increases at a higher temperature, so the electrical conductivity of the electrolyte increases with increasing temperature, the electrical resistance decreases. Semiconductors behave as non-conductors at low temperatures; with increasing temperature, the electrons gain energy and (via the "forbidden band") jump into the conduction band and can participate in current conduction. As the temperature increases, the concentration of electrons and holes increases and thus the electrical resistance of the semiconductor material decreases.
  Particles exert attractive and repulsive forces on each other, the magnitude of which depends on the distance between the particles :

Typical dependence of the forces F, acting between atoms or molecules in substances, on the distance r.   At a certain distance r0, the size of the attractive and repulsive force is the same - the resulting force is zero and the particles are then in an equilibrium position (for helium atoms it is 0.07 nanometers, for carbon atoms 0.15 nm, for water molecules 0.3 nm ).   For a distance r greater than r0, the force between the particles is attractive (attractive force outweighs repulsive force); at large distances it decreases to zero - the particles hardly interact with each other.   At distances r smaller than r0, the repulsive force (prevails over the attractive force) acts between the particles, and its magnitude increases sharply as the distance decreases.

The origin of those forces between atoms and molecules is electrical - Coulomb. Even though atoms and molecules are generally neutral on the outside, the distribution of electrons is often asymmetric, electric dipoles are created here. These are then polarized when the particles approach each other, and attractive or repulsive electrical forces arise depending on the mutual configuration of the dipole moments. The mutual force action of the particles causes the system of particles to have a certain internal potential energy; in the case of attractive forces, it is the binding energy (it is the work that we would have to do with external forces to break down the forces between the particles).
  At the usual temperatures of approx. 4÷3000 °K, atoms and molecules in substances collide elastically, so that the substance behaves according to the laws of thermals and thermodynamics outlined here. At very low temperatures near absolute zero, Bose-Einstein condensation occurs in some substances, which can lead to superconductivity and superfluidity (§1.5, passage "Fermions in the role of bosons; Superconductivity"). At high temperatures >3000°K, the kinetic energy of the atoms is already high enough that during their collisions, electrons are ejected from the atomic shells - to the ionization of the substance and the formation of plasma. Electromagnetic properties of electrons and ions are already fundamentally applied here (JadRadFyzika3.htm#Plasma). And at the highest temperatures >10¹² °K, even atomic nuclei and protons and neutrons break into quarks and gluons - a quark-gluon plasma is created for a moment (JadRadFyzika5.htm#KvarkGluonPlasma). Here the laws of thermodynamics are already debatable...
 Thermics
We call the disordered movement of particles thermal, because according to the kinetic theory of the structure of substances, these microscopic movements are the essence of heat and thermal phenomena. This is what thermals deals with. The basic physical quantity that describes the thermal state of matter and its internal energy (specific kinetic energy of the chaotic movement of atoms and molecules) is temperature. The absolute - thermodynamic temperature T is proportional to the mean kinetic energy of the disordered mechanical movement of matter particles (atoms, molecules) :
      < ¹/₂ m.v² >   =   ³/₂ k_B . T   ,
where m is the mass of the particles, v their speed of movement, k_B is the Boltzmann constant indicating the relationship between the thermodynamic temperature and the internal energy of the gas; curly brackets < > indicate mean value.
  Temperature is usually expressed in units called degrees " ^o ". The temperature degrees ^o are derived from the thermal state properties of water. One degree represents 1/100 of the temperature difference between the boiling point of water and its freezing point. So the freezing point of water (melting of ice) is 0 ^oC, the boiling point of water is 100 oC (at atmospheric pressure ....). Mainly two temperature scales are used. In everyday life, the mentioned degrees Celsius ^oC are used. In physical thermodynamics, the absolute temperature scale of Kelvin *) is used, where the initial-lowest temperature T=0 (^oK) is the temperature of "absolute zero", corresponding to -273.15 ^oC. Temperature differences in the Celsius and Kelvin thermodynamic scales are the same (Dt=DT), the difference is at the origin: -273.15 ^oC = 0 ^oK (absolute zero). Negative values in Kelvin are not possible. Occasionally, we can also encounter some other temperature scales (^oF - degrees Fahrenheit, derived from normal body temperature, were widespread mainly in the USA, where they are still used today; or the Réumour scale using the boiling point of alcohol). Conversion relationships between different scales are given in physical and chemical tables.
*) Note: For the absolute temperature in the Kelvin scale, the designations "^o degrees Kelvin ^oK" are now omitted and expressed only as "Kelvins K". However, we use the notation ^oK in our materials.
  The temperature is classically measured using the thermal expansion of mercury or alcohol in thermometers - thin glass tubes equipped with a scale, or using the different thermal expansion of layers of bimetallic strips. For electronic measurements, the temperature dependence of the electrical resistance of suitable conductors and semiconductors in thermistors is used. Another option is to measure the intensity and spectrum of infrared radiation emitted by heated bodies.
 Thermodynamics
Thermodynamics deals with processes in substances related to thermal phenomena (thermics), mainly from the point of view of the dynamics of energy and mass transfers in equilibrium and non-equilibrium systems, conversions of thermal energy into other types of energy, thermodynamics of phase transformations, reversible and irreversible events from the point of view of entropy. From a principled point of view, the explanation of thermal regularities using the statistical physics of a large number of particles using the methods of probability theory is important here.
  In addition to temperature T, heat Q (thermal energy) is also important here, which in thermodynamics is the total kinetic energy of all disorderly moving particles - atoms and molecules - in a system or body. It is part of the body's internal energy, which includes several components: above all the mentioned kinetic thermal energy of particles, potential energy of atoms and molecules, kinetic and potential energy of oscillating atoms inside molecules, energy of electrons of the atomic shell, nuclear energy, in particle physics and astrophysics sometimes even rest energy of matter according to Einstein's relation E=m.c².
  The basic unit of thermal energy is the Joule (in general, the unit of work and energy): 1 J = work done by a force of 1 N acting along a path of 1 m in the direction of movement. Sometimes the older unit of calorie is used: 1 cal = energy required to heat 1 kg of water by 1 ^oC (under standard conditions, 14.5^oC is stated). For small energies in atomic and nuclear physics, the unit electronvolt is used: 1 eV = 1.602x10^-19 J .
  Thermodynamics, as a comprehensive science of heat and its transformations, has developed several basic postulates and conclusions - laws or principles of thermodynamics, which are generalizations of observed experimental phenomena :
1. Equilibrium state: Every isolated system reaches an equilibrium state after a sufficiently long period of time, in which it will remain permanently (as long as it is not disturbed by external influences).
2. Zero thermodynamic law: In an equilibrium system, the temperature in all places is the same on average - thermal equilibrium is achieved.
3. First law of thermodynamics: The energy of a system can be changed only by exchanging heat Q, mechanical work W, or field or chemical energy. It is therefore the law of conservation of energy in a closed system, with the possibility of work and heat exchange, or transformations caused by excitations of physical fields. Energy can be transformed from one form to another, but it cannot be created or destroyed.
4. Second law of thermodynamics: Heat cannot flow spontaneously from a colder body to a warmer body. Heat cannot be transferred from a colder body to a hotter body without a certain amount of work required for this being changed into heat. Therefore, it is not possible to remove heat from a body and change it into useful work without a certain amount of heat passing from a hotter body to a colder body.
5. Third law of thermodynamics: The thermodynamic temperature of absolute zero T=0 °K cannot be reached by a finite number of steps.
 Entropy
In order to explain and quantify the 2nd law of thermodynamics, an important quantity of entropy S (Greek en=inside, tropo=change - change inside) was developed in thermodynamics. In classical thermodynamics, this quantity indicates the change in heat Q in relation to temperature T according to the Clausius formula :
     dS   =   dQ / T   ,
where dS is the change - increase or decrease - in entropy, dQ is the heat transferred to or removed from the system and T is the temperature. The 2nd law of thermodynamics states that for heat transferred by any possible process to any system for the change in entropy of the system, the inequality dS >= dQ/T holds. A small (infinitesimal) amount of supplied or removed heat dQ is considered here, at which the temperature T almost does not change (in the general case, when the temperature would change, integration over the temperature variable would be performed). From the point of view of thermomechanics, entropy also expresses the proportion of heat or energy of a system that does not have the ability to perform work. ......... ........
  In statistical thermodynamics, entropy is defined using the number of microstates that lead to a given macrostate of the investigated system - see below in the passage "Statistical thermodynamics". Both of these definitions of entropy are equivalent in the sense that they lead to the 2nd law of thermodynamics.
  The second law of thermodynamics can therefore be formulated using entropy as: In a thermodynamically closed (isolated) system, entropy cannot decrease.
 Statistical thermodynamics
Every system - substance, body - is made up of a large number of atoms or molecules that oscillate and move chaotically, collide and reflect each other, penetrate the spaces between them - they mix. This movement creates heat. From the point of view of classical physics, every atom and molecule must obey Newton's laws of mechanics, so their movements and collisions can in principle be measured and analyzed quantitatively. However, there are a huge number of atoms and molecules and they cannot be measured and evaluated individually. Only probability can be used here - statistical mechanics that connects and averages microscopic details into macroscopic behavior and overall outcome.
  Microscopic state - in short, the microstate of the investigated system represents detailed knowledge of the exact position and speed of each particle (atom, molecule) in this system at a given moment in time. As the particles move, collide with each other, change their positions and velocities, the microstate constantly changes chaotically during the temperature fluctuations of the system. Each microstate has only a certain probability of occurrence. Macroscopic thermodynamic description of the system - temperature, pressure, volume, represents its macro state. There are many different (mostly only slightly different) microstates that can globally provide the thermodynamically identical macrostate of the system. If the system is in an equilibrium state, despite the constant small fluctuations of the microstate, there are no changes in its macroscopic thermodynamic behavior - temperature, pressure, volume do not change.
  In statistical mechanics, the entropy S of a system is quantified as the number of all microstates that could provide the resulting macrostate of the investigated system, according to the probability relation of L.Boltzmann :
     S   =   k_B . ln W   ,
where S is the thermodynamic entropy, W is the number of all microstates that can provide a given thermodynamic macrostate. The constant k_B = 1,38049×10^-23 Joules per Kelvin is the Boltzmann constant between the average thermal energy of the particles in a gas and the thermodynamic temperature of that gas. The proportionality constant k_B serves to make the entropy value in statistical mechanics equal to the classical thermodynamic entropy in the Clausius formula.
Note: The natural logarithm of ln W is used in mathematical statistics, where it quantifies the information entropy of a random variable in a system.
  An example of such macroscopic behavior can be imagined in a simple experiment: We take a container of gas that is divided into two parts, separated by a partition (similar to the "Maxwell's Demon" picture below). We fill one part with a gas or liquid with a higher temperature (molecules move in it at a higher speed, with a higher kinetic energy) than in the other part. When we remove the partition, the molecules start to mix - the fast molecules diffuse and collide with the slower ones. There is an exchange of kinetic energy and after a certain time the gas or liquid reaches equilibrium with a constant average temperature. This is a widely known experience, in practice it never turns out differently.
  Newton's equations of motion are invariant with respect to time reversal. If, in a dynamic system, the motion of each of its particles is exactly reversed at the same time, then everything will happen backwards. If we observe the movements of individual atoms and molecules in microscopic detail, they behave the same when we watch them forward or backward in time (similar to when we project a movie forward or backward). But when we observe a container with gas or liquid, the mixing process macroscopically becomes unidirectional in time. We will never see that in a gas or liquid the atoms split into hot on one side and cold on the other. We cannot separate them from each other if time runs forward and never turns back.
  The reason for such one-way processes is probability - statistics. In general, the number of disordered states is incomparably greater than the number of ordered states. The 2nd law of thermodynamics is therefore a statistical consequence of the fact that there are much more disordered states than ordered states.
  "Maxwell's demon" particle sorter ?
In fundamental physics, nothing should in principle prevent particles from arranging themselves in a variety of uniform and non-uniform configurations. So nothing should prevent the gas from splitting into a cold and a warm part, it's just statistical randomness. J.C. Maxwell proposed the following thought experiment :
  We take two flasks connected by a tube with a separating partition that can be closed or opened using a valve (e.g. with a slide). With the partition closed, we pour into the left bulb a gas consisting of two types of atoms, or faster (red) and slower (blue) atoms - Fig.a). The right flask is empty, there is a vacuum. When the partition is then permanently opened, as a result of the chaotic movement of diffusion, an average of the same representation of both types of particles as in the left container (b) penetrates into the second container as well.
But if there was a valve in the separating partition that could be alternately opened and closed, it would be possible to "sort" the particles entering the flask on the right with it. Let's imagine that this valve would be controlled by some very fast and observant being with infinitely subtle senses - a "demon" who would be able to recognize whether the diffusing particles are fast or slow and could decide whether to let them pass through the valve (c). He could only allow faster (red) particles into the second flask, for example, while he would retain the slower (blue) particles by closing the partition from which they would bounce back. The demon thus replaces chance with purpose. Ultimately (after enough time), only "blue" particles would remain in the left flask and only "red" particles would remain in the right flask (d). This would violate the normal probability that all particles should mix.

"Maxwell's demon" able to sort particles. a) Initial situation: two flasks connected by a tube with a separating partition. The flask on the left is filled with a gas or liquid consisting of two types of particles. b) When the partition is opened, as a result of the chaotic movement of diffusion, the same number of both types of particles as in the left container penetrates into the second container as well. c) If there was a valve in the partition, which would alternately open and close a "demon" that is able to recognize diffusing particles, it could only allow e.g. faster (red) particles into the second flask, while it would retain the slower (blue) particles by closing the partition , from which they would bounce back. d) Ultimately (after enough time) only "blue" particles would remain in the left flask and only "red" particles would remain in the right flask.

In the idealized case, Maxwell's demon does not have to perform any work in the opening and closing of the valve when sorting particles, but it uses information that is not freely available when making decisions - "it's not free"! Information has a physical nature. This is information about the speeds and trajectories of individual particles. Each time the daemon decides between two particles (to release or retain them), it costs one bit of information. Each unit of information then brings a corresponding increase in entropy with the conversion factor k_B.log2. This restores compliance with the 2nd law of thermodynamics.
  "Maxwell's Demon" is of course just a fiction and does not exist in reality. However, its sorting role for atoms and molecules is performed in nature under certain circumstances by some physico-chemical and biological processes. During their function, growth, and reproduction, living cells create complex, ordered structures - as if they defy the 2nd law of thermodynamics. However, the cell is not an isolated system, so part of the energy used for its internal processes turns into heat, which is dispersed into the surroundings of the cell and increases its disorder. So the balance of the total entropy of the cell and the surroundings changes in accordance with the laws of thermodynamics. Cell membranes often exhibit one-way permeability and are able to regulate differences in ion concentrations during energy consumption. Enzymes can also act unidirectionally and can be metastable - after deactivation, their energy dissipates, turns into heat and increases entropy in the surroundings. During metabolism, cells and organisms successfully get rid of entropy, necessarily created during their life functioning.
  Herbivores and carnivores feed on organic substances that are in a highly organized state and return them to nature in a highly degraded state. But not completely, it can also be partially used by plants, which in addition obtain not only energy from sunlight, but also negative entropy. In short, "organisms organize" and at the same time suck "negative entropy" from their surroundings; and it gets it from sunlight.
  After all, we humans are probably the biggest "fighters" against entropy and the 2nd law of thermodynamics. In addition to the biological processes in our body, we constantly sort something, collect, collect, write literature, learn about nature and the universe, compose music, draw pictures, clean the apartment, ... etc. And human civilization builds colossal creations with a precise structure ...
The relationship between entropy and life is briefly discussed in the passage "Can the functioning of life and its evolution violate the 2nd law of thermodynamics?" work "Anthropic Principle or Cosmic God".
  All irreversible processes have the same physical-mathematical explanation: probability. The second law of thermodynamics is only probabilistic. Statistically, everything tends towards the highest entropy. In purely physical terms, it is not impossible for atoms or molecules in a container of gas to not mix and remain separate - it is just extremely unlikely. The improbability of heat passing spontaneously (without external help) from a colder body to a warmer one is similar to the improbability of spontaneous arrangement of order out of chaos. Both of these improbabilities are statistical in nature. The 2nd thermodynamic law means the tendency of physical systems of particles to flow from less probable - ordered - macrostates to more probable - disordered ones.
  The laws of thermodynamics play an important role in the behavior of substances in nature, composed of atoms and molecules - in our macroscopic world. However, thermodynamic concepts have been generalized even to other phenomena in the microworld and in the universe - see e.g. §4.7 "Quantum radiation and thermodynamics of black holes", or §5.6 "The future of the universe. Dark matter. Dark energy.", passage "The Arrow of Time" in the monograph "Gravity, Black Holes and the Physics of Spacetime"... Reflection on causality and randomness in nature and the universe is in §3.3, the passage "Determinism, chance, chaos?".

Electromagnetic fields and radiation
Before we begin to focus on the structure of atoms and the phenomena taking place inside, it will be useful to say a few words about one of the most important phenomena in nature - electromagnetic action and electromagnetic radiation. This is because all events in atoms and their nuclei are closely connected with electromagnetic interaction.
  Each electric charge Q excites around it an electric field of intensity E, proportional (according to Coulomb's law) to the magnitude of the charge Q and inversely proportional to the square of the distance r : E = r^o . k. Q/r², where r^o  is the unit vector extending from the charge Q to the test site and k is the coefficient expressed in SI in terms of vacuum permittivity e_o : k = 1/4p e_o . If the charge does not move (in the given reference system) , it is an electrostatic field. This electric field causes force effects F = q.E for every other charge q that enters this space. The electric field is generally springs, its source is the electric charges from which it emanates ("springs") and into which the electric field lines enter. However, even in the absence of electric charges, if the electric field is excited by electromagnetic induction with time changes of the magnetic field (as mentioned below), the electric field may be source-free.
What is the strongest electric field can be ?  
In classical (non-quantum) physics, the electric field in a vacuum can be arbitrarily strong, almost to infinity (in the material environment, however, this is limited by the electrical stability of the dielectric). From the point of view of quantum electrodynamics, however, even in a vacuum there is a fundamental limitation caused by the existence of mutual antiparticles of electron and positron : it is not possible to create an electric field with an intensity stronger than E_e-_e+ = m_e²c³/e.h = 1,32.10¹⁶ V/cm, where m_e is the rest mass of the electron or positron. When this intensity is exceeded, the potential gradient is higher than the threshold energy 2m_e c² and a pair of electrons and positrons is formed, which automatically reduces the intensity of the electric field. Such a strong electric field has not yet been created, with conventional electronics this is not possible; strong impulses from extremely powerful lasers could be a certain possibility in the future ...
  If the charge Q moves (electric current), in addition to the electric, a magnetic field also excites around itself. The moving charges, forming a current I in the longitudinal element dl, excite at a distance r a magnetic field of intensity B (unfortunately called magnetic induction for historical reasons) according to the Biot-Savart-Laplace law : dB = k .I .[dl´r^o]/r², where r^o is the unit directional vector of the measured point to the current element and k a proportionality constant expressed in SI units called via permeability of vacuum m_o: k = m_o/4p. The magnetic field shows force effects on each electric charge q moving at a speed v : F = q. (B x v); this so-called Lorentz force acts perpendicular to the direction of movement of the charge. The magnetic field is (unlike the electric field) always source-free, the magnetic field lines are closed curves - there are no so-called magnetic monopolies (magnetic "charges", similar to electric charges).
  During movement or time changes in the magnetic field, it arises according to Faraday's law of electromagnetic induction the electric field - in the form of a kind of "vortex", a rotating electric field around a variable magnetic field. An induced electric field can cause the movement of charges, eg electrons in a conductor - induced electric current. And the time changes of the electric field, in turn, cause a magnetic field (as if the so-called Maxwell shear current flowed), again of a vortex character. This dialectical unity of electric and magnetic fields finds its application in the concept of electromagnetic field, whose special manifestations are electric and magnetic fields. This field is governed by Maxwell's equations of the electromagnetic field, which were created by merging and generalizing all the laws of electricity and magnetism. The combined science of electricity and magnetism, including the dynamics of charge motions and the time variability of fields, is called electrodynamics.
Note: Details on the theory of the electromagnetic field can be found, for example, in §1.5 "Electromagnetic field. Maxwell's equations" of the book "Gravity, Black Holes and the Physics of Spacetime".
  Below, in the section "Atomic structure of matter" we will see that electromagnetic forces are decisive for the structure of atoms and for their properties - they are determining significance for the structure of matter at the microscopic and macroscopic level, including all chemical phenomena. Along with strong interactions, electric forces also play an important role in the structure of atomic nuclei (as we will see in the section "Structure of the atomic nucleus") and in the excitations and deexcitation of their excited energy states.

The electromagnetic waves
Maxwell's equations have a number of remarkable properties, but the following regularity is important to us here: The disturbance (change) in the electromagnetic field propagates in space at a finite speed equal to the speed of light. When electric charges move at a variable speed (with acceleration), they create a time-varying electromagnetic field around them, which leads to the formation of electromagnetic waves, which detach from their source and carry some of its energy into space. The electromagnetic field further propagates through space already independently of the source electric charges and currents in the form of a free electromagnetic wave - it is derived in §1.5, part "Electromagnetic waves" already mentioned monography "Gravity, black holes, and physics of space-time".
  From Maxwell's equations, by a suitable modification can be obtained two partial differential equations for the vectors E and B :
     ¶²E/¶x² + ¶²E/¶y² + ¶²E/¶z² = e.m .¶²E/¶t² ,  ¶²B/¶x² + ¶²B/¶y² + ¶²B/¶z² = e.m .¶²B/¶t² ,
which are wave equations describing the propagation of a time-varying electric and magnetic field in space at speeds c = Ö(1/em), where e is the electrical permittivity and m is the magnetic permeability of the given medium: E(x, y, z, t) = f(t - x/c) and analogously for B, if we consider for simplicity the waves propagating in the direction of the x-axis. The most commonly considered is the harmonic (sine or cosine) time dependence: E(x,y,z,t) = E_o.cos(w.(t - x/c)) and analogously for B, where w = 2p.f is the circular frequency; this is because waves are often caused by periodic oscillating movements of electric charges (eg in antennas powered by a high-frequency signal of frequency f); even in cases where this is not the case (eg braking radiation), the resulting waves can be decomposed by Fourier into harmonic components of different frequencies and phases.
  The highest speed is reached by electromagnetic waves in a vacuum, where c_o = 1/Ö(e_o.m_o) = 2,998.10⁸ m/s @ 300 000 km/s. In a material environment whose permittivity and permeability are greater than for a vacuum, the speed of electromagnetic waves is somewhat lower - in light this leads to known optical phenomena of refraction of light rays when light passes between substances with different optical densities (see below "Electromagnetic and optical optical properties substances").
  Thus, according to Maxwell's equations of electrodynamics, electromagnetic waves are transverse waves of electric and magnetic fields (mutually excited by their variability), where the vector E of electric intensity and vector B of magnetic induction oscillate with amplitude A constantly perpendicular to each other and perpendicular to the direction of wave propagation (see upper part of Fig.1.1.1), which in a vacuum travels at the speed of light c = 300,000 km/s. The electromagnetic wave periodically exerts a force on electrically charged particles - it sets electrons in motion in conductors and induces alternating electric current in them; the reception of electromagnetic waves by the antenna is based on this. Periodicity in space is given by wavelength, periodicity in time by frequency. The intensity (power) of an electromagnetic wave is given by the amplitude of the oscillating electric intensity E and the magnetic induction B, energy transfer by the so-called Poynting vector. There are simple relations between the speed of light c , the frequency of oscillation n and the wavelength l : l = c/n, n = c/l, l.n = c. The higher the oscillation frequency of the electromagnetic field, the shorter the wavelength. And it is on this frequency or wavelength that the properties of electromagnetic waves depend significantly.
Note: Wave propagation in material environments and especially physical fields is a general fundamental natural phenomenon - it is analyzed in the introductory part of §2.7 "Wave propagation - a general natural phenomenon" of the already mentioned book "Gravity, black holes and space - time physics".
Electromagnetic waves in atomic and nuclear physics
The general regularity of electrodynamics, that the temporal changes of electric and magnetic fields are capable to propagate in space as electromagnetic waves transmitting energy, play an important role in atomic, nuclear and radiation physics. First of all, it is the electromagnetic radiation of atoms during the jumps of electrons between the energy levels in the electric field of the nucleus (see below "Radiation of atoms"). Furthermore, it is the braking radiation generated generally during the accelerated motion of electric charges, in radiation physics especially during the impact of fast electrons on matter and their rapid braking during interaction with atoms of matter (§1.6, section "Interaction of charged particles"). The more subtle radiation effects are Cherenkov radiation and transient radiation, arising during the passage of fast charged particles through the material environment (§1.6, passage "Cherenkov radiation"). In the field of atomic nuclei, it is the deexcitation of nuclear levels by the emission of electromagnetic radiation - quantum gamma radiation (§1.2, part "Gamma radiation").

