Preface

In this book, the author opens up new possibilities for the main quantities in quantum physics – the statistical operator 𝜚̂ and the density matrix_𝜚𝑛𝑚. The meaning of the density matrix is that its diagonal elements 𝜚𝑛𝑛 are equal to the probability 𝑤_𝑛 that the system in the quantum state n. The point in this book is the Lindblad equation for the statistical operator 𝜚̂, where the main element of influence on the system of its environment is the dissipative operator :

This operator is written in the most General form. In order for the Lindblad equation to be solved, the operator must be specified. The author wrote down the dissipative diffusion and attenuation operators that will allow us to find the operator. Now, this operator depends on the temperature T and describes the effect of the thermostat R on the quantum system S. This new equation is not difficult to write for the particle density matrix in coordinate representation as compared to the Wigner equation, which coincides with the Fokker – Planck equation. This proved the equivalence of quantum physics and classical statistical physics.

The author wrote the Lindblad equation for a harmonic oscillator and inserted a dissipative attenuation operator into it. And without any approximation, he derived the equation of damped oscillations for the average value of the 𝑥̅(t) coordinate with absolute accuracy.

Bondarev based on the Lindblad equation with another operator developed the theory of the harmonic oscillator, in which he found the density matrix and proved the Heisenberg relation.

He further developed the theories of the light diode and ball lightning. In light diode theory, he used the diffusion and attenuation operators and derived the Fokker – Planck equations for electrons and holes. These equations present the terms that are responsible for radiation.

The theory of ball lightning is based on the assumption that the gas inside the ball is completely ionized and electrons, due to their lightness in comparison with nuclei, evenly fill this ball. The equation for the statistical operator 𝜚̂ nuclei contains operators of diffusion and damping. This equation is a second-degree equation with respect to the coordinate and momentum operators. The probability of distribution of nuclei over the volume of a ball lightning is found.

Bondarev derived von Neumann equation from the Liouville, which is valid for a non-equilibrium system S and an equilibrium thermostat R, the equations for the density matrix S of a single particle and a system of identical particles. These equations have a remarkable property. When the density matrix has a diagonal form, they get turned into quantum kinetic equations for probabilities, which are obtained in the wave graphical representation.

The book presents new theories of such experimentally discovered phenomena as step kinetics of bimolecular reactions in solids, superconductivity, superfluidity, energy spectrum of an arbitrary atom, laser, spaser and graphene.

Kinetics is called as a stepwise process, in which the reaction suddenly stops at a constant temperature even in the presence of a lot of reagents. But as soon as the temperature is raised, the reaction starts again. The reason for this reaction is the tunnel effect, which is observed only in solids, when there are molecules in the bodies that hold the reagents near them. In liquids, these molecules can move along with the reagents and enter into a reaction that goes all the way while there are reagents. The reaction in liquids always obeys the Arrhenius law. To describe stepwise kinetics, the author came up with a correlation theory.

So, when processing the results of the step kinetics experiment using correlation theory, it was found that the Arrhenius law is also fulfilled here. And there was also an increase in localization volume with increasing temperature, as predicted by the tunnel effect.

Superconductivity can be described by the law of changing the probability 𝑤𝒌 of filling the state of electrons with the wave vector k as a function of temperature. This law has long been known. It depends on the energy 𝜀_𝒌𝒌′ of the interaction of electrons with the wave vectors k and 𝒌′. When 𝜀𝒌𝒌′ = 0, the probability 𝑤𝒌 obeys the Fermi − Dirac law. Our goal was to find the energy 𝜀_𝒌𝒌′ of the interaction of electrons.

We denote the matrix elements of the interaction Hamiltonian of two particles as 𝐻_12,1′2′ , where 1 is the spin quantum number of the particles. If the particles are bosons, then the matrix elements must be antisymmetric, i.e. then the matrix elements must change the sign when replacing variables 1 and 2, or 1′ and 2′. This is possible if the matrix elements represent the sum of two terms of different characters. In the wave representation, the energy ε_(kk^') will also represent two terms of different signs. But in this case, it is very difficult to solve the equation. Therefore, we roughly denote these terms as

𝜀_𝒌𝒌′ = I 𝛿_{𝒌 + 𝒌′} − J 𝛿_{𝒌 − 𝒌′} ,

where I and J are positive constants, 𝛿_𝒌 is the Kronecker symbol. Now we can substitute this function into the equation and get

ln [ ( 1 − 𝑤_𝒌 ) ⁄ 𝑤_𝒌 ] = 𝛽 ( 𝜀_𝒌 + 𝐼 𝑤_{− 𝒌} − 𝐽 𝑤_𝒌 − 𝜇 ) .