Types of electromagnetic radiation
According to wavelength or frequency, we divide electromagnetic waves into several groups :

• Radio waves are the longest used electromagnetic waves.
  They are mainly used in communication technology (radio engineering). They arise in transmitting antennas powered by high-frequency electrical voltage from the oscillator - this high-frequency voltage causes a periodically oscillating electromagnetic field around the antenna, which generates electromagnetic waves. These waves propagate through space and, upon impact with the antenna of the receiver, induce a weak alternating electrical signal of the same frequency as in the transmitting antenna. The high-frequency voltage supplying the transmitting antenna is usually modulated by an audio or video signal (either its amplitude - amplitude modulation AM, or frequency - frequency modulation FM); in this way are then modulated the transmitted electromagnetic waves as well as the induced signal in the receiver antenna. After demodulation and amplification, the desired audio or video signal is then generated. Radio waves are divided into long waves (approx. 150-500kHz, ie 200-60m) *), medium (approx. 500-2000kHz), short (approx. 2-30MHz) and very short (30-1000MHz) waves; furthermore shorter centimeter waves used, for example, in satellite links and millimeter waves.
  *) The longest electromagnetic waves registered here on Earth are the so-called Schumann resonances. They are excited by electric discharges in the atmosphere (lightning during thunderstorms, especially tropical) and propagate in the atmosphere, reflecting off the electrically conductive surface of the Earth and also from the electrically conductive ionosphere. The space between the earth's surface and the ionosphere thus behaves like an electromagnetic waveguide. Due to the closed spherical shape, this space also functions as a spherical cavity resonator, in which standing electromagnetic waves can be established. The fundamental wavelength of this resonant standing wave is approximately given by the circumference of the globe: l » 40000 km. This corresponds to the fundamental frequency f = c / l » 7.5 Hz. Other harmonic frequencies are also created, given multiples of the fundamental frequency. This result would correspond to a linear resonator; in the case of spherical resonators are different harmonic resonance frequency given by the formula: f_n = ^c/_2pR .Ö[n.(n+1)] , where R is the mean radius of the resonator (between the earth's surface and the ionosphere) in which the waves move, c is the speed of light, n = 1,2,3, ... is the sequence number of the harmonic frequency. The actual values of the frequencies of Schuman resonances are f₁ = 7.8 Hz (fundamental frequency - the first harmonic n = 1, which is the strongest), then there are gradually fading higher harmonics 14.3Hz, 20.8Hz, 27.3Hz, 33.8Hz.
    Electromagnetic waves of Schuman resonances are received and registered using special cylindrical coil antennas connected to highly sensitive electronic signal amplifiers. Changes in the frequencies and intensities of these waves can provide important information about the global state of the Earth's atmosphere, in connection with solar activity.
  Note: Schumann waves are therefore not a kind of basic "pulse of the planet Earth", as is sometimes stated in esoteric literature (and various unsubstantiated conclusions are drawn from it), but it is a well-explained natural phenomenon in the Earth's atmosphere.
• Infrared radiation
  is known as thermal radiation, which is emitted by any body heated to a temperature higher than 0 °K. It has a continuous spectrum, the higher the temperature of the body, the greater the intensity of infrared radiation and the representation of shorter wavelengths. At higher temperatures (above approx. 600 °C), hot bodies also emit visible light.
• Visible light
  forms only a very narrow range in the spectrum of electromagnetic waves, but it is very important to us because our eye is sensitive to it and allows us to see. It extends from wavelengths of about 750 nm (we see it as red light) to about 360 nm (purple light). Energetically, it is in the range of about 1.6-3.5 eV. Visible light is most often produced during electron jumps (deexcitation) in the outer electron shells of atoms - then it has a discrete spectrum. It is also emitted by hot bodies (heated to a temperature higher than about 500 °C) - continuous spectrum; for us, the main source of daylight is mainly the hot gases on the surface of the Sun.
• Ultraviolet radiation
  (abbreviated UV) follows immediately behind (purple) visible light towards shorter wavelengths, or higher frequencies. It is formed by deexcitation in atoms when electrons jump from higher shells to lower ones in the middle parts of the electron shell, the energy of photons is in the range of about 4-100 eV.
• X-rays ,
  also known as Roentgen - rays, have even shorter wavelengths (and higher frequencies) than UV radiation. The energy of X-ray photons is most often in the range of about 1-200 keV. It arises either during electron jumps between the inner shells L-K of heavier atoms - characteristic X-rays, or as braking radiation X during the impact and rapid braking of electrons, accelerated by a voltage of tens of kilovolts, on the anode. The discovery of X-rays, its origin and use for diagnostics is described in more detail in §3.2 "X-ray diagnostics".
• Gamma radiation
  is the "hardest" electromagnetic radiation with the shortest wavelengths (approximately 10^-10 m and shorter) and the highest frequencies (approximately 10²⁰ Hz and higher). It arises mainly during deexcitation of excited energy levels in atomic nuclei during radioactivity - see §1.2 "Radioactivity", section "Gamma radiation". Furthermore, in the annihilation of particles with antiparticles (see §1.2, section "Radioactivity b⁺") and as braking radiation (bremsstrahlung) in interactions of particles accelerated to energies of the order of MeV and higher. Energy of radiation photons g ranges in a wide range, most often from tens of keV to several MeV, but the interaction of high-energy particles produces radiation g with much higher energies of the order of GeV, in cosmic radiation we encounter energies up to 10²⁰ eV!

   The last two types of shortwave radiation, X and gamma, partially intersect with their spectra (wavelengths or energies) and there are sometimes terminological ambiguities. In the mentioned §1.2, part "Gamma radiation", there is a terminological agreement on the division of shortwave electromagnetic radiation according to its origin - gamma radiation comes from the nucleus, X radiation from other regions of the atom outside the nucleus.

Units of energy, mass and charge in atomic and nuclear physics
In most areas of physics and natural sciences, a system of SI units is used, in which the basic units are: meter [m] as a unit of length, second [s] as a unit of time and kilogram [kg] for mass; decimal multiples are often used - centimeter or gram, etc. The basic unit of work and energy is the joule [J], the unit of electric charge coulomb [C].
  In atomic and nuclear physics, which examines phenomena at small spatial scales and very small values of absolute mass, energy and charge, some somewhat different habits have been established in the units of mass, energy and charge used. These alternative units are better "tailored" to the phenomena studied in the microworld than SI units derived from macroscopic phenomena.
   The unit of time second is to keep, the unit of length, meter or centimeter, is usually also to keep (of course using decimal fractions 10^-xx); sometimes the unit angstrom is used : 1A^° = 10^-10 m = 10^-8 cm (in atomic physics it is a typical dimension of an atom), or fermi : 1fm = 10^-15 m = 10^-13 cm ( femtometer, in nuclear physics it is a characteristic dimension of the nucleus).
   As a unit of energy in atomic physics it is not used too large 1 Joule, but 1 electron volt, which is the kinetic energy obtained by the charge of one electron in the electric field at accelerating potential difference of one volt: 1eV = 1.602x10^-19 J. In nuclear physics, where there are higher energies and energy differences, then decimal multiples - kiloelenktronvolt (1keV = 10³ eV), megaelectronvolt (1MeV = 10⁶ eV) and gigaelectronvolt (1GeV = 10⁹ eV).
   Also the usual unit of weight, kilogram or gram, is impractically large for atomic and nuclear physics. In nuclear physics, mass is usually understood as the rest mass of particles and it is customary to express it in energy units based on the Einstein relation E = m.c² equivalence of mass and energy, ie also in electron volts : 1eV = 1.783x10^-33 grams; and of course in their decimal multiples. The rest mass of an electron can therefore be expressed as: m_e = 9.1x10^-28 g = 511 keV. In addition to [MeV], the mass of heavier elementary particles is sometimes expressed in multiples of the mass of the electron m_e - eg the mass of a proton can be expressed in three different ways: m_p = 1.673x10^-24 g = 938 MeV = 1836 m_e .  
  For an electric charge, instead of an oversized unit Coulomb, as natural basic units of charge, the electron charge e is use (resp. the same size but the opposite charge of the proton), which is the elementary electric charge: e = 1.602x10^-19 Coulomb.  
  However, the currently used units of dosimetric quantities, characterizing the effects of ionizing radiation on matter and living tissue, are based on the SI system. Because these are cumulative effects of a macroscopic nature. The basic quantity here is the absorbed radiation dose, the unit of which is 1Gray = 1J/1kg (for more details see §5.1 "Effects of radiation on the substance. Basic quantities of dosimetry.").
A note on quantities and units in nuclear physics 
The terminology, quantities and units related to atoms, nuclei, radioactivity and radioactive radiation have undergone a long and complex development, which has left some illogicalities and ambiguities - to be specified below. After all, similar gnoseological inconsistencies also occur in other physical fields due to historical development. Recall, for example, the unfortunate introduction of an electric current as a basic quantity and its SI unit 1Ampere (using "the force of two infinite parallel conductors ..."), while the physically primary electric charge (and the Coulomb unit) is introduced as derived from current. Or in magnetism the terminological illogicality of the names "magnetic field intensity" and "magnetic induction" (for an electric field it is fine) ...

Excursion to high velocities - a special theory of relativity
Microparticles, of which matter is composed, usually move at very high velocities during processes inside atoms, atomic nuclei and mutual interactions, often approaching the speed of light. In experiments with these high velocities, it was found that the usual laws of classical Newtonian mechanics no longer apply exactly here. Albert Einstein in his research at the beginning of the 20th century, followed up on Galileo and Newton's classical mechanics, Maxwell's electrodynamics and the research of his predecessors (Lorentz, Michelson-Morley, ...) and created a new mechanics - the so-called special theory of relativity, generalizing classical mechanics to movements at high speeds close to the speed of light. A systematic interpretation of this certainly interesting theory is not possible here; it can be found in a number of book publications (on these pages it is eg §1.6 "Four dimensional spacetime and special theory of relativity" in the book "Gravity, black holes and physics of spacetime"). Here we will only briefly recall some basic phenomena of the special theory of relativity, which are of fundamental importance in nuclear processes and interactions of elementary particles.
  The special theory of relativity (STR) is based on two basic postulates :

• The principle of relativity , according to which all inertial frames of reference are equivalent for the description not only of mechanical phenomena (this was already expressed by Galileo's principle of relativity and velocity composition), but for all phenomena, including electromagnetic ones.
• The principle of constant speed of light independent of the state of motion of the source and the observer; the speed of light c is the maximum possible speed of motion of material bodies and the propagation of interactions, which is not composed with any other speed (symbolically: c + v = c, c + c = c).

Relativistic kinematics
From these two experimentally perfectly verified principles it follows, that the relationships between positional coordinates and time intervals of events in different inertial frames of reference with the laws of classical kinematics control only at low velocities, while in general they control so-called Lorentz transformations
     x´ = (x - V.t)/Ö(1-V²/c²) , y´ = y , z´ = z , t´ = (t - x.(V/c²))/Ö(1-V²/c²) ,
indicating the relationship between the spatial coordinates x,y,z and the time t in the inertial system S and in the system S´ moving with respect to S speed V in the direction of the x- axis.
Note: In non-relativistic physics, the relationship between these coordinates is given by a simple Galileo transformation x´ = x -V.t, y´ = y, z´ = z, t´ = t (time t´ here, of course, flows here as fast as t !).
  Important kinematic effects of the special theory of relativity follow from Lorentz transformations :
Contraction of lengths :
The dimension l of each body of (own) length l_o, which moves with velocity v, appears shortened in the direction of motion compared to its rest dimension l_o: l = l_o.Ö(1-v²/c²).
Time dilation :
The time on a moving body flows with respect to the time of the external resting observer the slower, the faster the body moves: Dt = Dt .Ö(1-v²/c²). Here Dt is the time measured by the external rest clock, Dt is the actual time measured by the clock moving together with the body at velocity v .
Einstein's law of velocity addition :
If one body moves with velocity v₁ and the other body with respect to it velocity v₂ in the same direction, then with respect to the initial inertial frame of reference, the result of the composition of both velocities will be v = (v₁+v₂)/(1+v₁.v₂/c²), and not v₁+v₂ , as would be was in classical mechanics.
  Of these kinematic effects of special theory of relativity is considerable importance for nuclear and particle physics, especially the dilation of time, thanks to which particles with a short lifetime can live many times longer if they move at a speed close to the speed of light. Thanks to this effect, for example, m - mesones (with a lifetime of 2.10^-6 sec) created by the interaction of cosmic rays in the high layers of the atmosphere, it is enough to reach the surface of the earth, where we can observe them. Or we can exploit the mesons p⁺, p^-, created during interactions of high-energy protons from the accelerator, to come out its in form of the beams and study their interactions for a time many times longer than their rest time of life 2.6x10^-8 sec.

Relativistic dynamics
Combining relativistic kinematics STR and (Newton) dynamics of body motion arises relativistic dynamics, the basic new finding of which is that the (inertial) mass of bodies m is not constant, but the mass depends on the velocity of the body v according to an important relation
     m = m ₀ / Ö ( 1-v ² / c ² ) ,
where m₀ is the rest mass of the body *), which it has in the inertial frame of reference in which it is at rest. The weight of the body therefore increases with speed, especially when the speed approaches the speed of light - then the mass of the body increases theoretically to infinity: lim_{v® c} m = ¥ .
  Another important result of relativistic dynamics is the relation for the total energy E of a body of rest mass m_o moving at a velocity v :
     E = m _o .c ² / Ö (1-v ² / c ² )
and the resulting findings of the equivalence of mass and energy expressed by the famous by Einstein's relation E = m.c² ; resp. DE = D m . c².   
  Both these relations of the dependence of mass on velocity and the equivalence of changes in mass and energy play a cardinal role in nuclear and particle physics, where there are mutual transformations of energies and particles moving at high velocities.
*) This relation cannot be used directly for particles with zero rest mass (m_o = 0) moving at the speed of light v = c - such particles are mainly quantum of electromagnetic waves - photons. The photon has energy E = h. n , given by its frequency n and can be attributed to the (relativistic) inertial mass m = E/c² = h.n/c² .

General Theory of Relativity
In addition to the special theory of relativity, Einstein also developed a general theory of relativity, which is a unified relativistic physics of gravity and spacetime. We will not deal with this here, because in atomic and nuclear physics the gravitational interaction does not manifest (if we omit the unitary field theory...). This very interesting theory is explained in detail in the monograph "Gravity, Black Holes and the Physics of Spacetime", especially in Chapter 2 "General Theory of Relativity - Physics of Gravity ", along with its implications in astrophysics and cosmology - Chapter 4 "Black Holes" and Chapter 5. "Relativistic Cosmology".

Corpuscular-wave dualism
In classical physics and in everyday life, we observe a diametrical difference between discrete particles or bodies with their motions described by mechanics, and between continuous waves propagating in a certain environment. However, in a microworld dominated by the laws of quantum physics, this difference is blurred in certain circumstances !

Corpuscular properties of waves
At the turn of the 19th and 20th centuries, physics explained all natural phenomena either using particles, or by means of an electromagnetic field and its waves - electromagnetic radiation, a special kind of which is light. Virtually all the properties of light known in optics at that time (laws of propagation, reflection, refraction, difraction of light, interference) could be very well explained by the wave concept. Huyghens's wave approach to radiation seemed to triumph over Newton's corpuscular notion. However, some of the properties of radiation that were recently discovered at the time could not be fully satisfactorily explained by the pure wave concept.
Black body radiation 
The first such phenomenon was the spectrum of radiation of a heated ("absolutely") black body *), which was examined in detail by M.Planck in 1900. To explain the observed shape of the black body's radiation spectrum as a function of its temperature, Planck hypothesized that the emission (and absorption) of electromagnetic radiation by individual atoms in the body does not occur smoothly and continuously, but after certain small precise doses - quanta of energy. Sources of electromagnetic radiation can be considered as oscillators that cannot oscillate with any value of frequency and energy, but radiate or absorb energy only in certain quantities. The magnitude of the energy E of these quanta depends only on the frequency of the radiation n and Planck established for it the relation E = h. n , where the proportionality constant h @ 6.626x10^-34 J/s was called Planck's constant. Planck himself initially considered this assumption only as an ad hoc working hypothesis (a kind of temporary "emergency trick" to explain spectrum discrepancies), which should later be replaced by a more acceptable explanation. In reality, however, this hypothesis proved correct and became the beginning of a new conception of the microworld - quantum physics.
*) Each body (composed of a substance of any state), heated to a temperature higher than absolute zero, emits electromagnetic radiation - thermal radiation, arising from oscillations and collisions of electrons, atoms and molecules due to their thermal movements. This radiation carries away part of the thermal energy supplied to the body from the outside or generated inside the body. For the model study of thermal radiation, a so-called absolutely black body is introduced, which absorbs all the radiation that falls on them. It can be realized with a closed box with heated inner walls provided with a small opening through which thermal radiation escapes into the outer space.
In 1879, Stefan and Boltznan discovered the radiation law for the intensity of black body radiation as a function of temperature: I = s . T⁴, where s = 5.67.10^-8 Wm^-2.K^-4 is the Stefan-Boltzman constant. However, a satisfactory and uniform law could not be found to determine the radiated spectrum of thermal radiation. Two laws were formulated for the radiated spectrum, which, however, only partially agreed with the experimentally measured spectral curve: Rayleigh-Jeans's law well described the spectrum in the long wavelength region, but did not agree (even diverged) in the short wavelength region; Wien's law behaved the other way around. Unify both spectral regions managed to M. Planck, who discovered a new radiation law that was in full agreement with experiments in all spectral regions.
Photoelectric effect 
Another phenomenon that resisted satisfactory explanation by the wave nature of light was the photoelectric effect, abbreviated as photoeffect. This phenomenon, first observed in the late 80s of the 19th century A.Stoletov (in experiments with electric arc radiation) and H.Hertz (in famous spark experiments demonstrating electromagnetic waves), consist in the fact that when certain substances, especially metals, light (or electromagnetic radiation in general) of sufficient frequency falls on it, electrons are released from its surface *).
*) We distinguish two types of photo effect, external and internal. Here we deal with the external photo effect, when the action of radiation releases electrons, which escape through the surface from the substance into the surrounding space - occurs electron photoemission. This phenomenon is used in special tubes - photons tubes and photomultipliers. During the internal photoeffect, the released electrons remain inside the irradiated material and contribute to its electrical conductivity (it is used mainly in semiconductor optoelectric components - photoresistor, photodiode). In §1.6, part "Interaction of gamma and X-rays", Fig.1.6.3, we will deal with a special type of photoeffect, where a high-energy quantum of X-rays or g- rays eject electrons from the inner shells of the atomic envelope; and mention also the so-called nuclear photoeffect or photonuclear reaction.

Photoelectric effect
Left: Experimental setup for the study of the photo effect. Top right: Irradiation with strong long-wave radiation does not lead to a photo effect, while irradiation even with weak short-wave radiation causes a photo effect.
Bottom right: Quantum mechanism of a photoeffect by absorbing photons of incident radiation and transferring their energy to electrons.

Detailed experimental tracking (using electron tubes for the left picture - a prototype of so-called photon tube) showed that the photoeffect has certain specific properties, some of which canot be explained by classical wave concept of electromagnetic radiation :
¨ 1. For each metal, there is some threshold minimum frequency n_min , in which a photo effect occurs; if n < n_min , the photo effect does not occur even at the highest radiation intensity. On the contrary, even weak radiation with a higher frequency will cause a photo effect (even if the number of emitted electrons is lower), and immediately; according to the wave idea, the electron would have to "wait" until a weak wave gradually brought it enough energy to release. It follows that if an electron is released, it cannot receive energy gradually and continuously, but must receive the necessary energy at once.
¨ 2. The number of emitted electrons is directly proportional to the intensity of the incident radiation (provided, however, that a photoeffect occurs).
¨ 3. The kinetic energy (velocity) of the emitted electrons does not depend on the intensity of the incident radiation. It depends somewhat on the irradiated material and is directly proportional to the frequency of the incident radiation.
    The classical wave concept failed to satisfactorily explain the independence of the energy of the emitted electrons on the intensity of the incident radiation and, conversely, its dependence (even direct proportionality) on the frequency. In 1905, A.Einstein studied in detail the properties of the photo effect and explained all the experimentally established facts by assuming that the absorption of radiant energy takes place not continuously, but in certain small doses, quantum. The electromagnetic wave of the frequency n and the wavelength l = c/n, during the photoefect behaves as a set of particles - light quanta with a certain energy E and momentum p : E = h.n, p = E/c = h.n/c = h/l. Thus, electromagnetic radiation (including light) not only radiates, but also propagates and interacts (absorbs) in individual quantities.
  The electron on the surface of the plate receives just the energy E_f = h. n of one light quantum - photon. Part of this energy is consumed for the work needed to release the electron from the metal (the output work is equal to the binding energy E_v the electron in the metal, which is relatively small - units of electron volts). The residue is converted into kinetic energy E_k = (1/2) m_e. v² emitted electrons of mass m_e, flying away at speed v. The law of conservation of energy then leads to Einstein's photoelectric equation h. n = E_k + E_v, which quantitatively describes the properties of the photoelectric effect in perfect agreement with the experiment. At longer wavelengths, ie lower frequencies, the energy of the photon is insufficient for the electron to be released from the bond in the metal (or in the atom) - no photo effect occurs.
Compton scattering
Particle nature of shortwave electromagnetic X and gamma radiation is indirectly reflected in some of their interactions such as Compton scattering of this radiation on electrons. The experiment shows that the higher the change in the direction of electromagnetic radiation after scattering on an electron, the lower its frequency. This dependence of frequency on the scattering angle is difficult to explain by the electromagnetic interaction of a plane wave with an electron. On the other hand, the idea that the interaction occurs by the mechanism of a photon collision with energy E = h.n with an electron, similar to the elastic collision of two bodies (such as "billiard balls"), in which the redistribution of directions of motion, velocities and energies (and thus wavelengths and frequencies) is governed by simple laws of classical mechanics of mass points, explains the observed results very well Compton scattering - is analyzed in §1.6, section "Interaction of gamma radiation and X", the passage"Compton Scattering".

The corpuscular-wave dualism of electromagnetic waves
is illustrated in Fig.1.1.1. At the top of the figure, a common electromagnetic wave of lower and higher frequency (i.e., larger and smaller wavelengths) is schematically shown first. If we increase the frequency n of electromagnetic waves, according to classical physics, nothing happens other than that the wavelength (l = c/n) will be reduced proportionally. However, at very high frequencies (of the order of n»10¹⁴ Hz, ie l»10^-7 m) we will observe that the wave will no longer have a constant amplitude, but its amplitude will fluctuate. This tendency will increase with increasing frequency and decreasing wavelength. At extremely high frequencies n»10¹⁸ Hz (already corresponding to radiation g) we finally find that the wave in the classical sense has disappeared - the radiation will be emitted and propagated in short doses - quanta (Fig.1.1.1 below) particle character, among which are relatively long irregular "gaps".

Fig.1.1.1. Schematic representation of corpuscular-wave dualism in an electromagnetic wave. The upper part shows an electromagnetic wave with a longer and shorter wavelength, the lower part shows a quantum image of the propagation of radiation in quantum - photons.

The quantum of electromagnetic waves are called photons (this name was proposed by American chemist G.N.Lewis) - we can imagine them as a kind of small "packages" or "balls" of electromagnetic waves of a certain frequency, which move at the speed of light c (lower part of Fig.1.1.1). Each photon contains a certain amount of energy E, which is greater the greater the frequency n : E = h.n, where h is the Planck's constant (h = 6.6251x10^-34 Js). This constant plays a fundamental role in all phenomena in the microworld. In quantum mechanics, the "crossed out" Planck's constant h = h/2p is often used. The photon is the basic object of the microworld, which has both particle and wave properties, but strictly speaking, it is neither a particle nor a wave.
  In general, it is observed that in the long-wave region of the spectrum, wave properties (bending, interference, scattering, refraction) are more pronounced, while in the short-wave part of the spectrum, particle properties are more pronounced (photo effect, Compton scattering, formation of new particles during interactions). Radiation g, even though it is inherently an electromagnetic wave, will only behave like a stream of particles - photons, and we will not prove its wave properties by any macroscopic experiment; only if we became "little green dwarfs" in the imaginary experiment, managed to reduce to the dimensions of the order of picometers and "entered" inside the photon, we would find out that the photon is actually an electromagnetic wave inside...
Physical processes of emission Þ wave or photon character of radiation
The wave or photon character of electromagnetic radiation is closely related to the mechanisms and spatial and temporal scales of the physical processes in which this radiation is emitted. In electrical circuits (LC oscillators and antennas) of dimensions millimeters - meters - hundreds of meters, electromagnetic oscillations of frequencies of the order of gigahertz, hundreds of MHz or kHz occur, which leads to the emission of continuous electromagnetic waves (wavelengths of millimeters, meters or hundreds of meters), in which the photon character does not manifest. Light created by deexcitation of the outer electron shells of atoms with dimensions of ~10^-8 cm, already bears significant traces of the quantum character of transitions between electron levels; it behaves like a wave as well as a stream of photons. And gamma radiation, which occurs during very fast quantum deexcitations in atomic nuclei of ~10^-13 cm in size, is already completely photonic in nature.
Detection of low-intensity high-energy radiation Þ manifestation of corpuscular character
How can we most easily, in normal laboratory conditions, prove the corpuscular character of electromagnetic radiation? Above all, instead of ordinary light, it is appropriate to use high-energy gamma radiation and to register this radiation with a sufficiently sensitive one-photon electronic detector - GM or scintillation detector (§2.3 "Geiger-Muller detectors", §2.4 "Scintillation detectors"). When measured in a high intensity beam, the signal at the output of the detector will be stable and continuous, in accordance with the idea of a continuous field of radiation. When the radiation intensity decreases, the output signal fluctuates - statistical fluctuations (they are analyzed in §2.11 "Statistical variance and measurement errors"). If we reduce the intensity of radiation even more significantly, the detector will register time-separated discrete pulses - responses to the passage of individual particles through the detector.
  Another possibility is the detection of radiation using a sufficiently sensitive photographic emulsion (§2.2 "Photographic detection of ionizing radiation"). At high radiation intensities, the film will be continuously blackened after development according to the degree of total exposure. However, the low intensity of the radiation will not cause a continuously distributed response in the volume of the photographic emulsion, but they will see individual separate traces, as if left by individual flying particles...

Wave properties of particles
So we see that electromagnetic waves can behave like a stream of particles - this is one side of corpuscular-wave dualism. But what about the behavior of (real) particles? According to classical physics, particles behave like discrete "pieces of matter" in all circumstances. However, experiments with the passage of electrons, which are typical particles in atomic physics, through fine grids *) showed that the electrons exhibited bending and interference similar to waves - as if the electron had "bifurcated", passed simultaneously through two adjacent grating holes at the same time, and then after bending, the two components interfered with each other, as is common with waves. Bending interference phenomena were also observed in other types of particles (corpuscular radiation). At the same time, these interference phenomena do not depend on the intensity of the particle flux - the pattern does not change, even if the intensity of the electron flux is so small that one electron passes through the system after another!
*) Davisson-Germer experiment
A suitable structure for performing diffraction and interference experiments in the microworld is a crystal lattice, the nodes of which are individual crystal atoms with typical mutual distances of about 10^-7 cm. Such electron diffraction measurements were first performed by C.J.Davisson, L.H.Germer, and J.J.Thomson in 1927. A beam of electrons accelerated by a voltage of U»50V was routed on the surface of a nickel crystal. The electrons reflected from the surface layer of the crystal atoms were registered by means of a detector which was adjustable at different angles ("goniometer"). It was observed that the electrons did not scatter approximately evenly in all directions, but significantly more scattered in some directions, significantly fewer scattered in other directions, and the minima and maxima of the scattered electrons alternated in the scattering pattern. When measuring the dependence of the intensity of scattered electrons on the direction of scattering, they observed distinct minima and maxima, corresponding to the Bragg condition of interference, the same as in the diffraction of X-rays on a crystal grating - that the path angle of two rays is an integer multiple of the wavelength l: n.l = 2.d.sin J; n = 1,2,3, .. (order of interference maximum), d is the distance between two adjacent nodes of the crystal (lattice constant), J is the scattering angle. The wave radiation scattered on the individual atoms of the crystal lattice is amplified by interference in directions in which the path difference of the waves scattered on the individual atoms is equal to an integer multiple of the wavelength. And the electrons for which the "wavelength" l » h/Ö(2m_e .e.U) was obtained, behaved in the same way, which corresponds to the so-called Broglie wavelength (accelerating voltage U gives the electrons of charge e and mass m_e kinetic energy E_k = e.U and momentum p = Ö[2m_e .e.U], so l = h/p).

Fig.1.1.2. The wave properties of the particles are manifested in the thought experiment of electron diffraction at the slit by the formation of interference patterns.

The essence of these experimental facts is clearly illustrated in Fig.1.1.2 in an imaginary experiment, which generalizes the results of many real experiments. The beam of parallel flying electrons impinges on an impenetrable wall (screen) with two slits, behind which a photographic plate is placed. If the electrons were classical particles with rectilinear propagation, two dark stripes would be displayed on the photographic plate after development as shadow images of both slits (Fig.1.1.2 on the left). In reality, however, we obtain an interference pattern - the alternation of light and darker stripes (Fig.1.1.1 on the right), exactly as it would occur when a plane wave with wavelength l = h/p passes, where p = m_e.v is the momentum of the electron. The interference pattern does not depend on the intensity of the incident beam, so it is not due to the interaction of electrons in the beam. If we attenuated the flow of electrons to such an extent that there would be only one electron in the system at each pass, each such passed electron would create its local ("spot") blackening on the plate. The resulting image, which is the sum of the spots caused by the individual electrons, would still have the character shown in Fig.1.1.2 on the right. Thus, the phenomenon occurs even when the individual electrons fall on the screen one after the other: as if a single electron passed through both holes of the screen at the same time - as if a wave was connected to each individual electron, which interferes with the two holes. We will return to this experiment briefly below in the paragraph on quantum mechanics, for the essence of which is of key importance.
  The analysis of these experiments (real and imaginary) led to the conclusion that any microparticle of mass m moving at velocity v , can behave as a wave of wavelength l = h/m.v = h/p, where p is the momentum of the particle (this wavelength is sometimes called the Broglie-Compton wavelength). And this is the other side of corpuscular-wave dualism. The acquired knowledge can be summarized in the following way :

Gateway to Understanding Quantum Physics
The duality between waves and particles can be a "gateway" to understanding quantum physics ! If some contemporary quantum physicists question this, it is a misunderstanding in the spirit of the proverb "Everyone is a general after a battle". Without the exploration of corpuscular-wave dualism, many newer concepts of quantum physics would not have emerged, or would have emerged much later, and would have been difficult to understand. Almost all phenomena in the microworld can be clearly explained with the help of this dualism. Particle-wave dualism substantially alleviates that unpleasant "incomprehensible-incomprehensibility" of quantum physics, discussed below in the section "Interpretation of Quantum Physics".

The quantum nature of the microworld
Classical mechanics, extended and generalized by Einstein's special theory of relativity, together with classical Maxwell's electrodynamics, can explain almost all the phenomena observed in the macroworld of our experience. Classical physics (especially Newtonian mechanics) is a generalization of our everyday experience, according to which material objects exist independently of the observer, have certain positions and velocities, and move along precisely defined paths.
  However, as we learned in the paragraph on corpuscular-wave dualism, and as we well'see even more in the next chapters on atoms, atomic nuclei, nuclear reactions, radioactivity, elementary particles and their interactions, the deeper we go into microworld of the structure of matter, the more the experimental behavior of microsystems differs from the laws of classical physics. To understand and describe the atomic and subatomic processes that take place in very small parts of space and in which particles with very small masses participate, it was necessary to fundamentally change or supplement the basic classical ideas and laws - to build new microworld physics, quantum physics.
Note: This "new physics" cannot be imagined in such a way that it would perhaps refute and destroy "old" classical (non-quantum) physics. In science (and in physics in particular) the continuity of scientific knowledge applies. Quantum physics does not refute, but complements, refines, and generalizes classical physics into phenomena that it is no longer able to explain; it contains classical physics as a limit case. The relationship between classical and quantum physics is formulated as the so-called principle of correspondence: In the limit of large quantum numbers, the difference between quantum and classical physics is blurred, quantum physics becomes classical. Or for large quantum numbers, quantum physics gives the same results as classical physics (will be shown below). Thus, although the atoms and subatomic particles that make up everything are governed by quantum physics, their large arrays of macroscopic bodies (including us) follow classical Newtonian mechanics with great precision. The larger the object, the less clearly its quantum nature...
Randomness and Probability in Quantum Physics
As mentioned above (passage "Classical and quantum models in the microworld"), the basic specific feature of the microworld is stochastic (probabilistic) character of quantum phenomena. The motion of particles and all other phenomena in the microworld show quantum fluctuations - chaotic variability in the positions of particles and their velocities, field intensities, energy values and other quantities. These physical quantities fluctuate around their mean values; the magnitude of quantum fluctuations is limited by the so-called uncertainty relations mentioned below. During quantum fluctuations, the classical law of conservation of energy and momentum, as well as other classical laws that apply exactly in classical physics, can be violated for a brief moment, or rather interrupted.
  Quantum physics is unable to predict with certainty the specific results of a measurement. It offers only individual alternatives of values and their probabilities. And the state of the studied system will change after the measurement.
  According to quantum physics, the result of a physical process in a given system cannot be accurately and unambiguously predicted - regardless of how exactly we know the initial state of the system and how exactly we can solve the relevant equations of system dynamics. The development of the system, as well as the result of the experiment, cannot be determined unambiguously, there are only a number of different possible results, each of which has a certain probability *). When we repeat a certain experiment many times, the frequency of different results corresponds to the probabilities predicted by quantum physics.
*) A.Einstein metaphorically likened it to the situation, as if God always threw a dice, and only according to the result that falls will he decide how the experiment will turn out. Einstein never came to terms with this idea ...
  Quantum theory thus points to a new form of determinism at a deeper microscopic level: if we know the state of the system at a given moment, the laws of physics do not unambiguously determine the future (or the reconstructed past), but only the probabilities of different futures (or pasts).
  The probabilities and randomities that occur in everyday life are a reflection of inaccuracies in the knowledge of initial conditions and other, often complex, influences. When we shoot an air rifle at a target, shots with different probabilities hit different places on the target around the center, depending on the dexterity of the shooter. This probability is not a property of the shots moving towards the target, but is caused by ignorance and variability of shooting circumstances. If we fastened the airgun in the vice, the shots would fall into one precisely focused place. However, probabilities in quantum physics express a principled randomness, that is inherent in the very nature of phenomena. The microparticles targeted and transmitted under the same initial conditions will always fall to a slightly different location around the center of the target.
  To the seemingly trivial question "Where something is ?" classical science responds in such a way that "Every thing is in one particular place", as our common experience in classical physics based on Newton's foundations shows. In the microworld, it turns out to be different: subatomic particles can be "simultaneously in several places". Their exact coordinates are subject to quantum uncertainty relations (they fluctuate) and obtain their specific values only at the moment when we start measuring them (at the moment of interaction). It's as if something only starts to exist the moment we "look at" it (discussed below in the section "Observations and Measurements in the Microworld") ...

Is quantum stochasticity fundamental, or is it caused by hidden unknown parameters ?
Anyone who begins to learn about quantum mechanics, upon deeper reflection, is surprised by a certain "inappropriateness" of the behavior of particles and systems, contrary to common sense and natural experience. Is this surprising and incomprehensible behavior really fundamental, or are there some other hidden and unknown influences behind it - hidden parameters ? In such a case, standard quantum physics would be an incomplete theory, and its stochastic character would be the result of ignorance of some hidden (yet unavailable to us) parameters that take on different values. The behavior of these parameters should be described by some future more "fundamental" theory, probably within the unitary "theory of everything".
  Let's go back for a moment to the simple example of shooting an air rifle. If we were to shoot in open space at a target approx. 50 meters away, we would observe a relatively large dispersion of hits. If we could not measure the variable speed and direction of the wind, small variations in the size and shape of the shots and other possible influences, we could in principle empirically build a kind of "quantum mechanics of the movement of the missiles" that would show the fundamentally stochastic nature of the trajectory of the shots..?..
  No such additional hidden parameters have been found in the quantum mechanics of microworld particles. System analysis of the possibility of influencing measurable quantities in quantum mechanics by hidden parameters was dealt with by J.Bell in 1964, who established limiting inequalities for the results of measuring the polarization of two photons from a quantum "entangled" pair. Experimental measurements were carried out in the 80s on photon pairs during cascade transitions in calcium atoms with correlated polarizations. The experimental results somewhat exceeded the values of ±2 in Bell's inequalities, which is some indication against the concept of hidden parameters...
  Current quantum physics is generally inclined to the opinion that the quantum state represents a complete description of systems, and quantum stochasticity and uncertainty relations have a fundamental character that cannot be derived from anything..?.. So it seems that randomness is somehow incomporated in the deepest essence of our world ?
  Other unusual phenomena related to quantum entanglement are discussed below in the passage "Quantum entanglement and teleportation. Quantum computers.".

Statistical fluctuations and noise in imaging
The emission of radiation quantum, as well as its interaction with atoms of the material environment (and thus the mechanisms of radiation detection) takes place at the microscopic level through events governed not by deterministic laws of classical physics, but by the laws of quantum mechanics. These quantum regularities are in principle stochastic, probabilistic. The transitions of electrons in atoms or the transformation of radioactive atoms is therefore largely a random process and the resulting radiation is emitted randomly, uncorrelated, incoherently *). Therefore, the radiation flow is not smooth, but fluctuating. The response will be just as fluctuating of any device detecting and displaying this radiation - these are fluctuations which cannot be eliminated by any improvement of the device or method, these fluctuations have their origin in the very essence of the measured phenomena.
*) The fact that LASER emits coherent photons is due to the fact that it is not a spontaneous, but stimulated production of photons.
  Statistical fluctuations (noise) in measurements and images are therefore a general phenomenon. They are also hidden in ordinary light during optical vision and photography, where we do not observe them due to the large number of photons (of the order of 10⁹), which are available here. At the top of the image is a photographic portrait exposed with varying numbers of photons of light. We see that if the image consists of less than 10³ photons, we do not recognize anything at all in the image except scattered clusters of dots. With the increasing number of photons, the quality of the image improves (above 10⁵ photons we begin to recognize the basic motif) and at about 10⁸ photons we get the usual photographic image with all the details, without noticeable noise.

Influence of registered number of photons on image quality in terms of statistical fluctuations (noise) - image quality improves with increasing number of photons.
Above: Photographic portrait exposed with different number of photons of light (computer image processing performed by Ing.J.Juryšek) .
Bottom: Gammagraphic image of a phantom (Jasczak, filled with ^99mTc radionuclide) accumulated by a scintillation camera with different numbers of g- photons in the image.

In practice, statistical fluctuations are unfavorably applied wherever we do not have enough number of quanta (photons) displaying radiation. This is especially true for radiation detection, spectrometric and imaging measurements. The influence of statistical fluctuations on the results of these measurements can be expressed simply (but concisely) by the following rule: If we measure N pulses on a radiation detector, we actually measured N ± Ö(N) pulses. These statistical fluctuations are reflected in all cells of the image and the only way to reduce them is to increase the accumulated number of pulses - the number of "useful" photons g, which results in a response in the image. An image is sharp and clear when it is created by at least 1 million photons/cm². This is difficult to achieve with gamma images, in practice we often have to settle for about 500-1000 photons/cm². Therefore, scintigraphic images tend to be quite "noisy".
  Statistical fluctuations degrade image quality mainly due to the loss of useful structural details. This can be seen at the bottom of the figure in the images of the phantom modeling "cold" lesions of various sizes in a solution of g- radionuclide ^99mTc. If the image consists of less than about 10⁴ photons g, no structure is seen, only statistical fluctuations. With a larger number of registered photons g of about 10⁵, larger lesions are visible, and only with 10^7-8 photons, even the finest structures with small lesions are shown.
  In addition to gammagraphic images, statistical fluctuations are also reflected in astronomical images of distant objects, from which very little light falls on us.

The essence and interpretation of quantum physics
A systematic interpretation of quantum physics is outside the thematic scope of this treatise *) and would also take up an enormous amount of space (reference can be made to standard textbooks and monographs, eg Landau, Lifšic: Quantum Mechanics). We will only take a brief excursion into the ideas and laws of quantum physics to outline some basic common principles applied decisively in processes with atoms, atomic nuclei and elementary particles.
*) After all, fully understanding the essence of quantum laws and internally identifying with them is not at all easy - if not impossible! It is said that the theory of relativity is "understandable - incomprehensible", but quantum physics is "incomprehensible - incomprehensible"..!.. Even the best professional specialists in quantum physics (including its builders such as Bohr, Schrodinger, Pauli, Feynman, Fermi, Dirac, Heisenberg, ...) admit, that they can very accurately analyze the results of quantum processes in atoms, nuclei or in particle interactions, but when someone asks them why it behaves like that, they don't know..

Quantum physics in the micro world - but also in the macro world ?
Quantum physics arose during the study of phenomena in the microworld - at the level of atoms, atomic nuclei, elementary particles. And it also has its main application there... Quantum mechanics, however, claims universal validity not only in the microworld, but also in the macroworld (even in the megaworld, see e.g. §5.5 "Microphysics and Cosmology. Inflationary Universe." in the book " Gravitation, black holes and the physics of spacetime"). The usual idea is that the application of quantum physics is determined by the size of the investigated objects: small objects behave quantumly, large objects classically. In reality, however, it doesn't always have to be this way. What makes an object quantum or classical is not so much the geometric size, but the number of degrees of freedom available to the object. With a small number of degrees of freedom, the object behaves quantumly. Small objects such as elementary particles (with a simple structure, or without structure) have a small number of degrees of freedom and behave uniquely quantum. Larger objects, composed of a large number of particles, have a large number of degrees of freedom and behave classically because the values of their state variables are averaged over many degrees of freedom.
  Under certain circumstances, even relatively large macroscopic objects, such as a laser beam of coherent photons, a laboratory container with superfluid helium, or a stream of Cooper electron pairs in a superconducting wire can have a small number of degrees of freedom. An extreme case of long-range quantum behavior can be states of entangled particles (see below "Quantum entanglement and teleportation. Quantum computers."). The specific situation is with quantum processes involving black holes (it is discussed in detail in §4.7 "Quantum radiation and thermodynamics of black holes" of the mentioned monograph).
  For ordinary bodies of macroscopic masses and dimensions, quantum effects are completely tiny and unmeasurable. However, in advanced sensitive experiments, quantum properties can be observed in increasingly large objects (such as macromolecules)...

Wave stochastic description of particles and systems. Interpretation of quantum physics.
In classical (non-quantum) physics, the state of a physical system is described using directly measurable quantities, such as positions and momentum of particles. In quantum physics, the state of a particle is described by a wave function (see below), the square of which indicates the probability that the particle is in a particular state. The wave function is not a directly measurable physical quantity, but rather a model idea. When a particle interacts with another particle, there is a decoherence (loss of mutual spatial and temporal connection of phase and amplitude) of the wave function, leading to the particles acquiring a specific measurable state as in classical physics. However, this is only a probability that we measure a certain value of a state quantity at some place and time. It is not possible to predict in advance which value will be measured in a specific case, it is only possible to determine the probability distribution of the occurrence of different measured values with a larger number of measurements (under the same conditions).
  From a gnoseological point of view, however, it must be kept in mind that all these are only mathematical models, enabling the description and quantification of phenomena in the microworld. Their physical nature is probably hidden somewhere deeper ..?..
  Simultaneously with the building of its own quantum mechanics and its mathematical formalism, have been created several ways of interpretation quantum laws and heuristic methods of creating a chain of conceptual structure of quantum physics. We will usually follow an inductive procedure based on a gradual analysis of experimentally established facts (we will analyzed it physically, without debatable philosophical speculations). The most common approach is the so-called Copenhagen interpretation quantum mechanics, founded by a physics group led by Niels Bohr. According to her, during the measurement or interaction, the wave function "collapses" *) (originally describing the superposition of possible states) - to reduce quantum alternatives. This forces the particle to probabilistically "choose" only one final state and position (which was uncertain until then), as it is in classical physics. During this collapse, the original information is not preserved, which can no longer be detected - the particle is described by a completely new wave function from the moment of measurement or interaction.
*) "Wave function collapse" is just a model mathematical abstraction, it is not a real physical process! If we were to take seriously the collapse of the wave function all at once in all space, it would essentially mean that we "throw away" one universe and replace it with another universe with a new wave function. Such an idea would already be close to the many-universe Everett hypothesis. Of course, nothing like that happens in reality.
  And no anthropomorphic "conscious" observer is necessary; interaction processes take place in innumerable quantities spontaneously in inanimate and living nature. Our "observations" are only rare accidental "probes" into these processes.
  In the passage "Mystical Quantum Physics? ", we mention H.Everett's somewhat bizarre multi-cosmic interpretation, according to which the measurement or interaction does not collapse the wave function, but creates parallel realities ("universes") in which all possible states of the resulting particles exist; these particles can then no longer interact with each other. At the end of this general part, we will briefly mention Feynman's approach of quantizing "path integrals", which gives some opportunity to understand the internal causes of quantum behavior.
 Superposition of different states and collapse of the wave function: sometimes a misinterpretation of quantum physics ?
However, the concept of "wavefunction collapse" is only a theoretical model that can be misleading. This is shown, for example, by a simple (Stern-Gerlach) experiment, in which neutral atoms, vaporized in a suitable source, are accelerated and allowed to pass through an inhomogeneous magnetic field formed by two pole extensions placed one above the other. After passing through, they will hit the detection plate. Atoms with different spin states will be in mag. fields behave differently. According to the polarity of the magnetic field, atoms with spin +1/2 will be deflected upwards, atoms with spin -1/2 will be deflected downwards. Under normal conditions, the probability of neutral atoms having spin +1/2 or -1/2 is 50%/50%, so an equal number of upscattered and downscattered particles will be detected. We then know with certainty that atoms absorbed on the upper side have spin +1/2 and those absorbed on the lower side spin -1/2. There was no "collapse of the wave function", only before the measurement we did not know the specific spin state of the particles, we only knew that it could be +1/2 or -1/2. After the measurement, we know the spin state of the detected particles - which were already emitted from the source with this state.
  The particle is not in all possible states at once before observation-measurement - it is only in one state, which we do not know. Before the measurement, we can only determine the probability of what state it could be in. And after the measurement, we know the specific condition for sure. We will discuss the frequently occurring misinterpretation of quantum mechanics further below in the passage "Schrodinger's cat".

Wave function
Let's go back to corpuscular-wave dualism (Fig.1.1.1 and 1.1.2), which is an important characteristic feature of the quantum understanding of the microworld - it suggests that the division of matter into waves and particles is only formal; in general, we must consider corpuscular and wave properties simultaneously. The particle does not move along a fixed localized path, but as if it "waves" along a blurred path, it behaves like a Broglie wave.
  What is the physical significance of Broglie waves associated with particle motion? The first straightforward notion that the particles themselves are waveforms does not hold up, because in some processes, especially scattering, we could in principle register "parts of the waves" as "fragments" of the particle, contrary to experimentation. Even the opposite idea that waves are formations composed of particles is unsatisfactory (no particles originating from the wave were observed, only the wave can behave as a quantum with the properties of the particle when interacting). A more adequate idea of the relationship between waves and particle motion can be obtained by studying the diffraction of electrons, which we register on photographic film (Fig.1.1.2). If only a small number of electrons pass, we get an irregular image scattered on the film, but after passing a large number of electrons, we get a smooth regular pattern analogous to the diffraction patterns of light waves. This fact leads to a statistical interpretation of Broglie waves: that the intensity of Broglie waves at any place in space is proportional to the probability of the particle occurring at that place. The classical trajectory of a particle is replaced by a kind of "probability cloud", representing a set of places where a particle occurs with different probabilities.
  In quantum mechanics, the state of a particle (or a set of particles and generally every physical system) is described by the so-called wave function y(x, y, z) (in the simplest case of an isolated particle, this wave function is identical to the Broglie wave). The physical meaning of the wave function is that the square of the modulus of the wave function ú yú ² determines the probability dW that the particle at a given time t is in the element of volume dV = dx.dy.dz around the point (x, y, z): dW = ú yú ² .dx.dy.dz. And the mean value of any physical quantity F(x, y, z), which is a function of the coordinates x, y, z, is then given by the relation ` F(x, y, z) =ò F(x, y, z).ú yú ² .dx.dy.dz, where it integrates over the whole field of variables x, y, z.
Note: The wave function y is generally introduced as a complex function (containing both real and imaginary components), so the square of the modulus ú yú² = y . y*, where y* is a complex associated function by y. For the simplest case of a free particle moving in the direction of the x- axis with momentum p_x , the wave function is written in the form y = exp[- ⁱ/_h (E.t - p_x.x)], representing a plane harmonic wave.

Observation and measurement in the microworld
"Things can be observed without disturbing them" - this is an experience of everyday life, especially from visual observation by an "uninvolved observer". However, the process of observation or measurement *) in the microworld differs diametrically in its nature and consequences from the processes of measurement and observation in classical physics describing the macroworld.
*) The terms "observation" and "measurement" are often not distinguished: quantitative observation is a measurement.
  In the physics of classical systems of the macroworld, it is tacitly assumed that the process of observation (measurement) does not disrupt significantly their movement or evolution. The relevant physical quantities can be measured with sufficient accuracy without disturbing their values or disturbing the development of the observed system. Or we assume that any failure caused by the measurement can be accurately corrected, at least in principle.
  E.g. when measuring the voltage in the electrical circuit, we use either a voltmeter with a sufficiently large input resistance, which practically does not affect the measured value, or if this is not possible, we can know the impedances in the circuit and the voltmeter to accurately correct the voltage change. However, experienced electronics experts know that when measuring extremely weak electrical signals (in which only a few hundred electrons participate), unremovable noise and fluctuations are applied, and all correction methods already fail here.
  The most common way to examine the position of an object is to visually observe it: we illuminate the observed object with light (unless it is itself a source of light) and our eyes register reflected photons of light. If the observed body has a macroscopic size and mass (such as an apple or a stone), the incident and reflected photons of light do not appreciably affect the position of the body and the basic premise of a "non-participating observer" is met. However, if the body is of microscopic size and mass, the impact of each photon can significantly affect its position and velocity - all the more so the more accurately we try to determine the position (accurate measurement of the position of a particle can completely "throw off" its momentum!). For a more accurate localization of the particle position, the wavelength of the irradiating wave must be short enough, ie the energy and momentum of the quanta is correspondingly higher - it causes a more appreciable disturbance of the observed system (position and velocity of the particle).
  Here we will no longer observe directly with the eyes, but through an instrument (eg microscope incl. electron microscope, particle detector, radiometer, spectrometer), while for observation of particles of very small dimensions it is necessary to use radiation with a correspondingly short effective wavelength. The subjective role of the observer is often overestimated in connection with quantum mechanics (it comes from the time of the origin of quantum physics) and sometimes even the objective reality as such is questioned. This is a misunderstanding! Natural processes with innumerable interactions of particles and fields are constantly taking place in nature and their results are independent of us. Only our occasional probes into the events of the microworld are burdened by fundamental quantum uncertainties. However, this is not a consequence of our subjective intervention as a conscious observer, but the influence of the interaction of objectively existing particles with sensors and instruments used for observation or measurement (it is briefly discussed below in the section "Mystical Quantum Physics?").
  The oft-cited claim that "observation creates reality" is misleading and erroneous: this reality has been there already before, the observation has only "illuminated" it for us - and may have changed it, due to quantum influence at the microscopic level.
  So in order to "observe" and measure a microparticle, we must let it bounce off it some other particle or quantum of radiation and observe only the result of this reflection - more generally the result of the interaction. The inevitable consequence of such a process is that the collision or interaction irretrievably changes the state of the monitored particle - it deviates it, changes its velocity, or even the internal structure. In general: in order to observe an object or system, we must interact with it. Only in this sense can the reality in the micro world be influenced by mere "observation" !
  Thus, the operations (processes) of observation or measurement necessarily affect the physical system (disrupt its evolution), while for small systems *) this disruption is considerable and irreversible, it has a principled character and cannot be eliminated or corrected in any way; and by no improved method - it is the very essence of things and processes themselves! Quantum mechanics deals with the behavior of such systems and the processes of measuring their physical quantities.
*) In the microworld, the term "small" loses its usual relative character and becomes an objective absolute attribute determining the quantum behavior of a given system.
 "Observer "
it tends to be one of the most misunderstood concepts in the interpretation of quantum mechanics (next to stochasticity, interference of states or collapse of the wave function...). From an objective point of view, an observer is some "classical" (non-quantum) object *) that interacts with a given system, here a quantum one, in order to investigate some of its properties. Depending on the type of phenomenon being investigated and the experimental configuration, this "observer" can be an electronic or optical measuring device, a mechanical body, or therefore a person (or even another biological organism...). As a result of the interaction, we can then obtain the proper value of a certain quantum number.
*) Each such "classical" observer is internally also composed of a huge number of quantum particles, but their fluctuations are averaged out, so for all practical purposes they behave like classical...
 Quantum theory: ontological physics + epistemological measurement
In quantum physics, two aspects are closely intertwined, in feedback, which from a philosophical point of view are called ontological (what is the actual physical reality) and epistemological (how to know this reality) :
--> On the basis of experience from classical mechanics and electrodynamics, we create models of the studied systems, generalized so that they correspond to the specific laws of atomic, nuclear and particle physics.
--> We let these particles and fields interact with a classic measuring system and monitor its response.
  This is the only possible method of measurement, that simultaneously produces the observed stochastic quantum regularities....