This equation has a remarkable property. For some areas of 𝒌 will this inequality be true

𝑤_𝒌 ≠ 𝑤_{− 𝒌} .

The property that is expressed by this inequality is called anisotropy. The appearance of this property here is superconductivity.

Solving this equation, we obtain for T = 0 functions that have five values for one argument value. Since this function describes stationary states, the lowest energy is the value of the function where the electrons remain indefinite. This will be a superconducting state.

In theory, the parameter represents f = ( 𝐽 − 𝐼 ) ⁄ ( 𝐽 + 𝐼 ). This parameter divides superconductivity into two kinds. If 0 ≤ f ≤ 1, then it is a I-type superconductor, and if −1 ≤ f < 0, then it is a II-type superconductor. Critical temperature is defined as 𝑇_𝑐 = ( 𝐼 + 𝐽 ) ⁄ ( 4 𝑘_B). All the main effects and properties of superconductors are covered by this theory.

In the theory of superfluidity for liquid helium, Не³ and Не⁴, all values that express the properties of this mixture are described by functions having multiple values in a certain temperature range. As a consequence, the heat capacity tends to infinity when the temperature approaches the temperature 𝑇_𝜆 of the lambda transition.

The theory of the energy spectrum of an arbitrary atom begins with determining the energy using statistical operators:

E = ∫𝐻⁽¹⁾ 𝜚̂⁽¹⁾ dq + 1 2 ⁄ ∫𝐻⁽²⁾ 𝜚̂⁽²⁾ d𝑞₁ d𝑞₂ ,

where 𝐻 ̂ ⁽¹⁾ is the Hamiltonian of one electron, 𝐻 ̂⁽²⁾ is the Hamiltonian of two interacting electrons, 𝜚̂⁽¹⁾ and 𝜚̂⁽²⁾ are the statistical operators of one and two electrons. The matrix 𝐻_{𝛼1𝛼2, 𝛼1 ′ 𝛼2 ′} of the Hamiltonian 𝐻 ̂ ⁽²⁾ must be antisymmetric. To do this, it is taken equal to

where

there is an antisymmetric Slater function. The eigenfunctions of electrons in the hydrogen atom are taken as functions 𝜑_𝛼(𝑞). After a series of calculations, an equation is obtained from which one can obtain the eigenfunction and energy 𝜀_{𝑛 𝑚 𝑙 𝜎} of electrons of an arbitrary atom.

In the following chapters, new theories of the laser and spaser are constructed, which are similar to each other in the content of the main quantum approaches to describing the phenomena occurring in them. The basis of these theories is the Lindblad equation. The equation for the density matrix will be written in a coordinate form with a known Hamiltonian and an unknown dissipative matrix. To find this matrix, we need to remember that we know the kinetic equation for active atoms, which follows from the equation for the density matrix in the representation where it has a diagonal form. So, a representation needs to be find out where the density matrix has a diagonal form. The closest to this representation is the representation in which the Hamiltonian of active atoms will also have a diagonal form. Thus, the Hamiltonian has two representations. One α- representation is a coordinate representation. The other is the 𝜅-representation, in which the Hamiltonian has a diagonal form. These two Hamiltonians are connected by the unitary matrix 𝑈_𝛼𝜅. The density matrices 𝜚_𝛼𝛼′ and 𝜚̃_𝜅𝜅′ will also be connected by the same unitary transformation:

𝜚_𝛼𝛼′ = Σ_𝜅𝜅′ 𝑈_𝛼𝜅 𝜚̃_𝜅𝜅′𝑈^∗_{𝛼′𝜅′} .

We find the dissipative matrix in the 𝜅-representation.

Now we need to create another equation in the α-representation. This is the most important equation in laser theory. This is the equation for the spectral energy density of radiation. To solve it, we will use the density matrix 𝜚_𝛼𝛼′ . As a result, we will have the spectral energy density of the radiation from the laser.

Almost free electrons wander along the surface of graphene. For every carbon atom, there is one such electron. To obtain the kinetic equation of these electrons, their Hamiltonian must be reduced to a diagonal form. After these transformations, we will have a system of two equations that are equivalent to the equation obtained in the theory of superconductivity.