Mystical quantum physics - "quantum mysticism" ?
Quantum physics is from a philosophically point of view often falsely interpreted. From the above fact that reality in the (micro) world can be influenced by observation, mystical claims are drawn that "the human mind creates reality", or "quantum mechanics connects the human mind with the universe", or "quantum physics creates the unity of human and cosmic consciousness", is the essence of "freedom of will" and the like. The basic mistake or misunderstanding here again results from the above-mentioned overestimation and misunderstanding of the role of the subjective "conscious" observer. It is not our mind that makes observations, but the very interactions of the basic particles (objectively existing), which take place even without us. They change the quantum state of systems and transmit information, our mind only registers and processes it. In fact, it is an objective reality, whose laws of which (including quantum ones) determine the behavior of our minds and the functioning of the entire universe. This approach corresponds to all the results of our observations so far, it is fully consistent with them.
  The finding that causality and determinism are disrupted in some phenomena in the microworld led to a hypothetical connection between quantum physics and free will. We have a certain legitimate sense of free will - that we can decide what we will do today or what we will plan for tomorrow; that it is mainly up to us, it is not only external circumstances that decide. However, from the point of view of science, real free will is only an illusion. Above all, we cannot do something that is contrary to the laws of nature; and other things prevented by certain other circumstances... "Freedom of will" emerges from an inexhaustible number of interactions and their results, we do not need quantum physics for it (see also the discussion in the section "Determinism-chance-chaos?" §3.3 in the book "Gravity, black holes and space-time physics") ..?..
Infinitely many parallel universes ? 
The stochasticity of the behavior of particles and systems according to quantum physics also gave rise to H.Everett 's somewhat fantastic hypothesis of an infinite number of universes : with each attempt to discover reality - with every interaction of particles - the whole universe splits, branches or "duplicates" into two or more "universes" in which the individual possible results of the interaction take place (with the appropriate probability). This creates a constantly infinite number of "parallel worlds", in which all possible alternative futures (and pasts) are "real"..?..
  In some such universes, for example, an asteroid would have missed our planet 60 million years ago, and dynosaurs would still rule the Earth (or even create a civilization instead of humans) ..?..
  However, available for us (ie "real") is only the the universe in which we are at the moment; observations can only be made in "our world". Alternative events in parallel universes can only be imagined ...
  There have also been sci-fi hypotheses that parallel universes can be influenced at the quantum micro-level (and therefore also affect our world). Their quantum (particles) can "seep" between individual universes and evoke some "bizarre" effects of quantum mechanics..?.. They are all just unsubstantiated assumptions ...
  On the astronomical and philosophical context of several universes - multiverse - see, for example, the work "Anthropic Principle or Cosmic God", or §5.7 "Anthropic Principle and Existence of Multiple Universes" monograph "Gravity, Black Holes and the Physics of Spacetime".
  In the microworld, the order of measurements is important. E.g. on whether we first measure the position of the particle (thereby disturbing the momentum) and then only its existing momentum, or vice versa (by measuring the momentum we first disturb the position). The more accurately we measure the position, the less accurately we know the momentum of the particle - and vice versa. This leads to the principle non-commutativity of quantum mechanics, expressed in quantum uncertainty relations (see below).
  The term "state" of a physical system means a situation where this system is in a configuration (state) with a certain value of a given physical quantity. In classical physics, for example, the state of a particle is described by entering the position and velocity resp. momentum (as a function of time). In quantum physics, the situation is more complicated, the so-called state vector, denoted by |y>, is introduced here, where y only symbolically denotes a state quantity, which, however, does not have a certain value, but can be a superposition of several states. E.g. the electron in terms of spin (see below) may be in the spin state |1> with spin-oriented "up" (z-component spin projection has a value of + ¹/₂ h ), or in the state |2> with spin - ¹/₂ h. However, it can also be in a more general state |y>, which is a "mixed" superposition of "pure" states |1> and |2>, which is written in vector: |y> = a₁ . |1> + a₂ . |2>. This state |y> means that the probability a₁² we measure a value of + ¹/₂ h and the probability of a₂² we can measure the value of - ¹/₂ h. In general, a superimposed state can be made up of multiple components: |y> = a₁.|1> + a₂.|2> + ... + a_i.|i> + ... General |y> is therefore a state where a given physical quantity does not have a certain value, but only the probability a_i² measurements of individual potential values "i". Quantum physics is a mathematical algorithm (computational scheme) that can determine these coefficients a_i - but does not specify the internal physical nature of these phenomena. In the state |y>, before the measurement, the system has the value of the given physical quantity indeterminate (potentially different values are possible) and only the measurement concretizes this value. Even under the same initial conditions, we always measure different values of physical quantities, statistically divided around the mean values determined by probability coefficients a_i².

Operators. Uncertainty relation.
Observation or measurement operations are modeled in quantum mechanics using so-called operators. Each physical quantity A is assigned in quantum mechanics the operator A^{^}, which satisfies certain mathematical conditions (it is linear and Hermitian). By ^{^}A we mean a rule that assigns to each function u(x) some other function v(x) - symbolically we write v = ^{^}A u. The operator ^{^}x assigned to the coordinate x is a simple multiplication of x, while the momentum operator ^{^}p is given by the derivative according to x coordinates :
^{^}x ® x, ^{^}p ® - i h . ¶ / ¶ x.
  The Planck constant h got here to hold the relationship between the momentum of a particle and the corresponding frequency of the Broglie wave in corpuscular-wave dualism. Other physical quantities - energy and momentum - will be discussed below.
  For operators in quantum mechanics, it is important that the sequential application of two operators does not have to be commutative, ie it may depend on the order. For two operators ^{^}A and ^{^}B, the so-called commutator is defined by the relation [^{^}A, ^{^}B] = ^{^}A ^{^}B -  ^{^}B ^{^}A, i.e. the difference between the application of the operator ^{^}A and then ^{^}B, minus the same operators applied in reverse order. This difference is generally not zero as in classical physics, as each observation (measurement) in the microworld can cause a system violation and thus affect the result of the second observation (measurement), so that the two procedures can provide different results. The coordinate and momentum operators satisfy an important commutation relation [^{^}x, ^{^}p] = i . h .
  This commutation relation is related to the key principle of quantum mechanics, the so-called Heisenberg quantum uncertainty principle, which states that the position x and momentum p particles cannot be determined exactly at the same time *), but that the uncertainties of these two (complementary) quantities are given by the relation Dx . Dp ³ h. Every measurement of a particle's position irreversibly perturbs its momentum - and vice versa. Therefore, it is not possible to investigate the exact specific path, along which the particle moves...
*) Quantum "blur", implied by uncertainty relations, is mostly negligible and unobservable in the macroscopic world, but on an atomic and subatomic scale it becomes absolutely decisive!
  This quantum uncertainty is an expression of the basic property of observation and measurement: that it is always an interaction, irreversibly affecting the parameters of the measured system. The same uncertainty applies sessions between other dynamically coupled quantities, eg. between time t and energy E : DE . Dt ³ h , further between potential and kinetic energy, etc. This complementarity, whose "prototype" is corpuscular-wave dualism, is characteristic of quantum physics.

Characteristic equations. Discrete values of physical quantities.
When applying operators to wave functions, are especially important cases where the result of the operator ^{^}A applied to the function y(x) results again are the same function y(x) multiplied by a certain number a : ^{^}A y(x) = a. y(x). In general, to each operator ^{^}A belongs a set of numbers a_n and a set of functions y_n, for which the so-called characteristic equation applies
     ^{^}A y_n (x) = a _n . y _n (x) .
Numbers (coefficients) a_n are called proper (characteristic) values and y_n corresponding own (characteristic) function of the operator ^{^}A. Eigenvalues a_n of operator ^{^}A then represent possible values, which may be a physical quantity a corresponding operator ^{^}A take. Said equation is a differential equation for the wave function of the state in which the quantity represented by the operator ^{^}A has the value a. Eigenvalues satisfying this equation generally do not take on all possible values, but only certain ones discrete values, in accordance with experimental knowledge about discrete (quantum) values of physical quantities in the microworld - energy of atoms, magnetic moments, spins... It turns out that energetically (field) bound particles in the microworld belong to discrete values of energy, momentum and other quantities - we call them quantum physical quantities. These discrete characteristic values, expressed as multiples of their respective elementary value (usually Planck's constant h), are called quantum numbers.

Quantum energy. Schrödinger equation.
Similarly to classical mechanics, also in quantum mechanics the key concept is energy E. Energy E (consisting of potential energy U and kinetic energy T: E = T + U) is assigned in quantum mechanics an energy operator called Hamilton's operator, which for the simplest the case of a particle of mass m has the form
     ^{^} H = - ( h ² / 2m). D + U,
where D s ¶²/¶x² + ¶²/¶y² + ¶²/¶z² is the so-called Laplace differential operator. The proper (characteristic) equation of the Hamiltonian operator
     ^{^} H y _n  =   E _n . y _n
is called the stationary Schrödinger equation. Its solution for a particle is the wave functions of the stationary states of the particle in the potential field, in which the particle acquires discrete energy values E_n (for continuous and discrete energy values, see the note below).   
  The time evolution (motion) of the quantum state of a microparticle is then described by the nonstationary Schrödinger equation
^{^} H y   =   i h . ¶y / ¶ t  ,
which contains the time derivative of the wave function.   
  Solutions of the stationary Schrödinger equation indicates what possible stationary physical states might particles in a given force field to acquire, via nonstationary Schrödinger equation can in principle determine the probability with which the particles pass from one quantum state to another. It can be said that Schrödinger's equation has a similar position in quantum mechanics as Newton's laws in classical mechanics. Among other things, all the quantum properties of the structure of atoms flow from it, which will be discussed below (especially discrete energy levels).
Continuous and discrete energy - quantized
In classical physics, energy can acquire all possible values continuously; by doing work, the energy of bodies in a certain system can be changed arbitrarily. In quantum physics, the situation is more complicated. The energy values are the solution of the (stationary) Schrödinger equation given above. In the simplest case of a free particle (U = 0), this equation has the form ( h²/2m). Dy + E.y = 0 and its solution are wave functions of the form y = const. e ^i/h(E.t-p.r), for any energy values E, where E = p²/2m. Each such function (plane wave) describes a state in which a particle acquires a certain value of energy E and momentum p , where the frequency of such a wave is E/h and the wavelength l = 2ph/p is the Broglie wavelength of the particle. The energy spectrum of a free-moving particle is therefore continuous, the energy can take values from 0 to ¥ - the energy of a free particle is not quantized.
  If the particle is in the potential field U(x, y, z), then the motion of the bound particle with energy E <0 has a discrete spectrum energy levels, while for positive energies the particle is not bound and its energy can take on a continuous spectrum. A typical model case of quantum motion of a bound particle is the motion of a particle in a potential well - in the simplest case a one-dimensional motion bound to a line of length L between two perpendicular walls (infinitely high), from which the particle reflects perfectly elastically. Such a line-segment motion of the particle corresponds the Broglie wave, which is reflected on the walls, while the superposition of the waves reflected from both walls creates a "standing wave". Thus, an integral number of half-waves of standing Broglie waves is formed on the line segment of length L, ie L = n. l/2, where n = 1,2,3, ... The motion of a particle in a potential well therefore correspond only to certain discrete values of the wavelengths of Broglie waves l_n = 2.L/n , n=1,2,3,... The Broglie wavelength is related to the momentum of the particle, l = h/p, so that the momentum of the bound particle p_n = h/l_n = n.h/L and its energy E_n = p_n²/2m = n² .h²/8m.L² will have discrete values *). The state of a particle in a potential field, which corresponds to a standing Broglie wave l_n , represents a certain stationary state of the particle. It is a state with a certain energy E_n - energy level of particles in a potential field in a given steady state. The number n is then called the quantum number of this steady state. The state corresponding to n = 1 is called the ground state and corresponds to the lowest energy level of the particle bound in the potential field. The change in the energy of a particle is associated with the transition (jump) to another stationary state, which is accompanied by the emission or absorption of a quantum (photon) with energy equal to the difference of energies of both stationary states (energy levels). These laws find their application below in Bohr's model of the atom.
*) At large values of quantum numbers n, the energy differences of individual quantum states with quantum numbers n+1 and n are small - the ratio E_n+1/E_n = [(n+1)²-n²]/n² is close to 1. Thus, the energy changes at individual higher quantum levels energies are negligible - energy can be considered continuous here; the results of quantum mechanics at higher quantum numbers basically correspond to the results of classical mechanics - the principle of correspondence .
  The actual energy spectrum of particles in the microworld can be discrete or continuous, depending on the process by which the particles form, gain energy and are emitted. The continuous energy spectrum has eg braking radiation, radiation b, Compton scattered radiation g. Other spectra are discrete, quantized, line - eg radiation a and g, radiation spectra of excited atoms, characteristic X-rays, conversion or Auger electrons. A certain intermediate species between continuous and discrete spectra are band spectra, where individual quantum states are separated only by very small energy intervals and the resulting spectrum appears to be continuous within the resolution of spectrometric instruments. The specific mechanisms of particle emission and energy recovery will be discussed in more detail in the description of the radiation of atoms and atomic nuclei in radioactivity and other processes.

Permitted and forbidden transitions
From the point of view of classical physics, only such events can take place in which the laws of conservation of energy, momentum, angular momentum are fulfilled at all stages. Of course, such processes, called permissible transitions, can also take place according to quantum physics in the microworld. They take place very "willingly", their speed (effective duration of transition, transformation, interaction) depends on the type of force that causes it. The fastest are transitions caused by strong interaction, followed by electromagnetic processes and finally transformations caused by weak interaction.
  However, some such events can take place in the microworld, in which these classical laws of conservation are violated at some stages - the so-called forbidden transitions. We can simply imagine that a particle (within its permanent quantum oscillations and fluctuations) constantly "tries" it again and again until it manages to "break through" the barrier of prohibition. The wave function of the particle is spread in the phase space and can partially interfere with areas where the transition can "bypass" the violation of the law of conservation. Prohibited transitions can in principle take place, but with less probability. A typical case is the tunneling phenomenon of the passage of a particle through an energy barrier described below, or forbidden transitions between the energy levels of electrons in the envelope (see "Excitation and radiation of atoms" below) or between the energy levels of nucleons in the nucleus (§1.2, part "Radioactivity gamma", passage "Nuclear isomerism and metastability") due to higher differences in angular momentum (multipolarity) than the emitted photon is able to carry.

Quantum angular momentum. Spin. Magnetic moment.
One of the important physical characteristics of the motion of material bodies in space is the angular momentum - a vector quantity quantifying mainly the rotational motion of bodies. The law of conservation of angular momentum *) provides a number of useful data on the properties of motion.
*) The law of conservation of angular momentum is a consequence of the invariance of physical laws (Hamiltonian of system) to spatial rotation by any angle - isotropy of space. This property applies not only in free space without fields, but also when moving in a centrally symmetric field, where, however, the invariance to rotation applies to rotation around the center of the field. The angular momentum therefore plays an important role in observation the motion of planets and in analyzing the motion of electrons around the atomic nucleus in its central field.
  The angular momentum of a particle (mass point) in classical mechanics is a vector quantity L, which is defined as the vector product of the position vector r and the momentum vector p : L = [r x p], or in components in the direction of the x, y, z axis: L_x= y.p_z - z.p_y, L_y= z.p_x - x.p_z, L_z= x.p_y - y.p_x. By replacing the components of coordinates and momentum with the above mentioned operators, we obtain the operators of the angular momentum components: ^{^}L_x= ^h/_i (y.¶/¶z - z.¶/¶y), ^{^}L_y= ^h/_i (z.¶/¶x - x.¶/¶z), ^{^}L_z= ^h/_i (x.¶/¶y - y.¶/¶x). In vector it can be written ^{^} L = [^{^}r x ^{^}p] = -i h [r x Ñ], where Ñ is the vector form of the Hamiltonian differential operator. The characteristic equation for the angular momentum is customary (without prejudice to generality) to investigate for the component z : ^{^}L_z y = l_z . y, in spherical coordinates r, J, j. Her solution is (we cannot break down mathematical details here) : y = f(r,J). e^{i. l}z^.j, where f(r, J) is an arbitrary function of the radius r and the angle J. In order for the characteristic function y to be unambiguous, it must be periodic with respect to j with a period of 2p, so it must be:
               l _z  =  m. h   ,  where m = 0, ± 1, ± 2, ....
The eigenvalues of the angular momentum l_z are therefore quantized- can be equal to positive and negative integer multiples of the Planck constant h including zero. This result is important in that it quantum-mechanically justifies Bohr's basic postulate model of the atom, which is discussed below ("Bohr's model of the atom").
  In addition to the components of the angular momentum L, its absolute magnitude L º |L| = Ö(L²) is also important in mechanics. The characteristic values K of the angular momentum square are determined by the equation ^{^}L² y = K.y. A relatively complex and lengthy mathematical analysis (again using here e.g. the requirement of uniqueness of the characteristic function y leading to the periodicity 2p) can be used to obtain the formula for the characteristic values of the square of the angular momentum
          K =  h ² . l (l + 1)  ,  l = 0, 1, 2, ...
Characteristic values of the operator of the absolute magnitude of the angular momentum |L| then they are :
          | L | =  h . Ö [l (l + 1)]  ,  l = 0, 1, 2, ...
At a given value of the number L, the angular momentum component L_z can take the values L_z = L, L-1, L-2, ..., 0, -1, ..., -L, ie a total of 2.L + 1 of different values, corresponding to different orientations of the angular momentum in space. All these rules apply, among other things, in the structure of the electron shell of the atom, where the energy level corresponding to the angular momentum L is (2.L+1)-times degenerate; in conjunction with Pauli's principle, this implies occupancy rules for electron levels, as described below.
S p i n
In classical mechanics, in addition to the mutual angular momentum of moving bodies, or the angular momentum of a body with respect to a given point, is also applied the own (internal) angular momentum caused by rotation bodies around its own axis. In quantum mechanics, the angular momentum determines the symmetry of the state of the system with respect to rotation in space, ie the way in which wave functions corresponding to different values of the angular momentum projection are mutually transformed when the coordinate system is rotated. The origin of the angular momentum no longer matters here. The analysis of the properties of particles shows that in quantum mechanics we must also attribute a certain intrinsic angular momentum to an elementary particle, which is not related to its motion in space. Own angular momentum of a particle is called spin and is denoted by s, while the angular momentum associated with a particle's motion in space is called the orbital moment (usually denoted by L or l ). This property of elementary particles has a specific quantum nature and cannot be completely explained by classical mechanical concepts (spin cannot be quantitatively explained, for example, by the rotation of a particle around its own axis!) *). In the quantum description of a particle with spin, the wave function must determine not only the probability of different positions of its occurrence in space, but also the probability of different spin orientations. Wave function must therefore depend not only on the three spatial coordinates, but also on the spin variable that indicates the value of the projection of the spins in a particular direction in space (selected axis z ), and acquires a limited number of discrete values.
*) Difference of spin from angular momentum
The spin of quantum particles differs considerably in some of their properties from the usual orbital or intrinsic rotational angular momentum of bodies :
-> It is quantized, takes on certain discrete values (mentioned below); however, this is generally consistent with quantum physics (so it's not surprising).
-> For a given type of elementary particle, spin has a precisely given value (in multiples of the Planck constant); the particle cannot be "forced" to rotate faster or slower. The spin value depends only on the type of particle and cannot be changed in any way (unlike the orbital angular momentum or spin direction) ...
-> The spin of quantum particles can take values 0, 1/2, 1, 3/2, 2, ... - a multiple of the Planck constant. In contrast, the orbital angular momentum can only take on the integer value of a multiple of the Planck constant.
  Like the angular momentum in general, spin is quantized. The eigenvalues of the square of the spin are equal s² = h². s(s + 1), where the spin number s can be an integer (including zero) or a half number; is an intrinsic characteristic of a given type of particle. At a given s, the spin projection can take values s_z = -s, -s + 1, ...., s-1, s, so a total of 2.s+1 values. In §1.5 "Elementary particles", passage "Indistinguishability of particles" - "Spin, symmetry of the wave function and statistical behavior of particles", we will see that there are two main groups of particles according to spin s : particles with half-number spin (most of them - electrons, protons, neutrons, muons, etc.) and with integer spin (photons, p and K mesons, hypothetical gravitons and others). This circumstance is closely related to the quantum behavior of sets of particles - the particles behave like fermions or bosons (§1.5, passage "Fermios-Bosons").
Magnetic moment
Every electrically charged body ("charge", "charged particle") generates an electric field in the surrounding space according to Coulomb's law (if the charged particle is at rest with respect to the reference system, it is an electrostatic field). When the charged body moves evenly in a straight line, it also generates a magnetic field according to Biot-Savart-Laplace's law. And if the charge moves unevenly - accelerated or along a curved path, it excites a time-varying electromagnetic field around itself, part of which propagates through space as electromagnetic waves. These basic findings of the unified science of electricity and magnetism - electrodynamics (see eg §1.5 "Electromagnetic Field. Maxwell's Equations." in the book "Gravity, Black Holes, and the Physics of Spacetime") work perfectly not only in classical, but also in relativistic and quantum physics.
  Leaving aside the translational motion (irrelevant here) and the emission of waves (which we will discuss below), the main mechanism of excitation of a magnetic field by charged particles is their rotational motion. The motion of a charged particle in a circular orbit generates a magnetic field, the direction of which is perpendicular to the plane of circulation and whose intensity (magnetic induction) is proportional to the charge of the particle and the angular momentum of its circulation. This magnetic field behaves like a fictitious magnetic dipole- miniature "bar magnet". Its force is quantified by the vector quantity magnetic moment m, expressing the moment of a pair of forces f, which would act on this magnetic dipole in the external homogeneous magnetic field B : f = [m x B]. A particle with charge q and rest mass m , which rotates with the angular momentum L, generates a magnetic dipole moment m = (q/2m).L according to classical electrodynamics. When exciting a magnetic field by the rotational motion of charged bodies, the so-called gyromagnetic ratio g is often introduced, which is the ratio of the excited magnetic moment and the mechanical angular momentum of the rotating body. For a classically charged rotating body, g = q/2m.
  Rotational motion of a charged particle excitation magnetic moment, can be of two kinds :
  - Orbital (circular) motion of charged particles in the field of bonding forces with other particles. This is the case of electrons orbiting an atomic nucleus. The electron of rest mass m_e , orbiting with angular momentum L, behaves like a magnetic dipole with moment m = g . (- e/m_e). L. However, it is usually expressed in the form m = -g . m_B .L/h, where m_B = e. h /m_e is the so-called Bohr magneton. The dimensionless correction factor g indicates the relationship between the actually observed magnetic moment of the particle and the theoretical value of the Bohr magneton.
  - The proper rotational motion of a charged particle, rotating around its axis - the above-mentioned spin of particle. The above classical relation m = (q/2m).L in principle applies to the excitation of a magnetic moment even if the angular momentum L it is created by the rotation of a body with an equally distributed density of mass and electric charge around its own axis of symmetry. The spin magnetic moment of an electron can be expressed as: m_s = -g_s.m_B . S/h, where S is the spin momentum of the electron (+, - 1/2 h), the g- factor is approximately equal to 2.
  The  more complicated situation is with nucleons - protons and neutrons. By a straight analogy with electrons, we would obtain the relation m_p = g_p.m_N . S/h for the magnetic moment of the proton, where m_N is the so-called nuclear magneton m_N = e. h/m_p. However, the correction factor g here has a relatively high value of g_p = 5.58; this suggests that the determination of the magnetic moment of a proton based on its spin is problematic (see below). Thus, the proton has a magnetic moment m_p = 1,41.10^-26 J/T and a gyromagnetic ratio g_p = 2,675.10⁸ rad.s^-1.T^-1. The gyromagnetic ratio also indicates the frequency Larmor precession of the magnetic moment of particles in an external magnetic field; for protons, the Larmor frequency is 42.577 MHz/T - nuclear magnetic resonance is used in the analytical and imaging method (see §3.4, section "Nuclear magnetic resonance"). A neutron, as an electrically neutral (uncharged) particle, should have no magnetic moment, it should be m_n = 0. In reality, however, a neutron has a non-zero magnetic moment m_n = -0,97.10^-26 J/T, which is only slightly smaller than that of a proton (and has the opposite sign). How is it possible?
  The origin of the magnetic moment of nucleons does not lie in their rotational angular momentum (spin), but comes from their internal structure - that they are composed of quarks "u" and "d" (§1.5, passage "Quark structure of hadrons"). For hypothetical or model quarks, the magnetic moment m_"u" and m_"d" is assumed, which in the first approximation can be modeled in a similar way as a nuclear magneton: m_q = q_q . h/m_q. The magnetic moment of a nucleon can then be considered to be composed of the vector sum of the magnetic moments of three charged quarks and the orbital magnetic moments caused by the motion of these charged quarks in the nucleon. With quark model magnetic moment of the proton (composed of two quark "u" of charge +²/₃e and 1 quark "d" having a charge of -(1/3)e) we can then be approximated as m_p = 4/3m_"u"-1/3m_"d" = 2,8.m_N = 1,41.10^-26 J/T. And the magnetic moment of a neutron (composed of 2 quarks "d" with charges -1 / 3e and 1 quark "u" with charge +²/₃e) then it is m_n = 4/3m_"d"-1/3m_"u" = -1,9.m_N = -0,97.10^-26 J/T. Quantum chromodynamics seeks a more detailed analysis, including gluon fields and virtual particles inside nucleons (not yet completed).

Quantum field theory
We have so far dealt with the quantum behavior of the microworld from the point of view of quantum mechanics of microparticle motion: probability waves (forming fields) are assigned to mass particles and specific quantum properties of particle motion are obtained by solving relevant wave equations, including discrete values of energy and other physical quantities. Quantization of this kind is sometimes referred to as "primary".
  In addition to particles, the main subject of the scientific description is the physical field. The physical field, which carries energy, momentum and other physical parameters, as well as particles, must also have a quantum character in the microworld. In the quantum description of fields, sometimes called secondary quantization, on the other hand, the field is expressed using particles - quantum excitations in the field. The transition from classical to quantum field theory consists of two basic stages :

1. The field equations are converted to a wave equation, so that the field in a given region of space can be expressed as a superposition of plane waves. Thus, the field is described by a discrete series of variables - amplitudes and frequencies of waves, the so-called harmonic oscillators of the field. 
2. In the quantization procedure itself, discrete quanta of energy are assigned to the field, corresponding according to the laws of quantum mechanics to the individual possible energy states of the harmonic oscillators on which the field was decomposed. At the same time, this quantum of energy is considered to be those particles, that mediate the interaction of the sources of a given field.

  The application of this method of quantization to the electromagnetic field is the basis of quantum electrodynamics (QED) and leads to the idea of the electromagnetic field as a set of particles - photons, each of which has energy h.w and momentum h.w/c; the rest mass of photons is zero, their spin (intrinsic angular momentum) is equal to 1 (resp. 1.h). At the same time, these electromagnetic quanta (photons) are interpreted as particles mediating the interaction of electrically charged particles. Radiation and absorption of photons by electric charges (especially electrons) is expressed by means of so-called creation and annihilation operators, that generate or take photons in a certain energy state in an electromagnetic field.
New - quantum concept of force: intermediate exchangeable particles
In classical physics, each kind of interaction of bodies is assigned a corresponding field - a space in which certain forces act on particles. In classical physics, it is an electric, magnetic, gravitational field. The magnitude of the field action at each point in space is expressed by the field intensity (force acting on the "unit test particle") or by its potential 
(work associated with the transfer of particles to a given place). The changes ("commotion") in this field propagate at a finite speed from place to place, which is accompanied by the transfer of energy, momentum, and other physical quantities. From the point of view of classical physics, these quantities, such as energy and momentum, are transmitted continuously during field changes. In quantum physics, it turns out that during changes (disturbances) in the field, physical quantities are transmitted discontinuously over certain "portions" - quanta.
  Quantum field theory, in its concept of secondary quantization, leads to a new concept of the field as a set of particles - quantum fields. And the interaction of particles (interactions) is caused not by the field force, but by the mutual exchange of these quantum field particles - the exchange of intermediate particles. The particles are constantly receiving and emitting quantums of fields, which causes them to interact with each other. This exchange mediating (intermediate) quantums, we interpret as a quantum particles - carriers of interactions. This introduces a new concept of force and interaction in quantum field theory. This concept plays a key role in the interactions of "elementary" particles - it is discussed in more detail in §1.5 "Elementary particles and accelerators", section "Interactions of elementary particles".
Virtual or real particles ?
Are these intermediate particles mediating the interaction real? The answer is yes and no ! Let's briefly discuss this problem in the electromagnetic field - quantum electrodynamics. According to it (as outlined above), the photons are quantum of the electromagnetic field, and the electric force between two charged particles is caused by the constant exchange of photons. However, if we looked at the space between two stationary charges, we would not register any flow of flying photons. It's just a model, those intermediate photons are virtual here ! In quantum electrodynamics, the force is only modeled by using photons: the static field is artificially decomposed into a superposition of waves (harmonic oscillators), these are quantized and the resulting photons are designated as the quantum of the field that mediates the interaction. Rather physically substantiated is the claim that "photons are a quantum of the electromagnetic wave", not than "quantum of the electromagnetic field". In the static case, nothing is radiated physically! The actual radiation associated with the transfer of energy and momentum - with the flow of photons - occurs only in the dynamic case - during the accelerated motion of the charges. Then the virtual photons turn into real ones. The mutual interactions (collisions, scattering) of particles in the microworld are always dynamic processes (often at high energies), in which the virtual intermediate particles, hidden in the vacuum are "liberated", transformed into real particles and actively participate in the interaction.
Note: An interesting exception to the radiation of virtual particles even in the static case is the so-called Hawking radiation of quantum evaporation of a black hole. It is created that one of the two particles is pulled below the horizon of the black hole and absorbed and the other particle thus becomes a real particle and can be emitted (it is analyzed in detail in §4.7 "Quantum radiation and thermodynamics of black holes", passage "Mechanism of quantum evaporation" monograph "Gravity, black holes and space - time physics").

Quantum fluctuations of fields
One basic postulate of quantum mechanics is known Heisenberg uncertainty principle Dx. Dp ³ h , where h º h/2p @ 1,05.10 ^-27 g cm³/s is the Planck constant. The relation of uncertainty applies not only between position and momentum in quantum mechanics, but between every two dynamically coupled -conjugated quantities, ie also in quantum field theory. If we observe, for example, a magnetic field in a small spatial region characterized by the dimension L, there will be energy proportional to B².L³ and the time required to measure the field will be L/c; uncertainty relation DE. Dt ³ h then gives (DB)².L⁴ l h.c, or DB l hc/L². Thus, it can be said that the quantum fluctuations of the electromagnetic field in size L are of equal order: DE ~ DB ~ Ö(h.c) / L².
  Thus, the field is constantly "oscillating" between configurations whose fluctuation range is greater the smaller the spatial areas we observe. The influence of these quantum fluctuations on the motion of an electron around the atomic nucleus (these quantum fluctuations "overlap" over Broglie waves in Bohr's model of the atom - see the passage "Bohr's model of the atom", Fig.1.1.6) is schematically shown in the figure :

Schematic representation of the motion of an electron around an atomic nucleus. A closer look at the Kepler trajectory of the electron would reveal small chaotic irregularities caused by quantum fluctuations in the electric field. The mean deviation from the global trajectory is zero, but the standard deviation leads to a small shift in the energy level of the electron. This shift was actually measured as part of the so-called Lamb-Rutheford shift.

In the above-mentioned quantum field description - secondary quantization - these quantum field fluctuations can be considered as quantum - particles. The vacuum thus becomes a highly dynamic environment in which virtual particles are constantly created and destroyed. These particles have an immeasurably short duration, they are undetectable, we say they are virtual. In addition, quantum fluctuations and virtual particles arise everywhere with the same density and impinge on matter from all directions, so that their force effects are balanced and canceled on a macroscopic scale. Under certain circumstances, however, the cumulative effects of a large number of these particles may still have a slight macroscopic effect.
  In addition to the afore mentioned Lamb shift we managed to measure the so-called Casimir effect :
We place two horizontal plates (electrically uncharged) right next to each other in a vacuum. In the gap between the boards may arise quantums - virtual particles with only short wavelengths (whose integer multiple is given by the width of the gap), while in the space outside the plates they can take on any wavelengths. The total density of the particles is thus lower in the gap and the pressure of the particles from the outside prevails on the plates. The plates are thus attracted to each other by a force which is greater the narrower the gap between the plates.
  Furthermore, the observed dipole magnetic moment of an electron is formed, in addition to the basic electrodynamic component, also by an anomalous magnetic moment, created by the interactions of the electron with virtual photons in quantum electrodynamics.
  Assuming the universal validity of the quantum principle of uncertainty, a similar situation should occur in the general theory of relativity as the physics of gravity and spacetime: should quantum fluctuations in the geometry of spacetime occur..?.. - §B4 "Quantum geometrodynamics". Some paradoxical interpretations of vacuum quantum microfluctuations are discussed in §B.5, the passage "The mystery of vacuum quantum energy <-> Cosmological constant".

Feynman quantization of path integrals
At the beginning of our fleeting excursion into quantum physics, we mentioned that it is by no means easy to understand the intrinsic causes of quantum behavior of microsystems based on our experience with classical macroworld behavior. For example, how is it possible that in the famous double-slit experiment (Fig.1.1.2 ) a particle can pass through both holes at the same time and then interfere "with itself"?
  Feynman's formulation of quantum theory is characterized by a very close relationship to classical physics *) expressed by the principle of least action. In classical physics (mechanics, electrodynamics, GTR), between a given initial x1 and final x2 state, of the investigated systen always make only such a movement for which the integral of the action S = _x1ò^x2 L dt is extreme. On the other hand, in quantum physics, as is well known, such processes also take place that do not comply with this principle and are impossible according to classical physics - for example, the tunneling phenomenon.
*) The transition from classical to quantum physics is so elegant and straightforward here, that J.A.Wheeler used this approach to persuade A.Einstein to revise his opposition to the stochastic principles of quantum mechanics. But to no avail: "I don't believe that God would play dice with the world ", Einsten persistently objected...
  In Feynman's approach, all trajectories leading from the initial state x₁ to the final state x₂, are considered equiventally and simultaneously, regardless of whether they are permissible or not according to classical physics. As if the particle were moving along each imaginary trajectory at the same time as it traveled between the two states - it is the set of all virtual trajectories ("history"). If the integral _x1n^x2Ldt is calculated for each trajectory, the probability of transition of the system from the initial state x₁ to the final state x₂ will be given by the square of the quantity

^,

obtained as sum and taken through all trajectories - the sum through all possible "histories". It is evident that the largest contribution to this sum is made by those trajectories that have a phase coefficient (i/h) ò Ldt almost the same (exponents add up), while for trajectories with large differences in (i/h) ò Ldt the exponents in the sum cancel each other out. The most probable trajectory (corresponding to close values of ò Ldt) will therefore be a classical trajectory with extreme behavior of the integral of the action. Trajectory here means "path" in the space of the given configurations systems; if it is a complex system described by a large number of parameters, it will be a trajectory in multidimensional space. Feynman showed that this formulation is equivalent to the usual Schrödinger and Heisenberg concept of quantum mechanics. Similarly to about the classic principle of least action, in practice in not immediately seek extreme integral ò Ldt, but derive Lagrange equation of motion, even when using Feynman method, the total sum over all trajectories is not directly calculated. Feynman's procedure is rather used as a means for deriving and elaborating quantum theories, as well as their physical interpretation.
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Gnoseological questions :
Is quantum physics a major obstacle to the knowledge and use of nature ?
We physicists believe in the recognizability of the world, and it is our professional duty to work for the best possible knowledge of the mechanisms and laws according to which nature "works". From this point of view, we are "horrified" that randomness, stochasticity, statistics, which are a reflection of our ignorance of the exact conditions and states in complex sets of many interacting particles, are "dabbling" into fundamental physics !
  Quantum physics is often considered a theory of fundamental constraints, according to which our observations and measurements are inevitably inaccurate, natural phenomena are ruled by chance, and we should give up hope that science can accurately describe our world. Quantum mechanics is often considered an insurmountable obstacle to the knowledge of the deepest microworld or the practical use of microscopic phenomena (eg further miniaturization of electronic circuits). Already in the early periods of the development of quantum physics, it turned out that corpuscular-wave dualism, the randomness of phenomena and their superposition, discretion and especially the quantum relation of uncertainty, fundamentally prevent us understand and use nature in such a way and to such an extent as we were used to in classical physics (mechanics, electrodynamics, ...). This somewhat misleading view has its roots in a time when physicists have developed, perfected and confronted quantum mechanics with classical theories and philosophical concepts.
  In recent decades, however, a different perspective has become increasingly common. What does the uncertainty principle say and what does it not say? They merely claim that not all observed quantities of a physical system can take on certain ("sharp") values at the same time. Not all quantum measurements are limited by the uncertainty principle. Although the position or velocity is indeterminate and "blurred", other properties can be quite "sharply" defined - for example, blurred electrons in an atom produce a well-defined energy of a given orbital. In some cases, we can ingeniously circumvent this dreaded obstacle, and at the quantum level we can use the special properties of microsystems in new advanced devices such as lasers, integrated circuits, nanotechnology, new possibilities in informatics and computers (see below "Quantum computers").
........ see also the discussion in "Natural laws, models and physical theories".................
At the deepest level, is the world discrete or continuous ?
This is an important gnoseological question on which opinions of physicists differ. In modern terminology, this question could be paraphrased: “Is the physical world essentially analog or digital ?”. The physical relationship between the continuous (fluent, smooth) and discrete nature of the microworld can be essentially twofold :
× Discretion is basic : ® secondarily generates continuity (apparent)
This is the most common approach in modern physics, based on atomistics, thermodynamics and statistical physics. From the point of view of atomic physics, all substances are composed of discrete ones atoms that have a specific integer number of protons in the nucleus and electrons in the envelope. And the spectrometry of the radiation emitted and absorbed by the atoms shows that the electrons in the atom occupy discrete energy levels, determined by integers. Bohr's model of the atom is based on this (see "Bohr's model of the atom" below). In sets of large numbers of atoms and molecules, the methods of statistical physics make it possible to derive the laws of gas kinetics and thermodynamics, which are continuous. However, the basic "input values" of the theory are discrete integers. The continuity "emerges" from the averaged a large number of discrete events. The water in the glass appears as a continuous medium, but if we look at it with high magnification, we will see the molecules and atoms of which it is composed. These atoms also have the internal structure of electrons, protons and neutrons.
At the most basic level of current knowledge, matter is made up of fundamental leptons and quarks (§1.5, part "Standard model - uniform understanding of elementary particles"), which are considered as discrete particles that can in principle be "counted down one after the other" *).
*) Gnoseological note: The role of integers is called metaphorically in classical mathematics that "God created only whole numbers, everything else in mathematics is the invention of humans". However, for modeling nature, mathematics introduced more general sets, especially real numbers (see §3.1 "Geometric-topological properties of spacetime", section "Sets and representations") for which it created extensive apparatus of differential and integral calculus.
× Continuity is essential : ® secondarily generates discretions (again only apparent?)
Quantum physics microworld based on a particle-wave ideas. The wave equation of quantum mechanics (Schrodinger equation) contain only continuous quantities. An illustrative example of this concept is Broglie's wave explanation of Bohr's quantum electron paths around the nucleus of an atom - see below, Fig.AtomBroglie.gif. Here, the discretion of electron paths is formed by the continuity of the wave functions of electrons - their "wave continuity".
  In the standard particle model, it is taught that the basic building blocks of matter are the basic discrete particles - leptons (especially electrons) and quarks. However, on a more fundamental level, in unitary field theory (§B.1 "The process of unification in physics" in the book "Gravity, black holes ...."), the basic building block of physical theories is  field - a continuous "fluid" substance distributed in space (the best known example is the electric and magnetic field). From the point of view of unitary field theory, "fundamental particles" are not fundamental, but are composed of continuous fields (and their waves or quanta) - see below. Particles are "precipitates" of a unitary field.
  Nature is probably a true continuum, in which we do not find any building elements that are no longer indivisible at any level of magnification. Physical quantities are generally not integers, but continuous real numbers, for which the number of decimal places is constantly increasing with the gradual refinement of the measurement. Integers retain only the significance of the number of types of significant particles in terms of the type of their interactions (e.g. 3 types of neutrinos, 6 types of quarks), the expression of the number of electrons in atomic shells, the number of protons and neutrons in nuclei, or order and number of excited states.
Is space and time continuous or discrete ?
In most disciplines of classical and quantum physics, space and time are considered to be a continuous, infinitely divisible continuum - a kind of "stage or arena", against the background of which physical processes, interactions of particles and fields take place. What if, however, the continuity of space-time is the same illusion as it was until the 19th century continuity of matter? As modern physics learns about the discrete quantum structure of matter, it is hypothesized that space-time is also quantized - it consists of a huge but countable number of very small already indivisible elementary "cells", a kind of "space-time dust". If these hypothetical "quantum geometries" are small enough, e.g size of the order of the Planck length 10^-33 cm, spacetime appears to be completely continuous, as no physical processes studied so far can distinguish finer distances than about 10^-15 cm.
  Thus, there is a possibility that continuous quantities could in fact be discrete in a closer (enlarged) view: they may lie on a dense grid of individual separate points, which in the view available to us gives the illusion of a continuum. It is similar to the pixels on a computer screen observed at basic magnification and zoom.
Note: It is interesting that a discretized version of quantum fields - a lattice field - was developed in quantum physics, where the continuous space - time is replaced (modeled) by an evenly arranged set of points, only in which the quantities of the fields are determined. However, it is only a model that facilitates quantum calculations, it does not follow that this is in fact the case.
  Thus, in addition to the generally accepted concept of continuous space, it is possible to alternatively postulate or axiomatically introduce the primary discretion of space : space is formed by individual separate "cells", only in which field values (potentials, intensities) are defined. These fields are therefore also discrete in terms of spatial distribution. If the spatial lattice (matrix) is sufficiently dense or fine (even in the dimensions of Planck lengths 10^-33 cm), space seems to us "illusory" as a continuum. However deepest mikromìøítcích the space could primarily be discreet - "pixellated" or "voxellated"..?..
  General relativity conceives gravitational field as a curved spacetime (see §2.2 "Versatility - a basic property and the key to understanding the nature of gravity" in the book "Gravity, Black Holes and the Physics of Spacetime"). If we want to quantize gravity, it is necessary to "quantize" spacetime. The combination of the general theory of relativity and quantum physics thus reveals (or postulates) a discrete structure in space-time itself, whether fundamental or induced - see §B4 "Quantum geometrdynamics" and §B5 "Quantization of the gravitational field", part "Loop theory of gravity " in the already mentioned monograph).
  Reflection of continuous versus discrete aspects of nature is probably necessary in creating a unitary theory of physics - the theory of everything - TOE (§B.6 "Unification of fundamental interactions . supergravity. Superstrings." monograph "Gravity, black holes and spacetime physics").
Author's note:
Personally, I slightly prefer the opinion that fundamental is a continuity (perhaps even causal?), which induces an apparent discretion (and perhaps also quantum stochasticity)..?.. However, even the concept of a discrete super-dense space-time lattice could perhaps be the main idea for understanding the microworld..?..
Is the world recognizable ?
This basic gnoseological question is often discussed from a variety of philosophical perspectives. From the scientific point of view, the cognition of our world can be reflected in principle on three levels :
1. Phenomenological cognition
The study of the specific course of individual natural processes is the basis of scientific knowledge. The accuracy of this knowledge is given by the level (resolution) of physical instrumentation, optical observation systems, chemical-analytical methods. The principal limitations in phenomenological cognition are imposed on us in the microworld by quantum relations of uncertainty (see, for example, the section "Quantum physics" above), in the macro- and megasworld then horizons of events of relativistic astrophysics (§3.3 "Cauchy's problem, causality and horizons" in the monograph "Gravity, black holes ....").
2. Knowledge of internal causes, mechanisms, laws
This is the main content of advanced scientific research. From a detailed analysis of the course of natural processes (phenomenological) under different conditions and comparisons from other processes, general natural laws are formulated, if possible with universal validity for a wider class of phenomena. This makes it possible to understand the functioning of nature (it is discussed in the section "Natural laws, models and physical theories" §1.1 in the already mentioned book "Gravity, black holes ...").
3. Absolute deterministic recognizability
The maximalist requirement of complete recognizability of the world would require that for all elementary particles, atoms, molecules and other structures we can predict their exact spatial positions at all times, as well as predict accurate values of fields (potentials, intensities) in all places of space. Our current knowledge shows that this is not possible ! In addition to technical impracticability, quantum relations of uncertainty prevent this at the microscopic level, and at all levels special irregularities in the behavior of sets of particles, called "deterministic chaos", which generate chance - is discussed in more detail in the section "Determinism-chance-chaos?" §3.3 in the book "Gravity, black holes and the physics of space-time". Furthermore, if space is continuous (see the discussion above "Is space and time continuous or discrete?"), an infinite and even innumerable set of points would require an infinite amount of data for each, even the smallest, district of the system under study!
  However, to deduce from the negative message of level 3 a categorical statement about the unknowability of the world is not substantiated and can be misleading *)! The world is basically recognizable in the sense that we understand the mechanisms of its functioning, on the basis of which we can often predict the behavior of many important systems in nature and space in the long run and with great accuracy. E.g. based on Newton's and Kepler's laws (gravity and mechanics), astronomers can predict the motions of planets in the solar system with high accuracy many centuries to come. In the horizon of millions to billions of years, however, minor gravitational disturbances will eventually result in chaotic deviations, which will significantly change the motions of the planets (some of them may even escape from the system ...). So it is not apt to say that "the world is unknowable", but that "the knowability of the world has its limitations".
*) Such a sharp statement that "the world is unknowable"could provoke skepticism, nihilism, agnosticism. It would also record various "alternatives" and charlatans who downplay the impressive achievements of serious scientific knowledge and claim that only they, thanks to their "miraculous abilities", have the gift of "true knowledge" and can control the world (or rather some trusting people...).

Some unusual and paradoxical consequences of quantum physics
Quantum tunneling phenomenon
If a particle moves in a certain force field, the law of conservation of energy - the sum of the kinetic energy of a particle and its potential energy in a given field - is fulfilled at each point of the trajectory when moving according to classical physics. An interesting case of particle motion is the motion in a force field, whose potential has the shape of the potential barrier - in the simplest case of movement in the direction of the X axis, such forces act on the particle that its potential energy E_p is everywhere zero, except for the area x₁ <x <x₂, where the value E_p = V_o. If the kinetic energy of the particle E_kin is less than the height V_o potential barrier, according to classical physics the particle should bounce when moving from the place of the barrier and move back against the original direction of motion - the particle is never able to overcome the potential barrier. The particle can overcome the potential barrier only if it has a sufficiently large kinetic energy E_kin>V_o.
  However, in quantum mechanics, where a particle is described by a wave function (according to corpuscular-wave dualism, it is a Broglie wave), there is a non-zero probability that the wave will "seep through" the barrier and the particle may suddenly be on the other side of the barrier. Waves, unlike conventional particles, can get behind the obstacle due to bending and then continue to move through space. Analysis of the wave function using Schrödonger's equation shows that a plane wave incident on the barrier wall partially bounces off it (and interferes with the original wave) and partially penetrates inside the barrier. If the width d = x₂ - x₁ of the barrier is small enough compared to the depth of penetration of the wave, the Broglie wave reaches the second wall of the barrier, where the potential suddenly decreases - the wave enters free space and continues to move away from the barrier as a plane wave, with a lower amplitude (expressing the probability of the particle passing to the other side of the barrier). The particle has passed potential barrier even if, according to classical physics, the energy of a particle is insufficient to overcome the barrier!

Symbolic representation of the quantum tunneling phenomenon.
Left: A ball rolling with the kinetic energy of E_kin against an elevated terrain wave (hill - gravitational potential barrier of height V_o) can overcome it only if it has sufficient energy E_kin > V_o. Middle: If a tunnel is pierced in a terrain obstacle, the body can overcome it even at a significantly lower energy than the potential height V_o. Right: Simplified representation of a rectangular potential barrier of height V_o, through which particles (~waves) can pass with a certain probability even at a lower kinetic energy than V_o - as if "hidden tunnels" led across the barrier.

From an energy point of view, the described phenomenon can be explained by the quantum uncertainty relation DE.Dt ³ h between energy and time (discussed above). The value of the instantaneous energy of a particle fluctuates in its short time around its mean size, up and down. Once a particle reaches a potential barrier, energy can (coincidentally) be momentarily increased for a short time, allowing it to cross the barrier. During the act of self-penetration, within the uncertainty relation, the law of conservation of energy may not apply. The shorter the moment of fluctuation Dt, the greater its possible range DE. If the particle does not manage to reach the other side of the barrier during this duration of sufficiently large fluctuation, it will return - it will bounce off; this, of course, happens even if the energy fluctuation is negative at the moment the barrier is reached. The wider and higher the potential barrier, the less likely the particle is to penetrate successfully; most of the particles penetrate the barrier only partially and eventually bounce off the potential wall. With a sufficiently narrow barrier, the particle is more likely to successfully overcome it due to a sufficiently high short-term energy fluctuation.
  This effect, when a particle crosses a potential barrier higher than the energy of the particle, is called a tunneling phenomenon - a particle that does not have sufficient kinetic energy (and therefore cannot "fly" over the potential barrier) can still penetrate the barrier with a certain probability, as if a hidden "tunnel" had been drilled in it. The particle, on the other side of the barrier, seemed to "tunnel". The tunneling phenomenon has a probabilistic character. It occurs either in a large number of incident particles, some of which manage to "tunnel", or the individual particle must be able to perform a series of "failed attempts" before it can "release" from binding in the nucleus (eg alpha-radioactivity) or in the material (eg thermoemission).
  Probability w of quantum tunnel passage of a particle with kinetic energy E_kin through a potential barrier of height V_o (> E_kin) and width d is approximately equal to
             w » exp ( - 2d. Ö [2m. (V_o - E_kin) / h ² ] ) .
This probability decreases exponentially with the width d of the potential barrier.
  One of the basic parameters by which we characterize physical systems is their energy content. If a system has a lower energy state available than the current one, its dynamics (acting forces) tend to bring it to this lower energy state. However, situations of more complex potential dependencies often arise, when reaching this lower energy state first requires a local ascent to a higher energy state - there is a potential barrier in the way that must be "climbed". In classical physics, such an obstacle is insurmountable and a lower state cannot be reached, the higher state is metastable. We must first "invest" some energy to overcome the energy barrier and only then can we "profit" from the released energy. The quantum system makes it possible to overcome this barrier even without the supply of external energy. What is impossible in classical physics is only less likely in quantum physics... When tunneling through a barrier, it is never possible to observe a system in a transition state with a higher energy than it had at the beginning, but only before tunneling with a given initial energy and after tunneling at the resulting state with lower energy and with the released energy in some other form (such as radiation, heat, ...). The law of conservation applies exactly to the overall energy balance!
  The tunneling phenomenon, which is typically a quantum-mechanical effect associated with the wave properties of particles, is significantly applied in many phenomena of the microworld - in atoms and atomic nuclei (eg in alpha radioactivity, nuclear reactions - especially in thermonuclear fusion), in electrical phenomena in conductors and semiconductors. In an electric field, electrons can be emitted from metals (thermoemission, photoemission) even if the kinetic energy of the electrons is lower than the corresponding output work; a tunnel scanning microscope is based on the quantum tunnel emission of electrons from the surface of conductive substances. Thanks to the tunnel effect, many processes at the microscopic level can take place even at significantly lower energies than would be necessary according to classical physics. Without it, we wouldn't have a chance to use natural energy - or a chance to live!

Schrödinger's cat - a wrong interpretation of quantum mechanics ?
It is a somewhat morbid *) and absurd thought experiment, which the Austrian physicist E.Schrödinger formulated in 1935 to show the paradoxical property of the superposition of states in quantum physics. We know that when some stochastic phenomenon is observed - measured in the micro world, it affects its state. If we have a particle, for example an electron, by measuring its position or speed we get a certain result. However, we perturbed its position and velocity a bit during the measurement itself, so the question is, what did it look like before we measured and perturbed it? Quantum mechanics offers the interpretation that when we measure a certain property of an electron (position or velocity), we get a certain result, but before we measured it, this electron existed in all possible states at once. And only the act of measurement is what forces him to accept a certain state. With this "cat experiment", Schrödinger tried to point out the absurdity of this interpretation, which, however, was later mostly accepted in quantum physics...
*) Unfortunately, the virtual morbidity of this experiment was replaced in a few years by real morbidity committed by his german compatriots, who used the same hydrogen cyanide to murder thousands of innocent people in concentration camps..!.. However, Schrödinger himself was a staunch opponent of german fascism.
The following objects are placed in a hermetically sealed opaque box :
1. Live cat; 2. Flask with poisonous gas - hydrogen cyanide (in the first version it was a rifle aimed at a cat) ;
3. A sample of radioactive material containing one radionuclide atom with a half-life of 1 hour;
4. A radiation detector electronically coupled to a mechanism capable open the flask. When the radioactive atom decays, the detection device registers it, the electrical device inside the box uncorks the flask of poison, and the cat dies.

Schematic representation of the "Schrödinger's cat" thought experiment.

The decay of a radioactive atom is a stochastic quantum phenomenon - we cannot predict the time when the nucleus will decay, only the appropriate probability; after one hour, there is about a 50% chance that the nuclide has decayed. According to the notions of quantum mechanics, a radionuclide that is not observed, is in a superposition of a decayed and an undecayed nucleus, as if it were in both states at the same time. So the whole follow-up system in the box should be in a superposition of states: [decayed radionuclide - dead cat] and [undecayded nuclide - live cat]. However, if we open the box, we will of course see only one of these states.
  The paradox is that in interaction with a suitable quantum system (radionuclide), a cat can get to a state where it seems to be alive and dead at the same time. According to quantum mechanics, the system in the box is described by a wave function that contains a combination of these two possible mutually exclusive states - that is, at every point in time, the cat is alive and dead at the same time. It is only when we make an observation (by opening the box) that we force the system to accept one option or the other. So when we open the lid of the box, external observation decides whether the cat really lives or not. The imaginary experiment points to the imperfection (incompleteness) of thid interpretation of quantum mechanics in the complex understanding of nature, in the transition between the micro- and macroworld.
  This imaginary experiment was popular especially in the period of building concepts of quantum mechanics, when it played a certain heuristic role. From today's point of view, he is no longer very convincing... Rather than the contradiction of physical reality, it expresses the paradox of its formal description. It is an erroneous, frequently occurring interpretation of quantum mechanics (cf. also the above discussion of the Copenhagen Interpretation of Quantum Mechanics "Wave functions. An interpretation of quantum physics." and their problems "Superposition of different states and collapse of the wave function: ever an erroneous interpretation of quantum physics?" ).
  In reality, the particle before observation-measurement is not in all possible states at once - it is only in one state that we do not know. Before the measurement, we can only determine the probability of what state it could be in. And after the measurement, we know the specific condition for sure.
Note: A possible bizarre solution to this paradox is sometimes considered from the point of view of the hypothesis of an infinite number of parallel worlds in which all possibilities are realized (it is discussed in §5.7 "Anthropic principle and existence of multiple universes", passage "Concept of multiple universes"). In one universe the cat is dead, in the "neighboring" universe the experiment survived ...

Quantum entaglement and teleportation. Quantum computers.
An interesting and for classical physics completely surprising consequence of the fundamental nonlocality of the quantum description of particles by means of wave functions is a phenomenon called quantum entanglement. It consists in the fact that two particles, whose quantum state is "intertwined" originally by a common wave function, remain in a sense still connected by a kind of "invisible bond", even at any distance. If the state of one of the entangled particles changes, the state of the other particle also changes, "immediately" - a kind of "teleportation" of information occurs, according to some opinions "at superluminal speed"..?.. (will be discussed below) .
  As mentioned above, the evolution of a quantum system is described by a wave function. It is a (thought.) wave propagating through space, with the object ("particle") "occurring", in a non-local sense, everywhere at the head of this wave. When an object interacts with another quantum object or measuring instrument, a "wave function collapse" occurs, and the object is temporarily localized and can be described in particle form. The collapse of the wave function takes place non - locally - the wave function, according to this interpretation, suddenly disappears from the whole space..?.. It is only a theoretical-hypothetical idea....
  According to the usual so-called Copenhagen interpretation of quantum mechanics, the investigated system consists of the quantized objects themselves and of classical measuring instruments or observers. The collapse of the wave function locates the information that the observer obtained by measuring.....
  If we have a pair of spatially separated quantum subsystems that form part of a single system, these subsystems are bound to each other through a common original wave function. The measurement (interaction) of one subsystem thus bound forces the other bound subsystem to immediately go to the corresponding (complementary) state, regardless of the spatiotemporal distance. This phenomenon is referred to as EPR-nonlocality (Einstein-Podolsky-Rosen) or EPR-paradox .
  A.Einstein and his collaborators B.Podolský and N.Rosen formulated the thought experiment outlined below to show the internal contradiction and incompleteness of quantum physics. It seems paradoxical that without the presence of exchangeable (mediating) particles or fields, it is possible to immediately influence a particle that is, for example, at the opposite end of the universe - a kind of "haunting effect from a distance"! According to the special theory of relativity, it can be expected that only places between which the space-time connection is limited by the speed of light can be in causal contact. However, quantum mechanics, due to its nonlocality of wave functions, can in a sense temporarily violate this causal requirement of relativistic physics. Now, quantum entanglement is no longer considered paradoxical. By performing measurements on one particle, no mass or energy is transferred to the other particle. And as for information, both observers must use "classical" communication using a signal with sub-light speed to confront the measured results; this ensures the STR causality of both measurements (see also the commentary to Fig.1.1.3 below).
  Mutual nonlocal interconnection or "intertwining" of quantum states is referred to in English as entanglement. It is a quantum correlated state of a system of two or more particles, in which the state of one particle cannot be measured without influencing the other (and therefore it makes no sense to talk about the states of individual particles). Both particles have a common non-local wave function.
  This can be illustrated by the example of a particle initially at rest, with zero momentum, which breaks up into two identical particles flying apart, each with a spin of 1/2. The law of conservation of the total momentum implies that if we measure a spin projection 1/2 to a certain axis for one particle, the other particle must have a spin projection to the same axis -1/2 (and vice versa). Therefore, if we measure the spin on one particle, immediately we learn (we judge) the spin value of the second particle, no matter how far. As if this information was spreading immediately, contrary to the special theory of relativity *)! From a quantum-mechanical non-local point of view, when measured on one particle, the common wave function "collapses" in the whole space (in imaginary idea?), which is reflected in the state of the other particle. However, the actual verification and reconstruction of the measured quantum state of the second particle is only possible with a classical communication channel with sublight speed!
*) Therein lies the oft-stated misconception about the superluminal speed of quantum teleportation. If we know that both particles are quantum linked and have oppositely oriented spin, measuring the spin of one particle naturally implies the opposite spin of the other particle, without the complexity of its measurement and teleportation of information. And if we don't know it, it is generally necessary to transmit information about the state of particles with a communication signal at sublight or light speed.
  Another typical example of "entangled" particles is a pair of photons generated simultaneously in a quantum process. It can be a pair of photons emitted from an atom after its excitation, the polarization of which can be correlated (in the simplest case to the opposite, perpendicular to each other). If we measure a single photon polarization e.g. in the direction of the horizontal X axis, the second photon polarization is in the direction Y. This may occur with radiation gamma produced during annihilation particle and antiparticle (see §1.2 and 1.5). Quantum entangled photons are also formed in nonlinear optical crystals by the impact of coherent monochromatic radiation from a laser, where some incident photons "split" into two bound photons with lower energy, the polarizations of which are complementary to each other.
Quantum teleportation
The term "teleportation" (Greek tele = distance , Latin portare = carry, transfer ; ie transfer, relocation in distant) often found in science fiction, generally refers to the process by which a given object (even a person in science fiction) disappears in its original place and appears in another place (in science fiction, for example, at the other end of the universe, immediately or at superlight speed). In a more sophisticated embodiment, it is an indirect relocation : the object is disassembled and analyzed in one place, the obtained complete information about its construction is transferred to another remote location, where then an exact copy is created (reconstructed) using this information.of the original object. This copy is not created from the original matter, but from particles of the same kind (eg atoms) in a new place, which are assembled into the same structure as the original object - it is not a physical transfer of objects - their matter (substance), but information transfer. And it definitely doesn't go through infinite or superlight speed!
  Quantum interconnected (entangled) particles can in principle be used for the so-called quantum teleportation of information about the state of another particle that interacts with one of them. In 1993, Ch.Bennet proposed the following indirect method (Fig.1.1.3) :

Fig.1.1.3. Simplified arrangement principle for quantum teleportation.

Let us have an O1 observer (sender's laboratory) and an O2 observer (recipient's laboratory). At the O1 observer, we create a pair of entangled particles A and B so that the A particle remains at the O1 and the B particle is sent to the O2 observer. The O1 observer then realizesthe interaction of particle A with the third particle C carrying the information (state) to be teleported; measures the resulting states of particles A and C after interaction. The original state of particle C is deleted, but thanks to entanglement, this information appears (in coded form) on the distant particle B, whose state B´ is measured by the observer O2. In order for the O2 observer correctly determined the original state of particle C which was in the place O1 , the observer O1 must connect with the observer O2 using a classical (non-quantum, causal) communication channel (eg electromagnetic signal) and tell him what result of the states of particle A and C after measured the interaction. The O2 observer then confronts the result of his measurement of the state of particle B with the data communicated by the observer O1 (using them to decode his result by linear transformations of the type of rotations in the vector base...), while the final result is to determine (reconstruct) the original state of particle C in place O1 - this corresponds to the teleportation of this information.
  Thus, both a nonlocal "EPR" channel of entangled particles and a normal (causal) communication channel are required to perform quantum teleportation. This is the only way to decode teleported information. It is this necessity of classical communication that effectively makes it impossible to send information at super-light speeds. Thus, quantum teleportation does not violate the principles of causality of the special theory of relativity - EPR is no longer a paradox...
  The process of quantum teleportation in its current understanding can only work within elementary particles and cannot be used for teleportation of macroscopic objects. There is no known way in which a set of quantum-bound states could interact with a macroscopic object in a targeted manner. In addition, this method transmits only the value of one observable quantity, not the complete quantum state.
  Quantum teleportation of the polarization state of UV photons was first performed experimentally in 1997 in the laboratory of quantum optics and spectrometry in Innsbruck. Later, quantum teleportation was performed on excited states of calcium ions ⁴⁰Ca⁺ (in the same laboratory in Innsbruck) and beryllium ⁹Be⁺ (at the National Institute of Standards and Technology in the USA). In 2017, we managed to teleport photons between the ground station and the Chinese satellite Micius to a distance of 1400 kilometers.
Quantum electronics, quantum computers
An important quantum property of particles, electrons, protons and whole atoms is spin - the intrinsic rotational momentum of a particle. Particles with non-zero spin can act as magnetic dipoles that respond to an external magnetic field. The spin of an electron is oriented in two opposite directions (spin projection) , which are referred to as upward-pointing states |á> and down |â>. When such a particle passes through a suitably configured (inhomogeneous) magnetic field, the particles with spin |á> deflects to one side, while particles with spin |â> tilts to the opposite side. They therefore fall into different places of electronic detectors. On this principle, the so-called spintronics develops - electronics, which, in addition to the charge of electrons, also uses the orientation of their spin. Digital application of the principles of this quantum electronics leads to quantum computers.
  On these remarkable quantum properties have therefore recently "rushed" some experts in the field of computer and cybernetics, who have reformulated them into their digital terminology and, together with physicists, have begun to work on the possibilities practical applications in this area. In current (already classical) digital and computer technology, the basic unit of information is "bit" - an electronic signal whose state takes two values (it is digital), which are expressed as "0" or "1". It is usually realized by two normalized well-distinguishable values of voltage in the gate circuit. Groups of combinations of these bits ("bytes", groups of 8 bits) then express the codes and numerical values of all data in the binary system.
  In quantum informatics, the so-called quantum bit or qubit (quantum bit) is introduced as the basic unit as a quantum bit version - digital units of information. While the classical bit is always in either the |0> or |1> state, the qubit also carries arbitrary values between 0 and 1 during the process - it also includes all superpositions of probabilities of these states. In the wave function, information about all superposition coefficients is carried in parallel.
Qubit state |q> is written as: |q> = A . |0> + B . |1>, where A and B are complex probability coefficients of states |0> and |1>, for which A² + B² = 1 holds.
  The complex superposition state "between 0 and 1" is implicitly contained only in the free state, without any interaction. The specific explicit value |0> or |1> is acquired by qubit only at the moment of measurement (interaction). During the interaction (when we "look" at it, detect it, decode it), the state of the qubit "flips" - in quantum terminology, "collapses" - to one side or the other, whichever is most likely at that moment.
  Quantum computers mainly use three specific phenomena from the quantum microworld: quantum superposition of states, quantum interconnectedness and interference (constructive or destructive) of quantum states. The quantum computer is based on three basic principles :
1. Technical realization of qubits ; 2. Quantum entanglement of qubits ; 3. Detection and decoding of quantum states (logical operations with qubits) .
Ad 1 :
Suitable two-level quantum-mechanical microsystems can be used for technical realization of qubits (photons, electrons, atoms). So far, four methods for realizing qubits have been tested :
¨ Utilization of photon polarization ;
¨ Use the spin of particles, especially electrons ;
¨ Use excited atoms,  especially hydrogen atoms ;
¨ Use of superconducting conductors arranged in so-called Josephson junctions (they are described in §2.5, section "Microcalorimetric detectors", passage "SQUID") .
  Several other special quantum phenomena could be used (.....), which have not yet been realized.
Ad 2 :
Quantum entaglement of qubits can be performed by mutual interaction of particles, atoms or ions. Interconnected quantum states are often realized by means of a laser pulse. If we managed quantum interconnected N qubits, the number of superposition coefficients is 2^N. Operations with these interconnected qubits are parallel (it takes place with all superposition coefficients), which gives the potential for high performance of electronic storage, transmission and analysis of information.
Ad 3 :
For the recognition and decoding of quantum states of qubits and for logical operations with them, use methods depending on the type of qubits used. For polarized photons, these are optoelectronic methods. The measurement of the plane of polarization can be performed by placing a polarization filter in the path of the photon, through which only photons with a certain plane of polarization (state |1>) pass, while photons polarized perpendicular to the plane of the filter do not pass (state |0>). Photons polarized in other planes will behave as qubits in the superimposed state - according to the angle of rotation of the polarization, the amplitude will change the probability that the photon will pass or not pass through the polarization filter. Magnetoelectronic methods are used for spin orientations.
Perspectives of quantum computers ?
The application of the laws of quantum properties in informatics and computer technology promises attractive possibilities for quantum computers that some computational tasks might be able to solve many times faster than conventional computers. Often mentioned, but in fact marginal, is the possibility of perfect quantum cryptography (protection of transmitted data using quantum keys).
  The problem is to build quantum computers with a large enough number of qubits. Large sets of particles no longer behave according to the laws of quantum physics and begin to follow the laws of classical mechanics and electrodynamics. Possibilities of creating so-called modular quantum systems are being explored : construction of many small quantum processors and their interconnection by a small number of nodal quibits, which would not disturb their quantum properties (the modules themselves remain relatively isolated from each other). Quantum chips are developed in "high-tech" microelectronics laboratories, in which pairs of entangled ions (in ion traps) or entangled photons (in optical crystals or micro ring resonators) are generated and analyzed. These could become essential elements of truly usable quantum computers.
  A big problem is the prevention of quantum decoherence of qubits, for which it is necessary to completely isolate the system from the disturbing influences of the environment, including thermal oscillations. Therefore, the operating temperature is a significant technical obstacle to the practical use of the quantum computers being developed so far. For the correct function of qubit circuits, it is necessary to cool them to a very low temperature close to absolute zero (of the order of milliKelvins), which requires a complex cryogenic technique.
  So far, quantum computers are in the stage of experimental verification and improvement of basic physical and technical principles. Manipulating qubits is incomparably more difficult than bits in electronic computers. There will be a long and difficult journey to create truly usable and powerful quantum computers ..!..
  After initial enthusiasm, many computer experts are now somewhat skeptical of quantum computers. They will definitely not be "self-saving"! They will not be universal, so they can no replace classic digital electronic computers (we will probably never have quantum PCs at home...). They will be suitable only for some special areas (eg searching for information in large unsorted data files, factorization decomposition of numbers into coefficients, fast Fourier transform, ...), where they can significantly speed up classical algorithms. Due to the quantum-stochastic nature of qubit states, quantum computers use probabilistic algorithms (individual processes are repeated a million times), with an effort for fast repairability and convergence. Whole pure quantum (100-%) computers they are not feasible - and it probably wouldn't even make sense. Rather, they will be quantum coprocessors for large specialized computing systems.

Brief recap :
What are the basic differences between classical and quantum physics
  Before we start dealing with the properties of the microstructure of matter - atoms, molecules, subatomic particles, for the sake of clarity, we will briefly recap here what the most important differences are between classical physics and quantum physics and how these differences affect our understanding of events in nature (both on earth and in space) :
-> Scales, sizes :
Classical physics deals with macroscopic objects - objects around us and their systems, their movement, forces acting between them, mainly electric and gravitational.
Quantum physics focuses on the behavior of particles on microscopic scales - molecules, atoms, subatomic particles. Their movements, mutual interactions, physical fields and their quanta.
-> Duality between particles and waves :
Classical physics deals with particles as distinct, localized and separate objects.
In quantum physics, particles can exhibit properties belonging to both particles and waves. The particles are not localized. This can cause diffraction and interference.
-> Determinism - probability :
Classical physics is deterministic - it is assumed that the future behavior of the system is precisely determined by the initial conditions in the past, and it is possible to predict exactly what the outcome will be. Randomness is considered only as a manifestation of a large number of interactions in sets of particles, within the framework of statistical physics.
Quantum physics introduces a stochastic nature - quantum fluctuations - into the description of the movement and interactions of particles. We can only predict the probability of a certain outcome, which will basically be slightly different each time, even given the same starting conditions. With multiple repetitions or a large number of particles, the various stochastic results are averaged, the fluctuations are smoothed out and, on the contrary, a very accurate result is produced.
-> Uncertainty Principle :
In classical physics, we can measure all physical quantities of particles, such as position, velocity, momentum, energy, in principle with absolute precision; apart from instrumental precision, nothing prevents us from doing so.
In quantum physics, however, the so-called uncertainty principle (Heisenberg's) is applied, according to which certain pairs of physical quantities, such as the position and momentum of a particle, cannot be measured simultaneously with absolute precision. The more precisely we measure one quantity, the more uncertain the other quantity becomes. This uncertainty here is a consequence of the wave nature of the particles.
-> Concept of physical forces :
In classical physics, each interaction of bodies and particles is assigned a corresponding field - a space in which certain forces act on the particles. In classical physics, it is an electric+magnetic and gravitational field. Changes in intensity in the field propagate with a finite speed (c), which is accompanied by a continuous transfer of energy, momentum, angular momentum.
In quantum theory, the field is modeled as a set of particles - a quantum field. And mutual force action - particle interaction - is caused by the mutual exchange of these quanta - intermediate particles. Particles constantly receive and emit quanta of intermediate fields, which causes their force action. For electromagnetic forces, the intermediate quanta are photons. In the static case, intermediate photons are virtual, under dynamic action, virtual photons are transformed into real ones, real radiation occurs. Physical quantities are transmitted discontinuously in certain "portions - quanta".
  On a macroscopic scale, quantum processes are basically negligible, classical physics remains highly accurate in the vast majority of phenomena, all mechanical and electrical machines in industry and households, cars, airplanes, space rockets work according to it. And natural phenomena on land, in the sea, in lakes and forests, in the atmosphere, the orbits of the planets around the Sun, etc.
  However, quantum physics has enabled us to understand the hidden atomic and subatomic phenomena that are the inner essence of all matter. In space, it participates in thermonuclear reactions in stars leading to their radiant energy and the creation of elements ("we are all descendants of stars"), the creation of the universe and the primordial cosmic nucleosynthesis. She also led or assisted in the development of new technologies such as transistors, integrated circuits, lasers, for the construction of advanced electronic devices. It also contributed to the construction of equipment for obtaining energy - photovoltaics and, above all, nuclear energy.
  Quantum and classical physics are thus increasingly intertwined, for the future a combined approach will be needed...

Atomic structure of matter
The question of the structure and composition of matter is one of the most basic and important questions that people ask nature - along with questions about the origin, size and structure of the universe, or questions of the origin of life. In earlier times, when humans did not have the means to gain a deeper insight into the microscopic dimensions of the interior of matter, it was by no means easy to make any credible claims about the invisible microstructure of matter. Scholars therefore resorted to various assumptions and hypotheses, formed by analogy with what was seen with the naked eye.
   In this context, a fundamental question arose: does matter have a continuous or granular structure? In other words: matter is infinitely divisible to smaller and smaller particles, or in this division do we finally come across the smallest, further indivisible isolated particles?
Intricate historical development of the concept of atoms
The Greek ancient philosopher Demokritos (5th century BC, partly followed the views of his teacher Leukippos of Miletus) was a supporter of this second possibility of the smallest indivisible particles, arguing that if matter were indefinitely divisible, it would not remain in the end, nothing that would carry the properties of the substance. Therefore, each substance must be composed of indivisible particles which carry the properties of that substance. He called these smallest indivisible particles "atomos" - Greek. "indivisible".
Note: It would be a complete misunderstanding to consider Demokritos as the discoverer of atoms or the creator of atomic theory! Demokritos knew nothing about real atoms, his opinion was just one of many speculative hypotheses, mutually equivalent at the level of knowledge at the time; other scholars at the time rightly considered matter to be infinitely divisible ...
  By the way, the above-mentioned philosophical argument about the bearers of the properties of matter would no longer stand. We know that an increase in quantity, or a combination of more quantities, can create a new quality. The properties of the system are created only by a combination of the properties of its subcomponents. And it's the same with matter. A specific substance may not have any elementary carrier of its properties - these properties are created only by a specific "construction" of a substance from particles, which themselves have completely different properties ...
   The idea of atoms then fell into oblivion for a long time. It was not until the turn of the 17th and 18th centuries, when feudalism and the church lost their absolute power, that the former alchemists - charlatan and often fraudulent "goldsmiths" in the service of the rich and powerful - gradually was replaced serious scientists who no longer wanted a recipe for making gold or various elixirs, but they tried to penetrate into the true essence of the construction of matter. It was then that the idea of the basic building blocks of matter (Descartes, Hook) began to appear again in connection with the study of the behavior of substances, especially gases (the dependence of pressure on the volume of gas). From the 18th century, with the gradual liberation from earlier alchemical superstitions and prejudices, chemistry emerged as an independent discipline. Through a series of experiments, R.Boyl and A.L.Lavoisier came to terms chemical element as a substance that can no longer be broken down into two or more other substances. During chemical experiments, important laws of chemical processes were determined :
- The law of conservation of mass and energy of substances entering the reaction and resulting reaction products in a closed system (M.V.Lomonosov 1748, A.L.Lavoisier 1774) ;
- The law of constant merging ratios (J.L.Proust and J.Dalton 1799), the law of multiple merging ratios (J.Dalton 1802) and the law of constant volume ratios (Gay-Lussac 1805), observed in reactions in the gaseous state.
  These laws became the experimental basis for clarifying the question of the internal structure of the elements. J.Dalton gave a natural explanation of all these important laws in 1808 in his atomic hypothesis, according to which each element is composed of a large number of mutually identical atoms, indivisible particles characterized by a certain characteristic mass and other properties. The atoms of the same element are the same, the atoms of different elements differ in mass and other "chemical" properties. These atoms are the basic indivisible building block of matter that participates in chemical reactions - the merging of elements consists in the joining of two or more atoms. In this concept, the law of conservation of mass in chemical reactions is an outward manifestation of the indestructibility (and also the "uncreatability") of atoms. The new bound whole, created by merging an integral number of atoms, was called a molecule (lat. Moles = mass); the name comes from A.Avogadra from 1811, who also discovered the first relationships between molecular, weight and volume amount of substances.
Note: We now know that the law of conservation of mass and merging ratios differ slightly from ideal values. This is in connection with the relationship between equivalence of mass and energy E = mc ²given by the binding energy of the reaction, the mass defect of atoms and nuclei, the difference between the mass of a proton and a neutron. These aspects will be discussed below where appropriate.
  Electrolysis played an important role in understanding the composition of substances and in discovering a number of elements: the decomposition of substances (mostly their aqueous solutions) by the action of an electric current from Volt electric cells (batteries). First it was the electrolysis of water into hydrogen and oxygen, then the electrolysis of various alkalis, acids and salts into hydrogen, oxygen, sodium, potassium, calcium, copper, zinc, chromium and other elements. An interesting question arose: "If electricity can decompose substances into elements, could it even combine elements into more complex substances?". She already anticipated later knowledge of electrical origin of all chemical reactions (see "Atomic interactions" below).
Periodic Table of the Elements
In 1869, D.I.Mendelev systematically dealt with the chemical properties of various elements. He found that the chemical properties of elements periodically depend on their relative atomic mass (atomic weight). He proposed to arrange the elements in a table, in order of increasing atomic weight into horizontal rows (forming periods) so that elements of similar properties get under each other. The definitive explanation of this Mendeleev's periodic table of elements was made possible by the development of the physics of atoms - see below "Bohr's quantum model of the atom", passage "Occupancy and configuration of energy levels of atoms" and "Interaction of atoms - Chemical merging atoms - molecules".
  A similar story, explaining the essence of the observed regularity and repetitive structure and properties at different scales is repeated in other areas of science :
- In particle physics - §1.5 passage "Unitary symmetry and particle multiplets" and part "Standard model - unified understanding of elementary particles", passage "Preon hypothesis".
- In astrophysics of stars - §4.1"The role of gravity in the formation and evolution of stars"- The Hertzprung-Russell diagram, in the book "Gravity, Black Holes and the Physics of Spacetime".
  At the turn of the 19th and 20th centuries, when a sufficient amount of experimental data from the field of chemistry and physics was collected, is was realized that pure elements are composed of "indivisible" basic particles - atoms (which bear their properties), which can combine - merge - into molecules in compounds. Next experiments in the early 20th century showed that even the atom is not an indivisible (unstructured) elementary particle, but has its own complex electro-mechanical structure. In terms of construction materials, atoms are not the last, smallest and most fundamental particles of the substance, but only one of the important hierarchical units of substance structure.

The structure of atoms
Although physics and chemistry during the 19th century increasingly convincingly showed that substances consist of atoms and molecules, the nature and structure of the atoms themselves practically nothing was known until the end of the 19th century. Experiments with electrolysis carried out by M.Faraday in 1836 have already shown that chemical compounds have a lot in common with electrical phenomena. The first significant penetration into the structure of the atom was the discovery of the electron, made in 1895 by J.J.Thomson in the study of electric discharges in gases *) and the discovery that all atoms contain electrons.
*) Electric discharges
Electric shocks in the air, known as spark jump among bodies sufficiently electrified by static electricity, have been known for a long time (for the development of knowledge about electrical phenomena, see also §1.1 "Historical development of knowledge about nature, space, gravity", passage "Electricity and magnetism" in the book "Gravity, black holes and space-time physics"). In 1743, M.Lomonosov suggested that lightning and aurora borealis are manifestations of electric discharges in the air (he was right about that lightning; aurora borealis is more about the interaction of high-energy particles from the Sun with the upper layers of the Earth's atmosphere). In the last decades of the 19th century a number of researchers have studied high-voltage electric discharges in dilute gases (as early as 1838, M. Faraday observed a strange fluorescent arc between the cathode and the anode, connected to an electrical voltage in a tube with dilute air). Glass tubes or flasks with sealed electrodes were used for this - discharge lamps, filled with air or other gases, diluted by means of a vacuum pump to a pressure of about 10^-3 atmospheric (approx. 100 Pa). The most famous were the so-called Geissler tubes, used since 1850. These differently shaped lamps, beautifully lit in different colors (according to the type of gas charge), were very attractive; they evolved into "neon" tubes. We now know that the light manifestations of an electric discharge are caused by ionization and excitation of gas atoms, caused by the impact of electrons (accelerated in an electric field), followed by deexcitation with the emission of photons.

Discharge lamps - gas-filled tubes with electrodes to which a high voltage >100 V is applied	Cathode ray tubes - very dilute gas-filled tubes with electrodes (discharge lamps - Crookes tubes), to which a very high voltage > 1-10 kV is applied

   Later, electric discharges were studied in even more dilute gases at a pressure of about 10 ^-6 atm., when the visible discharge ceases. In 1859-76, J.Plücker, J.Hittorf, and E.Goldstein observed a faint fluorescence of the flask opposite the cathode: as if some radiation came out of the cathode, hence the cathode radiation. In 1880, W.Crookes designed a special glass flask with sealed electrodes, the so-called Crookes cathode ray tube, into which various objects, screens, and minerals were inserted between the cathode and the anode. At voltages of about 1000V and higher, Crookes found that with sufficient dilution of the gas, invisible so-called cathode rays emanate from the direction of the negative electrode, causing the bulb to fluoresce in places opposite the cathode. The objects and screens inserted between the cathode and the anode cast sharp shadows in this luminescence, some of the minerals exposed to the cathode rays fluorescent. Vacuum tubes, screens, and X-rays tubes have evolved from Crookes cathode ray tubes, however, the diluted gas was replaced by a vacuum and the cold cathode by replaced by heated cathode, which thermoemission supplies the necessary electrons ("cathode rays").

1895: J.J.Thomson - discovery of electrons => first model of atom

In 1895, J.J.Thomson studied the deflection of these cathode rays in electric and magnetic fields and found that cathode rays are made up of very light negatively charged particles, whose charge corresponded to an elementary electric charge (roughly determined from Faraday's laws of electrolysis and later specified by Millikan's experiments). In this way, he discovered the first elementary particles of the microworld - electrons - and revealed the corpuscular nature of cathode rays, which are formed by a stream of fast-flying electrons. We now know that these electrons came from gas atoms ionized by the impact of other electrons accelerated in an electric field. Name electron (Greek electron = amber; static electricity was observed on amber objects in ancient Greece) comes from G.J.Stoney, who in 1891 dealt with Faraday's laws of electrolysis in connection with Dalton's atomic concept and concluded that the electric charges needed to exclude individual types of atoms are integral multiples of a certain small basic, elementary charge, representing a kind of "atoms" of electricity (electricity was until then considered a continuous "fluid"). Further experiments with cathode ray tubes led to the discovery of X-rays (§3.2 "X-rays - X-ray diagnostics").  
Note: The electrical phenomena in discharge lamps and cathode ray tubes were dealt with by a number of researchers in the given period, either independently or in mutual connection or cooperation. It is also probable that researchers "sunken patriots" who did not penetrate the general consciousness also came to new discoveries of knowledge in their remote laboratories. Therefore, arguing about the particular primacy of individual specific researchers can be problematic - and it is basically useless... It is important that by joint research they significantly contributed to revealing the laws of electricity and the microworld (cf. also the passage "Significant scientific discoveries - chance or method?" in §1.0).
  Electrons have (conventionally) negative electric charge and according to the first experiments were more than 1000 times lighter than electrically neutral atoms; we now know that electrons are 1837 times lighter than a hydrogen atom. Thus, each atom must contain a sufficient amount of positively charged mass to balance the negative charge of its electrons, and this positively charged component represents almost the entire mass of the atom. Based on these findings, J.J.Thomson proposed in 1898 the idea that atoms are a miniature homogeneous sphere of positively charged matter, into which electrons are nested - Fig.1.1.4 on the left. This Thomson model of the atom was also called the "pudding model", due to its resemblance to the English pudding with baked raisins.

Fig.1.1.4. To develop ideas about the structure of atoms.
Left: Thomson's "pudding" model of the atom. Middle: Rutheford experimental arrangement of scattering a -particles by metal foil. Right: Difference of dispersion a particles by atoms for the case of Thomson model and model of atom with nucleus.

A more detailed experimental investigation of the structure of atoms was undertaken in 1909-11 by E.Rutheford, who, together with his collaborators H.Geiger and E.Marsden, performed important experiments with alpha particle scattering (a maximum energy of 7.7 MeV emitted by the natural radionuclide ²²⁶Ra and its decay products, especially polonium) during their passage through a thin gold foil (thickness about 3.10^-4 mm, which corresponds to about 10⁴ atomic layers) - Fig.1.1.4 in the middle; alpha particles after passage and scattering of the foil are labeled as a'. These particles were observed visually by Rutheford and co-workers according to the flashes in the scintillation layer (zinc sulfide) with which the flask surrounding the irradiated foil was coated from the inside.
Note : Scattering experiments (mostly with high-energy electrons and protons on accelerators) are generally the most important method of investigating the structure of the microworld and the properties of particle interactions - see §1.5 "Elementary particles".
  According to Thomson's model of the atom, it was expected that heavy and fast alpha particles would easily "pierce" the thin gold foil - they would pass through the foil either directly or with only a small scattering (Fig.1.1.4 at the top right); the uniformly sparse distribution of charge and mass inside the "pudding" atom causes only weak electrical forces when passing heavy alpha particles. The passed alpha particles should then leave their light traces only on a small area on the back of the flask, in a direct direction from the emitter.
  However, the experiment showed, that in addition of this, a number of particles a' scattered by a large angle, some were even reflected in the opposite direction - Fig.1.1.4 in the middle *). In order for heavy alpha particles (more than 7,000 times heavier than an electron) moving at high speed (almost 2.10⁷ m/s) to disperse in this way, large forces had to act on them inside the atoms, which would not be possible with the Thomson model with a relatively light, sparsely dispersed positive mass in which light electrons are embedded. Although most alpha particles easily penetrated by the fringes of atoms, some of them had to bounce off "something" small, heavy, and positively charged inside the atom.
*) Most flashes, as expected, appeared on the back of the flask in a straight line from the emitter, which corresponded to the passage of alpha particles through "gaps" between atoms, far from the nuclei. However, the particles passing through the inner part of the gold atoms showed considerable angles of deflection.
  To clarify these experimental results, Rutheford abandoned Thomson's model and proposed an image of an atom composed of a very small nucleus (less than ten thousandths of the diameter of the whole atom *), in which the positive charge and almost the entire mass of the atom are concentrated, and from electrons located at a certain (relatively large) distance from the nucleus. Namely nearby of this extremely small, heavy, and the positively charged nucleus, around which, according to the Coulomb law, very high electric field intensities acts, the alpha-particles that fly just around the nucleus are effectively scattered (Fig.1.1.4 bottom right). If the alpha particle flew relatively far from the nucleus, it did not disperse almost at all. The closer the path a- particle approached the nucleus, the more it dissipated (due to the greater electrical repulsive forces).
*) Later measurements showed that the nucleus is even 100,000 times smaller than an atom (!) and clarified the structure of the atomic nucleus (described below in the "Atomic Nucleus" section).
  However, the electrons in this Rutheford model of the atom cannot be at rest, because the electrostatic force would draw them to the nucleus and the atom would collapse - they must move, orbit the nucleus *) along paths where the electric attractive force is balanced by centrifugal force, analogous to this is the case with planets in the solar system - a planetary model.
*) In quantum mechanics, an alternative explanation of the structure of atoms is sometimes given : electrons cannot fall on the nucleus due to quantum-mechanical uncertainty relations (discussed above in the "" section) and the fermion character of electrons. Electrons cannot acquire a smaller distance, or a lower energy level in the electric field of the nucleus than the lowest basic one; if we tried to "push" them even closer to the nucleus, they would "defend" themselves with an intense repulsive force - electrons as if their wave nature "did not fit " into such a small space - as if they suffered from a "claustrophobic effect". The so-called Fermi pressure of degenerate electrons arises here, which counteracts the electrical attraction of the nucleus. Electrons in the atomic shell according to Pauli's exclusion principle "pair" into pairs with opposite spin in areas of space called orbitals and the quantum-mechanical oscillations resulting from corpuscular-wave dualism prevent them from occupying smaller volumes. In our interpretation, however, for better comprehensibility and continuity between classical and quantum physics, we will stick to the usual "planetary" explanation - the orbit of electrons around the nucleus.
  Only in the quantum model of the atom will we use corpuscular-wave dualism to explain the quantization of electron orbits (Fig.1.1.6) and use the "claustrophobic effect" of quantum uncertainty relations to explain the repulsive forces in the nucleus at subnuclear scales (see below "Strong nuclear interactions").
All nature is almost absolute emptiness ! 
Our whole nature is only emptiness, "polluted" by an almost negligible amount of matter. Everything around us is made up of only a small amount of real - "solid", concentrated - matter. This somewhat paradoxical statement clearly follows from the knowledge of the structure of atoms. An atom is not some solid mass ball, but it consists of a very dense nucleus of only 10^-13 cm and an almost empty electron package. The nucleus, bearing more than 99.9% by weight of the atom, is about 100,000 smaller than the whole atom. This can be compared to a large sports stadium (representing an atom), in the center of which is a small children's ball representing the core; a few electrons would circle on the area of the stadium and the rest would be completely empty. As can be seen from Fig.1.1.4 on the right, the energy particle, in collision with the atom, in most cases flies through it as if through empty space; only if it accidentally hits the nucleus will there be a reflection or interaction.
  Thus, an atom is actually an empty space, "polluted" by several protons, neutrons and electrons. About 99.98% of each atom is an empty vacuum. Even our body, which is built of these atoms, is mostly formed by emptiness: the whole "real" mass of our body could theoretically be compressed into a ball with a diameter of about 1 mm, the rest would be emptiness. The same emptiness forms and all objects around us ...
Matter is mainly the field  
Thus, if atoms are predominantly empty space, at the mechanistic point of view there is a paradoxical question: "How is it possible that the most individual material objects do not penetrate with each other? Why do we see clear boundaries fixed objects?"- why don't we go trough the wall, or why don't we "penetrate the wood" while sitting and fall through the chair due to the force of gravity? The answer to this paradoxical question of why the material normally does not pass, is the electromagnetic field - a force evoked by charged protons and electrons in atoms. If the bodies get close to each other, their atoms (due to the deformation of the electronic configuration) begin to electrically repel each other and under normal circumstances there can be no interpenetration of atoms. When sitting on a chair, we are actually hovering ("levitating") over the upper layer of atoms of the chair's material, on the "pillow" of the electric field. The "penetration" of atoms occurs only at higher forces and energies, as discussed below in the section "Interaction of atoms".
  It is similar with weight. The sum of the rest masses of the basic building blocks - quarks and electrons - of our body would represent only about 1% of the mass of our body (§1.5, passage "Quark structure of hadrons"). The predominant part of the mass of our body (and of each material object) is formed by the kinetic energy of the building particles and the energies of the fields, according to the relation E = mc².

Planetary atomic model
E.Rutheford therefore based on the above-mentioned scattering experiment with alpha particles during their passage of thin metal foils, formed the first realistic model of the atom - now commonly known planetary model according to which the atom consists of a positively charged core, around which orbit the negatively charged electrons (Fig.1.1.5 - where, however, the planetary model is already draw in an improved version of Bohr's model). The attractive electric force acting under Coulomb's law between the negative electrons and the positive nucleus is balanced by the centrifugal force created by the circular circulation of the electrons.
  For the motion of an electron with charge -e and mass m_e in the electric Coulomb field of a nucleus with charge +Z.e (Z is an atomic number, now called a proton number- see below "Atomic nucleus"), according to 2. Newton's law of force and Coulomb's law of electrostatics, the equation of motion apply
     m_e.d²r/dt²   =  F  =   -(1/4p.e_o).(Z.e²/r²).r^o   ,
where r is the position vector from the nucleus to the electron location, r is the instantaneous distance of the electron from the nucleus, r^o is a unit radius-vector pointing from the nucleus to the electron. The nucleus is considered to be motionless and infinitely heavy compared to the mass of the electron m_e . This equation of motion expresses the motion of an electron in the central field of the nucleus along Keppler orbits (generally an ellipse, hyperbola, parabola), similar to the motion of planets in the central gravitational field (for a detailed mathematical analysis in §1.2 "Newton's law of gravitation" in book "Gravitation, black holes and space - time physics"). In the simplest case of a circular orbit path of radius r we get a simple equation of motion
     m_e.v²/r  =  (1/4p.e_o).Z.e²/r²  ,      
indicating the orbital velocity v the electron depending on the radius of orbit r. This basic equation of the planetary model of the atom can also be easily obtained as a condition of the balance of the centrifugal force m_e .v²/r, acting on the electron in a circular motion, and the attractive electric force (1/4p.e_o).Z.e²/r² of nucleus from the Coulomb law.
  However, the original planetary model had the drawback of being in conflict with classical electrodynamics: According to Maxwell's equations of electrodynamics, every electric charge moving with acceleration, emits electromagnetic waves. So every electron orbiting the nucleus (circular motion is uneven - the direction of the velocity vector changes - centripetal acceleration) should generate a periodically changing electromagnetic field, which would be manifested by the emission of electromagnetic waves carrying away the kinetic energy of the orbiting electron - see §1.5 "Electromagnetic field. Maxwell's equations.", Larmor formula (1.61'), monograph "Gravity, black holes and space-time physics". An electron braked in this way would orbit in a spiral and fall closer and closer to the nucleus, the intensity and frequency (equal to the frequency of the circular motion of the electron) of the radiation would increase, until the electron finally hit the nucleus *). Such an "electric collapse" the planetary atom would run very fast, in about 10^-10 seconds for the hydrogen atom.
*) By substituting into the mentioned Larmor radiation formula -(dE/dt) = (2/3).(1/4p.e_o).q²a²/c³ of the charge of the electron q = e and the acceleration of its circular motion a = v²/r = (1/4p.e_o)Z.e²/m_e.r² we get for the time change of the radius of circulation r (decreasing r , descending along a spiral) differential relation dr/dt = -(4/3).(1/4p.e_o).(Z.e⁴/m_e²c³r²). By integrating its inverse form from r(t = 0) = r_at - original radius of the atom r_at » 10^-10 m in the initial time t = 0, to r (t = t_col ) = r_nuc - impact on the nucleus of radius r_nuc » 10^-14 m, we get for the collapse time t_col the value t_col = (4p².e_o²m_e²c³/Z.e⁴).(r_at³ - r_nuc³) » 10^-10/Z [sec].
  Fortunately, we do not observe anything like that - atoms exist here and are stable! In addition, atoms with electrons in different orbits would emit different frequencies of electromagnetic radiation continuously, in contrast to experimentally observed discrete spectra of atomic radiation composed of individual spectral lines of precisely given wavelengths (frequencies and energies) characteristic of different atoms (see below "Radiation of atoms") .

Fig.1.1.5. Schematic ilustration of the planetary structure of an atom in which negative electrons orbit a positively charged nucleus. According to Bohr's model, electrons orbit the nucleus only along quantized discrete orbits on which they do not radiate. When an electron jumps from a higher to a lower path, the corresponding energy difference radiates as a quantum (photon) of electromagnetic radiation.

Is it an atom ? Looking at this picture, almost every educated person says that "it is an atom". This answer is only partially correct, because in reality it is only a model of the atom. If we could reduce ourselves in the sci-fi concept to the size of one picometer and penetrate the atom, we would not see any "balls" - electrons orbiting another "ball" - nuclei. At most we could see only fluctuating and rippling fields, with different densities distributed around the orbitals.   However, this drawing is quite apt (cf. the passage "Ball Model" in §1.5 "Elementary particles "), can clearly display and understand most important processes in atoms - excitation and deexcitation, emission of photons of characteristic radiation, chemical fusion of atoms, processes of interaction of atoms with ionizing radiation, internal conversion of gamma and X radiation with emission of conversion and Auger electrons, other accompanying phenomena in radioactivity... That's why we will use it often.

Bohr's quantum model of the atom
The mentioned serious shortcomings of the planetary model of the atom was repaired in 1913 by the Danish physicist Niels Bohr, who based on experimental knowledge and in the spirit of the ideas of the emerging quantum mechanics supplemented the original planetary model of the atom with three important postulates :

• 1. Bohr's postulate :
  An electron cannot orbit the nucleus in arbitrary paths, but only in fixed (quantized) paths with precisely determined discrete values of radius. There is a basic state of the atom - a stable state of the atom with the smallest radius of circulation and the lowest possible energy.
  According to the above-mentioned laws of quantum mechanics, this selection rule can be formulated as follows: The angular momentum of an orbiting electron cannot take on arbitrary values, but only on such values that are integral multiples of the number h/2p .
• 2. Bohr's postulate :
  Such orbits (electron orbits) are stationary and the electron does not emit electromagnetic waves in orbit on them .
• 3. Bohr's postulate :
  Only during the transition from a higher orbit to a lower one is a quantum of electromagnetic waves (photons) emitted, carrying away the energy difference between the two orbits.

The planetary model, supplemented by these three postulates, represents the famous Bohr model of the atom (Fig.1.1.5), which successfully explains the most important quantum properties of atomic structure, including discrete line spectra of radiation emitted by atoms (see below). Bohr's model has retained its validity to this day (with the relevant generalizations mentioned below).

The Atom and the Planetary System: Similarities and Differences
After discovering that the atom is a system of positively charged nuclei and negatively charged electrons bound by an electric force, the well-researched solar system bound by gravity became the inspiration for clarifying the structure of this atomic system. There is an obvious analogy on three points :

• Both electric and gravitational forces decrease with the square of the distance ;
• Both the attractive gravitational force and the attractive electric force (between charges of the opposite sign) can be permanently compensated in a vacuum by centrifugal force during orbital motion ;
• The same Kepler's laws apply to motion in the central gravitational and electric fields .

Based on these analogies, Rutheford's planetary model of the atom was created. However, there are also fundamental differences between the planetary system and the atom :

• The difference in the properties and strength of electric and gravitational forces. While the orbits of the planets are stable for a long time *), the orbital motion of an electron in an atom according to Maxwell's electrodynamics results in the intense emission of electromagnetic waves, which quickly carry away the kinetic energy of the orbit.
  *) According to the general theory of relativity, gravitational waves are emitted even during the orbit of the planets, but their energy is completely negligible and does not affect the orbits for many millions of years.
• Huge difference in size and weight. The planetary system (masses »10³⁰ kg, diameter »10⁸ km) can be fully described by Newton's classical mechanics, while the atom (diameter »10^-8 cm) is typically a quantum system.

  These differences have forced Bohr's above-mentioned modification of the planetary model of the atom. Nevertheless, the planetary concept of the atom is still used in some illustrative qualitative considerations...
  One of the main differences between the classical electromechanical and quantum understanding of atoms is the mechanism of radiation of atoms. Radiation from atoms is not emitted continuously, but after quantum, and the frequency of the radiation f not given by the frequency of a periodic electron circulation, but the energy difference E of stationary orbits of electrons, combined with the relationship E = h.f between the energy of electromagnetic quantum (photons) and the frequency f corresponding electromagnetic waves (cf. above-mentioned "Corpuscular-wave dualism").
Atoms are very empty !
If we compare the typical size of an atom 10^-8 cm - the orbits of electrons - and the size of a dense nucleus 10^-13 cm, we see how extremely empty the atom is: as much as 99.9999999999999% of the volume of the atom is empty space (vacuum)...

Why don't atoms glow at rest? - wave mechanism of quantization
The mechanism of quantization in Bohr's model of the atom can be most clearly understood by the idea of the corpuscular-wave behavior of the electron as it moves in orbit around the atomic nucleus. We will first consider the simplest case - the hydrogen atom.
  From a corpuscular point of view, an electron of mass m_e and charge -e, orbiting a proton of charge +e along a circular path of radius r with velocity v , is acted upon by a centrifugal force F_C = m_e.v²/r and Coulomb attractive electrostatic force F_E= (1/4p.e_o).e²/r². The condition of equilibrium (stability) of the path is then F_C=F_E, e.g. m_e.v²/r = (1/4p.e_o).e²/r², from which the relations for the radius of the path and the orbital velocity of the electron follow

However, if these relations are fulfilled, the electron could orbit according to classical ideas at any distance r from the center of the atom.
  From a wave point of view, a circulating electron can be considered as a wave whose Broglie wavelength is l = h/m_e.v . In order for such a "electron wave" to orbit continuously along a path of radius r , an integral number of wavelengs l must be "deposites" on this path, ie either one complete Broglie electron wave l , or 2 wavelengths per circumference 2pr, 3l/circumference, 4l/circumference, etc. - Fig.1.1.6, in above. Only then do all the electron waves stack and follow each other smoothly along the entire circumference of the path. If less than a few wavelengths occur along the path (Fig.1.1.6 in below part), the wave continuity is broken and the path is not stable, there will be discontinuity and disturbing interference, which is formed into a quantum of electromagnetic radiation - a photon is emitted, which carries the appropriate amount energy and the electron goes to the nearest stable orbit with an integer number of Broglie wavelengths.

Fig.1.1.6. Above: An electron orbits a nucleus along a stable orbit indefinitely and without radiation, if its orbit contains an integer number n of Broglie wavelengths of the electron. Bottom: With a non-integer number of wavelengths, the "wave continuity" is broken and the orbit is unstable - a photon is emitted and the electron goes to a stable orbit with an integer number of wavelengths.

A circular path of radius r has a circumference of 2p r, so the condition for the stability of the path is
                2 p r _n = n. l , n = 1,2,3,4, ..... ,
where r_n denotes the radius of the path which contains n wavelengths l = h/m_e.v. Substituting for the orbital velocity from the planetary model v = e/Ö(4p.e_o.m_er) we get that only those electron paths are stable, whose radius is given by the relation

For n = 1 we get the lowest (basic, unexcited) orbit of an electron in a hydrogen atom with radius r₁ = 0.529.10^-8 cm. This value is called the Bohr radius and is also considered as one of the model values of the electron radius r_e (see the discussion in §1.5, passage "Size of elementary particles ...").
   The integer n is called the main quantum number and determines not only the order of the "allowed" quantum path, but also the energy of the electron in a given quantum path: The total energy E of an electron in orbit is given by the sum of its kinetic energy E_k = (1/2) m_e,v² and the potential energy v = e/Ö(4p.e_o. m_er) in the Coulomb electric field of the nucleus (we choose the zero potential point at infinity; the sign "-" means that the force acting on the electron is attractive). Thus E = E_k + E_p = m_e.v²/2 - e²/(4p.e_or), which after substituting v = e/Ö(4p.e_om_er) gives E = e²/(8p.e_or). For the allowed paths of the orbital radius r_n they then discrete values of energy E_n are obtained :

which are referred to as energy levels or shells. These levels are all negative (related to the fact that we have chosen the potential of the electrostatic field to zero at infinity), which means that the kinetic energy of the electron in the quantum path is not enough to free the electron from the attractive force of the nucleus and escape from the atom. The absolute value of electron energy |E_n| indicates the work (energy) that we would have to supply the electron to transfer it from a given quantum path n to infinity, ie to free it from the attraction of the nucleus, ie to release it from the atom.
   If an electron orbits on the lowest quantum path n = 1, we say that it is in basic (unawakened) state. The transition to at higher quantum path is possible only by supplying energy - excitation of the atom, which may occur either photon absorption or action of Coulomb electric forces when passing charged particle impact or another atom (e.g at higher temperatures). During the transition from this higher energy level n to the lower energy level n-1 , ie during deexcitation , the energy difference is emitted in the form of a quantum (photon) of electromagnetic waves of energy E = E_n-1-E_n and wavelength ... ..
If the energy supplied to the electron is higher than the binding |E_n|, the electron is released from the field of the nucleus and flies out - ionization of the atom occurs.

An improved Bohr model ; quantum numbers
The original Bohr model applied to a hydrogen atom and considered only circular orbits of electrons. With the improvement of experimental spectrometric methods, it has been shown that the spectral lines of atoms are not simple, but double and multiple - the spectra show a fine structure. To explain this fine structure, Bohr's followers, especially A.Somerfeld, supplemented and perfected Bohr's original model of the atom.
   In addition to circular orbits, elliptic orbits of electrons with a longer (major) half-axis given by the principal quantum number n have been proposed, while the minor (shorter) half-axis is characterized by a second quantum number l , which can take discrete values from 0£ l £ n-1. This quantum number 1 , formerly referred to as the minor quantum number, is now called the orbital quantum number and determines the magnitude of the angular momentum M₁ of the electron in a given orbit. Quantum mechanical analysis gives a quantum value for the angular momentum :
              M _l  = (h / 2p) . Ö[ l (l-1) ] , l = 0, 1, 2, ......, n-1 .
The duplication and fine structure of the spectral lines can then be explained by the transitions between energy levels with different quantum numbers n to different sub-levelss differing in the value of l , on which the total energy E depends only little.
   An electron orbiting at a velocity v along a circular path of radius r represents, from an electrical point of view, a miniature current loop flowing through an electric current I = e.v/2p r (the v/2p.r indicates how many times an electron with charge e has passed through a given path point per unit time). This current loop generates a magnetic field and its magnetic moment is m_e = pr².I = r.e.v/2 = (e/2m_e).m_e.r.v = (e/2m_e).M, where M is the orbital angular momentum of the electron. Since the angular momentum M is quantized (M_l = l.h/2p, l = 0,1,2, ......, n-1), the orbital magnetic moment of the electron m_e on a given quantum path is
          m _e  = m _l . e.h / 2m _e  = m _l . m _B , m _l = 0, ± 1, ± 2, ....., ± l ,
where m_l is the magnetic quantum number and the constant m_B is called the Bohr magneton - represents the smallest, elementary quantum of magnetic moment.
   In addition to the orbital magnetic moment caused by the movement of an electron in orbit, the electron also has its own so-called spin magnetic moment and its own "rotational" angular momentum - spin.
These properties are often simply explained by the rotation of an electron around its own axis - a rotating electron would have its rotational angular momentum and corresponding magnetic moment. However, this explanation is not entirely consistent, as the "circumferential speed" of the electron would have to significantly exceed the speed of light (contrary to the special theory of relativity) and it would not be possible to explain what force compensates for the enormous centrifugal force and holds the electron together. Spin must be considered as a purely quantum property of a particle, for which we do not have an exact classical model.
   For own angular momentum, i.e. the spin of the electron, then is hold that its projection to the axis of rotation can take only two values: either -¹/₂ h , or +¹/₂ h; the spin magnetic moment of the electron is then given by the Bohr magneton: ± m_B. For the momentum of the electron M_s and the spin magnetic moment m_s the following applies: M_s = s . h , m_s = -(e/m_e).M_s = ±m_B, where s = 1/2 or -1/2. The number s is called the spin number, and in the electron can take values ±¹ / ₂ (in §1.5 "elementary particles" encounter with particles, e.g. mesons p , for which three values of the spin number are possible: -1, 0, +1).
   The interaction between the magnetic fields excited by the spin and orbital angular momentum of electrons, the so-called spin-orbital interaction, leads to the splitting of energy levels of electrons in atoms into nearby "subsurfaces", which is reflected in the spectra of radiation from atoms by splitting spectral lines into fine structure.
E.g. for hydrogen, the lowest energy level of the electron n = 1 is split into two subsets with the consensus and dissent spin of the electron and proton. The transition between these two states corresponds to the absorption or radiation of electromagnetic radiation with a wavelength of 21 cm. The emission and absorption of this atomic hydrogen radiation is very important in radio astronomical observations of outer universe.

a - the constant of the fine structure
For the construction of atoms (as well as atomic nuclei), the force with which particles interact with electromagnetic fields is of particular importance. In general, this force is expressed by Coulomb's law of electrostatics and Lorentz force acting on a charge moving in a magnetic field. In quantum physics, where the electrical charge is quantized in multiples of the elementary electron charge e , performs an interesting ratio that represents electrical, quantum and relativistic properties of electromagnetic interaction of charged particles in a vacuum: it is called fine structure constant *)
                         a = e ² /2 e _o h c = 0.0072973525376 = 1 / 137.0359996868 ,
where e is the elementary charge of the electron, h is Planck's constant (reduced), c is the speed of light, e is the electric permittivity of the vacuum. The constant of a fine structure is a dimensionless quantity, its numerical value does not depend on the choice of units.
*) The name comes from the fact that this constant appears in the relations for the splitting of spectral lines of radiation of atoms into a fine structure due to the so-called spin-orbital interaction.between spin and orbital angular momentum of electrons, resp. between the magnetic fields excited by them. This constant was first used in 1916 by one of the pioneers of atomistics, A. Sommerfeld, in the study of the fine structure of electron levels in an atom. He extracted this dimensionless value from earlier Rydberg constants expressing the wavelengths of spectral lines at electron jumps between levels in an atom and interpreted it as a measure of relativistic deviation of spectral lines about Bohr's model (ratio of velocity v₁ electron in the first orbit of Bohr's hydrogen atom in vacuum a = v₁/c).
   This important physical constant characterizes the strength of the electromagnetic interaction, acting as coupling constant in quantum electrodynamics. It co-determines the properties of atoms, molecules and substances composed of them, as well as the properties of atomic nuclei, including nuclear reactions. Possible variability of basic natural constants during the evolution of the universe is sometimes discussed , and the fine structure constant could be a suitable tool for sensitive spectrometric analysis of radiation from outer space (see also the passage "Origin of natural constants" in §5.5 "Microphysics and cosmology" monography "Gravity, black holes and space - time physics").

Occupancy and configuration of electron levels of atoms
Let us now move from a hydrogen atom to more complex atoms with more electrons. Imagine that we have created a nucleus with Z protons and we place it in a space containing free electrons. By electric forces, this nucleus will attract electrons, which will gradually occupy the individual "allowed" quantum orbits around the nucleus until an electron shell formed by Z electrons is formed and the atom becomes electrically neutral.
Electron orbits, shells, levels, orbitals 
The electron orbits are quantized, so from an energetic point of view it would be most advantageous, if all electrons occupied the lowest energy level with the main quantum number n = 1. However, such "crowding" of electrons to one level does not take place. Conversely, according to the so-called Pauli exclusion principle *), only one electron can be in the same quantum state, so if the lowest energy levels are occupied, other electrons must occupy ever higher and higher levels.
*) This exclusion principle was derived by the Swiss physicist W.Pauli in 1925 on the basis of a series of experimental studies of the distribution of electrons in atoms. Later, this exclusion principle was theoretically justified as a consequence of the quantum-statistical behavior of particles with antisymmetric wave functions (relative to particle transposition) - the so-called fermions, which include also electrons (see §1.5 "Elementary particles").
   Using Pauli's exclusion principle, we can determine how many electrons can simultaneously orbit in the paths (subshells) corresponding to the main quantum number n : there are possible n-1 values of the orbital quantum number l , with for each l there are 2.1 +1 different values of magnetic quantum number m_l and two possible values of spin magnetic number m_s (+1/2, -1/2). Thus, each subshell can contain a maximum of 2.(2.1 + 1) electrons and each shell with the main quantum number n a maximum of n-1 of these subshells, ie a maximum total of
                _{l = 0} S^n-1 2. (2.l - 1) = _{l = 0} S ^n-1 4.l + 2.n = 4. (n-1) .n / 2 + 2n = 2 n ²
electrons. Said set of electrons forms the n-th shell (sphere, level) of the atom. These energy levels, corresponding to the discrete values of the principal quantum number n , are denoted by letters (in the direction from the inside of the nucleus): K, L, M, N, O, P. The number of electrons that can orbit at a given level is therefore not arbitrary, but is limited by the maximum number 2n² :
K (n = 1): max. 2 electrons, L (n = 2): max. 8 electrons, M (n = 3): max. 18 electrons,
N (n = 4): max. 32 electrons, O(n = 5): max. 50 electrons, P (n = 6): max. 72 electrons.
Electrons occupy paths gradually, starting with the K shell.
   This system of occupying electron shells and subshells in atoms, together with the analysis of the binding electric force of electrons, makes it possible to understand the most important laws of the chemical behavior of elements. If we sort the chemical elements in order by atomic number, elements with similar chemical and physical properties are repeated at regular intervals. This empirically determined periodic law was formulated by D.I.Mendelev in 1869 in his periodic table of elements, which was supplemented by his followers and specified in its current form (see below the section "Interaction of atoms", passage "Periodic chemical properties of atoms").
   An analysis of the electron configuration mainly shows, that the electrons in a fully occupied shell, termed closed shell, are strongly bound, because the positive charge of the nucleus significantly exceeds the negative charge of the internal electrons causing electrically "shield". Distribution of the effective charge in of an atom containing only closed shells is perfectly symmetrical, the atom has no dipole moment, does not attract other electrons and its own electrons are strongly bound, such atoms do not enter chemical bonds, are chemically inert - this manifests itself in helium ²He, neon ¹⁰Ne, argon ¹⁸Ar, krypton ³⁶Kr, xenon ⁵⁴Xe, radon ⁸⁶Rn *).
*) The fully occupied sphere K is for helium and the fully occupied sphere K and L for neon. However, in the case of heavier inert gases Ar, Kr, Xe, Rn, the filling of the outer shell is only 8 electrons, which is related to the dependence of the binding energy on the orbital quantum number, as a result of which the filling of some subshells can become energetically disadvantageous.
   In contrast, atoms with one electron in the outer shell easily lose this electron because it is weakly bound: it is relatively far from the nucleus, whose charge is shielded by internal electrons to an effective value of only +e - this explains the high chemical reactivity of alkali metals (and also hydrogen) with valence +1. On the contrary, atoms that lack one electron in the outer shell for closure try to obtain this electron by the attractive force of an incompletely shielded nuclear charge, which explains, for example, the increased reactivity of halogens. For chemical reactions, see the section "Interaction of atoms" below, passage "Chemical fusion of atoms - molecules".
Orbitals 
Due to the wave-stochastic laws of quantum physics, electrons in the atomic shell do not orbit along a predetermined path (precise orbit), but it is possible to determine only the area in which the electron is located with a certain probability. The region in which the electron is most likely to occur (> 95%) is called the orbital. The electrons in the shell are grouped into finer spatial configurations, orbitals, depending on the minor (orbital) quantum number l , which determines the type of orbital and the value of the magnetic moment of the electron (discussed above). In a given shell, according to the increasing minor quantum number, the orbitals are denoted sequentially by the letters s, p, d, f. On the basic shell n = 1 there is only one orbital 1s, on the shell n = 2 there are orbitals 2s and 2p, ..., on the shell n = 4 can be orbitals 4s, 4p, 4d, 4f respectively. Due to the quantum exclusion principle, electrons gradually occupy the s- orbital shell K, followed by s- and p- orbitals of the shell L, s- , p- and d- orbitals of the shell M, etc ... Each orbital is filled with one and then the other electron with opposite spins. The way orbital is occupied is sometimes called Hund's rule. Orbitals can be differently oriented in space, according to the magnetic quantum numbers m (takes values -l, ..., 0, ... + l). These orbitals of the same type, which differ only in spatial orientation, have the same energy - they are energetically degenerate.
   When an atom is placed in an external magnetic field, the originally uniform energy of the electrons within the orbital spreads to several different (though close) energy levels, depending on the different orientations of the orbitals. The motion of each electron inside the shell creates a magnetic moment (the vector of which depends on the spatial arrangement of the orbitals), to which a force acts in the external magnetic field. This interaction leads to a fine splitting of the energy levels of the electrons. A very fine distribution of energy levels inside the orbitals also occurs due to the opposite spins (+1/2, -1/2) of electrons  (Stern-Gerlach experiment).
Designation of atoms of elements 
In connection with the structure of atomic nuclei (see section "Atomic nuclei" below), atoms are characterized by two basic parameters :
- Proton number Z (formerly called atomic number A), indicating the number of protons in the nucleus - and for a neutral (non-ionized) atom, also the number of electrons in the envelope. This also gives the position of the element in Mendeleev's periodic table of elements.
- The nucleon number N (formerly called the mass number) indicates the total number of nucleons in the nucleus of an atom - the sum of protons and neutrons. Characterizes weight of atom (since the nucleons in the nucleus represent more than 99.99% by weight of the atom).
   A commonly used way to write these numbers in certain element X is via the upper and lower index: ^NX_Z. E.g. hydrogen ¹H₁, nitrogen ¹⁴N₇, sodium ²³Na₁₁,... It will be discussed in more detail below in the "Atomic Nuclei" section.

Excitations and spectra of atomic radiation
According to Bohr's model, electromagnetic radiation is generated in the electron shells of atoms when electrons pass from higher levels to lower ones. Since the energy levels of atoms are quantized, photons of radiation with very specific energies are emitted from the shell of the atom - the spectral distribution of energies and wavelengths is not continuous, but discrete.
Excitation, deexcitation and ionization of atoms
Excitation of atoms
In order for an electron to transition from a higher to a lower level, the atom must first be supplied with energy leading to its excitation *) - to transition the electron to a higher energy level. This energy can be supplied either by a Coulomb electromagnetic interaction of an incoming charged particle (electron, proton, collision with another atom), or by photon radiation.
*) We are considering "already finished" atoms here, not a situation where atoms are just emerging - the formation of atoms is, of course, also accompanied by quantum excitations and radiation.
   Under normal circumstances, electron levels in atoms with the principal quantum number n are mostly excited to values of n+1, n+2 or n+few values of the principal quantum number. However, with the help of strong pulses of the electric field, electrons can also reach highly excited states with values of n~100 and more. Such atoms with highly excited electrons are sometimes called Rydberg atoms (after J.R.Rydberg, who measured the spectra of excited atoms, especially hydrogen, at the beginning of the 20th century). The diameter of such atoms already reaches macroscopic dimensions of the order of micrometers. Using sophisticated methods with a combination of laser and microwave radiation, it was possible to prepare excited atoms with dimensions of almost one millimeter!

Strong electrical impulses or radiation can cause electrons in atoms to reach highly excited states - Rydberg atoms are created.

An important property of Rydberg atoms is their higher sensitivity to the electric field gradient. This makes it possible to "operatively" manipulate atoms in demanding atomic and particle experiments - see e.g. §1.5, passage "Artificial production of antimatter".
   Excited atoms themselves are usually very unstable. They deexcite either one-time to the ground state with the emission of visible or UV light, or jumps between higher levels with the emission of microwave radiation. The Rydberg valence electrons are excited into high orbitals considerably distant from the nucleus. A somewhat unexpected consequence of the high excitation of electrons to large values of the principal quantum number is the extended lifetime of the atom. This is due to the fact that an electron can only absorb or emit a photon with an appropriate energy equal to the difference between the levels (in the case of Rydberg electrons, these are energies in the deep infrared region). The lifetime of a Rydberg atom is proportional to n⁴.
   Highly excited Rydberg atoms can bind to each other by sharing electrons when they approach and form a regular arrangement into hexagonal planar clusters, the so-called Rydberg mass. During the formation of this cluster, valence highly excited electrons are delocalized and trapped in the potential wells created between the atomic nuclei. This confinement prevents electrons from leaving the cluster and causes the Rydberg matter to have a long lifetime. Another property is the very weak interaction of this matter with electromagnetic radiation - its "darkness". These properties make Rydberg matter a certain candidate for dark matter in the universe, or for some part of it (§5.6, section "Future development of the universe. Dark matter" in the book "Gravity, black holes ...).

Deexcitation of atoms
To dexcitation of the atom and emission of radiation then has mostly occurs spontaneously (according to some concepts of quantum field theory, spontaneous deexcitations and emission of radiation is initiated by constant quantum fluctuations of vacuum). Electrons in atoms can only go over between existing discrete energy levels. Even here, however, there are certain limitations caused by the law of conservation of angular momentum. The photon carries the released angular momentum difference between the respective levels through its spin, which is equal to 1. The transition between levels whose angular momentum differs, for example, by 1/2, is therefore not possible by photon deexcitation - we say that this transition is "forbidden". If an electron occupies such a higher energy level, it cannot spontaneously go into a lower (basic) energy state - it remains "trapped" at a higher level for a longer time: such a state is called metastable or isomeric (atoms that are stuck in a metastable state differ from the original atoms only in the energy state). Deexcitation can be induced either Coulombically (non-radiation) by interaction with surrounding atoms or particles, or through a higher level by radiation.
   A similar mechanism of "allowed" and "forbidden" transitions - gamma-deexcitation - can be found even in the atomic nucleus in §1.2, part "Gamma radiation", passage "Nuclear isomerism and metastability".
Metastable states of atoms are used in some radiation applications :
- Quantum light generators - LASERs, by irradiation with light, they "pump" electrons to metastable levels and also, by means of light, trigger a mass return transition, avalanche deexcitation, leading to an intense flash of light.
- Thermoluminescent dosimeters, in the sensitive substance of which electrons are excited to a metastable state due to exposure to ionizing radiation, and during later evaluation, deexcitation is induced by heating (§2.2 "Photographic detection of ionizing radiation", passage "Thermoluminescent dosimeters").
   The very process of deexcitation of the excited energy level and formation of the emitted photon is very fast, but not immediate. According to the laws of quantum electrodynamics, deexcitation process of the electron in the atomic packing is about 10^-16 seconds.
Ionization of atoms 
If the atom, or any of its electron, energy supplied is higher than the binding energy of an electron at a certain energy level, this electron is released from the atom and wiil fly out - occurs ionization of the atom, from which it becomes an ion (Greek ion = pilgrim, going). When an electron is ejected, the neutral atom becomes a positively charged particle, a cation. If there are enough electrons in the environment, the electron will be recaptured - the electron will recombine with the ion, creating a neutral atom and photon radiation of the electron's binding energy. Similarly, a "extra" electron (in an electric discharge, by another atom) may be passed to a neutral atom; the result is a predominant negative electric charge, an anion (negative ion) is formed.
   Ionization occurs by the impact of fast-flying particles - ionizing radiation - into atoms, by electric discharges, by the collision of fast-moving atoms in a substance heated to a high temperature (several thousand degrees), by dissolving salts in water, by mechanical friction of substances ("static electricity"). Whenever an atom or molecule loses or gains an electron and the number of electrons no longer corresponds to the number of protons in the nucleus, an ion is formed - the atom or molecule carries a positive or negative electric charge. Atoms and molecules that have one or more free or missing electrons have an increased tendency to chemical reactions(see below "Interaction of atoms") due to electrical interactions with other atoms. At this reactions, atoms and molecules can formed, that othewise are already electrically neutral (they have the same number of protons and circulating electrons, so they are not ions), but they have unpaired electrons in their orbits: they are called radicals because they are chemically very reactive, with each other and with others surrounding atoms and molecules (their essential significance for radiation effects on matter and living tissue is discussed in more detail in Chapter 5 "Biological effects ionizing radiation").
Emitted radiation  
The energy of the emitted photons is given by the energy difference between the levels of electrons in the atom. The energy distribution of electron levels is completely characteristic of the atoms of a given element, so by measuring the spectrum emitted by a certain substance, we can determine the element whose atoms are located there - this forms the content of the atomic spectrometry.
   Spectral distribution of wavelengths, resp. the frequency or energy of the photons of the electromagnetic radiation emitted by the substances can, in extreme cases, have diametrically different shapes :   

• Discrete (line) spectrum ,
  containing radiation of only certain (discrete) wavelengths - called spectral lines, while the wavelengths lying between the lines are not represented. This line spectrum is a direct consequence of the transitions of electrons between discrete energy levels in the atomic shell of the excited atom.
  In order for an electron to transition spontaneously between two quantum levels under photon emission, two conditions must be met :
  1. Energy conservation law - it must be a jump from a higher energy level to a free level with lower energy, with the photon taking the difference these energies.
  2.The law of conservation of angular momentum - the emitted photon carries a angular momentum equal to the difference Dl = l₂ - l₁ between the angular moments of both levels l₂ and l₁, while the actual angular momentum (spin) of the photon is equal to 1. The law of conservation of angular momentum thus leads to the so-called selection rules for permitted transitions:
  D l = ± 1 , D m _l = 0, ± 1 ;
  the change of the main quantum number n is not limited. Thus, only those transitions occur in which the orbital quantum numberl changes by +1 or -1 and the magnetic quantum number m_l either does not change or also changes by +1 or -1.
     Electron jumps between the outer shells give rise to optical spectra in the visible light range. When electrons pass between internal levels, where there are higher energy differences, a characteristic X-ray radiation is created. In order for this to happen (the inner shells are occupied!), a more intense intervention than normal excitation is needed - eg the impact of a considerably accelerated electron or the absorption of a quantum of gamma radiation (photo effect), which leads to the ejection of the electron from the inner shell (eg K ); an electron from the higher shell (eg L ) then jumps to the place released in this way, and the difference in energies is carried away by the X-ray photon. If the selection rules are followed, there may be transitions between different shells and subshells - a series of X-ray lines. The K-series is formed by the transition from higher spheres to the released shell K : it consists of the line K_a formed by the electron jump from the shell L to the shell K and the nearby line K_b accompanying the transition from the shell M to the shell K (both lines are still split into a fine structure). Similarly, the L-series is formed during the transitions of electrons to the emptied place in the L shell, for heavier atoms we can also find M-series, etc.
  The spectrum of characteristic X-rays is shown in Fig.1.2.7 in §1.2 "Radioactivity", part "Gamma radiation", passage "Internal conversion of gamma radiation and X"; it is also shown here that the characteristic X-rays can undergo internal conversion with Auger electrons emission.
• A continuous spectrum , 
  in which all wavelengths from a wide range, albeit with different intensities, are continuously represented. Continuous-spectrum radiation is emitted when the atoms in matter are so close together, that their interactions lead to the formation of a large number of adjacent quantum states with very small energy intervals, effectively indistinguishable from the continuous band of energies.
     A typical example of a continuous spectrum is radiation of a solid substance,  heated to a high temperature, when the continuous spectrum of radiation has its maximum frequency the higher the temperature of the body (higher kinetic energy of atoms excites higher energy states, in the deexcitation of which higher frequency radiation is emitted, ie shorter wavelengths).
     The band spectrum also occurs in excited molecules due to the rotational and vibrational motion of the atoms forming the molecule, where the individual quantum rotational and vibrational states are separated only by very small energy intervals.   

In terms of the positional relationship between the primary energy source, radiating atoms and the spectrometer, we encounter two types of spectra :

• The emission spectrum ,
  showing the spectral distribution of the radiation, emitted from an excited source (the method of excitation, external or internal factors, is not important here). If this spectrum is line, it is formed by light lines on a dark background.
• The absorption spectrum ,
  arises when radiation from an external source (usually with a continuous spectrum, eg white light), passes through a given substance environment (usually a gas) containing atoms of certain elements. These atoms absorb radiation of certain wavelengths (the same ones that occur in the emission spectrum), causing the atoms to excite to higher quantum levels *). In the spectrum of transmitted light, dark absorption lines corresponding to missing (absorbed, attenuated) wavelengths appear on a bright background.
  *) During dexcitation, this absorbed radiation is emitted again, but on average in all directions, so that only a small part is emitted in the direction of the connection between the radiation source and the spectrometer.

Interaction of atoms
The atomic structure of matter makes it possible to explain naturally and from a uniform physical point of view a number of important phenomena at the atomic and subatomic level, from which all properties and manifestations of matter are derived - chemical reactions and molecular properties, structure and properties of solids, liquids and gases, all thermal phenomena (kinetic theory of heat), electrical, magnetic and optical properties of substances.

Chemical fusion of atoms -> molecules
Each atom binds a number of electrons in its electron shell exactly equal to the number of protons, so that the atom is electrically neutral. However, this electrical neutrality of the atoms is fully manifested only at greater distances, where the field of the positively charged nucleus is perfectly "shielded" by the negative electrons in the envelope. In the close vicinity of the atom, however, we can encounter residual manifestations of electric forces *), caused by vector folding of electric field intensities from protons in the nucleus and from electrons located in different places of the electron configuration of the envelope. When twoo atoms are closely approached, these electric forces (initially repulsive) can lead to such a rearrangement of the configuration of electrons on the outer shells (e.g., to the sharing or transfer of electrons) that electric attractive forces can arise that permanently bind the atoms together to form a molecule - Fig.1.1.7. We say that there has been a chemical fusion of atoms. The combination of rearranged electron orbitals of individual atoms creates common molecular orbitals. From the energetic point of view, chemical bonding results in such a rearrangement of electrons (electron density) in the outer valence layers of nearby atoms, which has a lower energy than isolated atoms and is therefore more stable.
*) Interestingly, although electric forces have a long (unlimited) range, their "residual manifestation" - the "chemical" forces between atoms - are short-range. In the vector composition of electric forces from protons in the nucleus and electrons in the envelope, these forces are canceled at greater distances, but at short distances a non-zero "residue" remains. A similar mechanism is encountered in the atomic nucleus in short-range nuclear forces between nucleons, which are a residual manifestation of long-range strong interactions between quarks - see below "The structure of the nucleus", part "Strong nuclear interactions".
   Due to the energy released during the chemical bonding of atoms, molecules are formed in an energetically excited state. Deexcitation occurs either by emitting infrared radiation or by direct electromagnetic interaction with surrounding atoms and molecules. Radiation deexcitations is applied in reactions in thin gaseous medium, while in the dense medium of liquids and solids is dominant direct dexcitation with the participation of surrouding atoms and molecules. In both cases, the energy released during the chemical fusion is eventually transferred to the surronding atoms and molecules of the substance in the form of kinetic energy of motion - the substance is heated, the heat of reaction is generated (we mean here exothermic reactions, see below).
  When atoms approach each other, they are initially electrically repelled (eponymous charged electrons in envelopes). That is, so that the atoms can get close enough to each other - such that their orbitals blend and a chemical bond can form - a certain electrical repulsive barrier must be overcome. Appropriate activation energy must be supplied to the atoms. This is done by the kinetic energy of the thermal motion of atoms - a certain minimum temperature is required to carry out chemical reactions in reaction mixture. At low temperatures, chemical reactions do not take place *). At high temperatures, chemical reactions take place faster, but the mean kinetic energy of atoms and molecules can exceed the binding energy of atoms in molecules - during collisions, the molecules break down, the chemical compound again decomposes.
*) Another possibility of stimulating chemical reactions is irradiation with ionizing radiation. In the irradiated substance, electrons are released from the atoms and positive ions are formed. The resulting electric forces allow the merging of atoms without the need to impart kinetic energy to overcome repulsive forces. Radiation stimulation of chemical reactions plays an important role in the cold gas-dust clouds in space (see "Cosmic radiation"). However, already "finished" molecules are decomposed by ionization radiation - occurs radiolysis of compounds.
Yet unexplored possibility of chemical reactions at low temperatures is the mutual intersection of the wave functions of atoms trought a tunneling phenomenon ...

Energy and the kinetics of chemical reactions
As mentioned above, to effect a merger of two atoms it is necessary to supply them with certain activation kinetic energy Q_A. On the contrary, during the actual merger, the binding energy of the atoms in the molecule Q_R is released. From the point of view of energy balance, their difference Q = Q_R -Q_A - reaction energy is important. According to the sign of the reaction energy, chemical reactions are divided into two groups :
¨ Endothermic (endoenergetic) reactions Q <0 ,
wherein the binding energy of the atoms in the molecule is less than the kinetic energy of the interacting atoms, "consumed" to overcome the repulsive electrical forces. Endothermic reactions cannot be maintained spontaneously, the activation energy must be supplied continuously from the outside; the rate of such reactions is then given by the "supply" of this energy. An example is the formation of carbon disulphide in the passage of sulfur vapors through a hot coal: C + 2 S ® CS₂.
¨ Exothermic (exoenergic) reactions Q> 0 , where there is a "release" and gain of energy, which is drawn from the binding energy of atoms in molecules. For exothermic reactions, there are several possibilities for their kinetics. Time course - kinetics exothermic reactions - decisively depends on the concentration of interacting atoms in the reaction mixture, pressure, temperature, the presence of other types of atoms or molecules.
   With a sufficiently high concentration of reacting atoms, a situation may arise where the released reaction energy during the merging of two atoms is efficiently transferred by electromagnetic interaction to the surrounding atoms. These atoms thus gain kinetic energy, causing them to merge immediately, releasing more energy - which is passed on and causes other atoms to merge. Upon delivery of the initial (initiating) activation energy, a chain chemical reaction is formed *). If the reaction mixture contains a large number of atoms in a sufficiently high concentration, this chain reaction has the character of an explosion: its velocity increases exponentially, in a small moment (of the order of ms) practically all atoms in the reaction mixture combine. Suddenly released heat of reaction heats the mixture to a high temperature (of the order of thousands of degrees), which causes a rapid expansion - the explosion of the reaction mixture. A well-known example is the ignition of a mixture of hydrogen and oxygen, a small spark of locally elevated temperature is sufficient. If the concentration of one of the components is lower, or the individual components are fed to the reaction space gradually, an equilibrium chain reaction having the character of a continuous combustion can be established.
*) The nuclear chain reaction has a similar kinetics, but a different mechanism fission of heavy nuclei of uranium or plutonium by neutrons - see §1.3, section "Fission of atomic nuclei".
   At low concentrations of reacting atoms, the chain reaction does not occur. When atoms are combined into a molecule, binding energy is released, which is emitted in the form of infrared photons. However, these photons fly away, the probability of their absorption by other distant atoms in a sparse environment is negligible. For chemical reactions to take place in a sparse environment, the activation energy must be supplied externally continuously (the situation is similar to endothermic reactions).  

Fig. 1.1.7. Symbolic representation of the mechanism of joining atoms and their electrical bonds in molecules.
Left: Covalent bond of two atoms caused by electron sharing. Right: An ionic bond of atoms caused by the handover of an electron from one atom to another.

Types of chemical bonding
When two atoms approach each other, there are basically three diferent possibilities of their interaction :

• 1. Electron sharing - covalent bonding
  One or more pairs of electrons (with oppositely oriented spins) from the valence shell begin to orbit together both atoms in such a way that the increased probability of their occurrence near the junction of the nuclei of both atoms leads to an attractive electric force (Fig.1.1.7 left).
  An example is the hydrogen molecule H₂ (HH), whose two electrons belong together to two protons, or the oxygen molecule O₂ (O = O). In these cases of diatomic molecules of identical atoms, the electrons are shared "equally" (the common molecular orbital lies midway between the interacting atoms) - we say that the bond and the resulting molecule are non-polar. In most cases, however, the shared electrons are attracted more to one of the atoms involved in the bond than to the other - such a molecule then behaves like an electric dipole, we are talking about bonds and polar molecules.
• 2. Transition of electrons between atoms - ionic bond
  One or more electrons can pass from the valence shell of one atom to the electron shell of the other atom, and the resulting positive and negative ions are electrically attracted to each other. This process occurs when one of the atoms has weakly bound electrons in its valence shell, which it tries to "get rid of" (we say are electropositive), while the other atom lacks one electron (or a few electrons) in the valence shell closure, characterized by strong bond and therefore tends to accept electrons from other atoms (such an atom is called electronegative). .........
     A typical example of an ionic bond is Na + Cl ® Na⁺ + Cl^- ® NaCl (Fig.1.1.7 in the right). Separate NaCl molecules occur only in the gaseous state (salt vapor), in the crystal lattice of the solid state of the salt, Na ⁺ cations and Cl ^- anions alternate regularly; in aqueous solution, NaCl is dissociated into ions.
  In a sense, an ionic bond can be considered as a borderline case of a strongly polar covalent bond.
• 3. Repulsion of atoms - no bond is formed
  If the interacting atoms in their envelopes do not have electronic configurations suitable for the realization of any of the two mentioned mechanisms, deformations and interweaving of electronic structures occur when the atoms approach each other, which reduces shielding of positive charge nuclei by envelope electrons electrostatic repulsion - no chemical bonding of atoms occurs. In particular, atoms with all closed shells that are chemically inert (helium, neon, argon, etc.) behave in this way.

In addition to the above-mentioned purely covalent and purely ionic bonds, many molecules undergo a mixed type of bond in which the atoms share electrons unequally.
Metal bond 
In addition to the covalent and ionic bond between two atoms, there is another type of bond in which the electrons of the outer valence layer are not shared by two nuclei or atoms, but by a large number of atoms. This applies to metals. Metal atoms are characterized by a small number of electrons in the outer shell, usually one or two electrons. These external valence electrons from a larger number of nearby atoms can then form a single continuous cloud - the so-called electron gas, in an array of regularly spaced nuclei surrounded by electrons of the inner layers. A metal crystal is a kind of huge "molecule", made up of regularly distributed cations, between which binding electrons move freely. The electrostatic attractive forces between these cations and the electrons in the cloud then form a bond called a metal.