Searcharxiv⌕ Search

arXiv subjects

B. I. Sadovnikov

Publications and source records attributed to B. I. Sadovnikov.

At least 19 recordsLinked to original sources
arXiv2026

Kinematic properties of the Pauli equation

Based on the Wigner-Vlasov formalism, this paper investigates the kinematic properties of the Pauli equation. It is shown that the probability current associated with the Pauli equation can be represented as a superposition of two currents with certain expansion coefficients. Each of these currents corresponds to a particular component of the spinor. The expansion coefficients effectively serve as weighting functions that determine the probability contribution of the corresponding spinor component. Therefore, each spin projection corresponds to its own probability flux. A new system of the Hamilton-Jacobi equations and also a system of motion equations in electromagnetic fields are obtained, taking into account the interaction between the spin and the magnetic field. To illustrate how these equations can be applied we have investigated the quantum system kinematics in detail using an exact solution of the Pauli equation in the presence of a uniform magnetic field and an asymmetric quadratic potential.

quant-ph↗
arXiv2026

The Maxwell class exact solutions to the Schrödinger equation and continuum mechanics models

By applying the nonlinear Legendre transform to the continuity equation, this paper derives exact solutions to the Schrödinger equation and the equations of continuum mechanics. A generalized Maxwell distribution has been used as the momentum density function. Explicit expressions for the vector fields of time independent flows, density distributions, quantum and classical potentials have been found, and a detailed mathematical and physical analysis of the results obtained has been carried out.

math-ph↗
arXiv2025

Exact solutions for the relativistic dynamics of a self-consistent system with electromagnetic and gravitational interaction within the Wigner-Vlasov formalism

This work derives exact solutions to the problem of interacting particle density evolution in relativistic and quasi-relativistic approximations for electromagnetic and gravitational interactions. Two types of radial symmetry for the initial density distribution are considered: spherical and cylindrical. It is shown that the relativistic effect delays the onset of the shock wave moment, and in some cases removes it entirely or, conversely, can facilitate it. The analysis of the system's dynamics is carried out within the Wigner-Vlasov formalism, which makes it possible to extend the obtained solutions to quantum systems, including those with gravitational interaction. The derived exact solutions can be directly used as a cross-check for modeling and optimizing nonlinear problems of beam dynamics with account for space charge, astrophysics, plasma physics, and quantum systems with a shock wave.

math-ph↗
arXiv2025

Characteristic solutions of the chain of Vlasov equations

A new method has been presented of constructing a class of exact solutions of an infinite self-linking chain of the Vlasov equations for distribution functions of kinematic quantities of all orders. Using the characteristic transformation of variables proposed in this paper, any equation from the Vlasov chain can be reduced to the mathematical form of the first Vlasov equation. Since the solution of the first Vlasov equation can be found by the solution of the Schrödinger equation, the authors have proposed an algorithm for constructing characteristic solutions for an arbitrary equation from the Vlasov chain. The proposed method of construction of exact solutions has been successfully implemented on an example of time-dependent quantum system with thermodynamic parameter in the form of inverse temperature. These found exact solutions are also applicable to quantum dot systems.

math-ph↗
arXiv2025

New exact solutions of the 3D Schrödinger equation

Previously we found a unique quantum system with a positive gauge-invariant Weyl-Stratonovich quasi-probability density function which can be defined by the so-called «quadratic funnel» potential [Phys. Rev. A 110 02222 (2024)]. In this work we have constructed a class of exact solutions to the 3D Schrödinger equation for a two-parameter «quadratic funnel» potential based on the -model of micro and macro systems. Explicit expressions for the energy spectrum and the set of eigenfunctions have been found. Using gauge invariance for scalar and vector potentials, a solution to the electromagnetic Schrödinger equation has been obtained, with a magnetic field in the form of a «Dirac string» defined by a singular vortex probability flux field. Superpositions of eigenfunctions leading to various types of vortex and potential probability current fields have been investigated in detail. The analysis of the quantum system's properties has been carried out within the Wigner-Vlasov formalism.

quant-ph↗
arXiv2024

Wigner function properties for electromagnetic systems

Using the Wigner-Vlasov formalism, an exact 3D solution of the Schrödinger equation for a scalar particle in an electromagnetic field is constructed. Electric and magnetic fields are non-uniform. According to the exact expression for the wave function, the search for two types of the Wigner functions is conducted. The first function is the usual Wigner function with a modified momentum. The second Wigner function is constructed on the basis of the Weyl-Stratonovich transform in papers [Phys. Rev. A 35 2791 (1987)] or [Phys. Rev. B 99 014423 (2019)]. It turns out that the second function, unlike the first one, has areas of negative values for wave functions with the Gaussian distribution (Hudson's theorem). An example of electromagnetic quantum system described by a non-Gaussian wave function has successfully been found. The second Wigner function is positive over the whole phase space for the non-Gaussian wave function. This result is analogous to the Hudson theorem for the gage-invariant Wigner function. On the one hand, knowing the Wigner functions allows one to find the distribution of the mean momentum vector field and the energy spectrum of the quantum system. On the other hand, within the framework of the Wigner-Vlasov formalism, the mean momentum distribution and the magnitude of the energy are initially known. Consequently, the mean momentum distributions and energy values obtained according to the Wigner functions can be compared with the exact momentum distribution and energy values. This paper presents this comparison and describes the differences. The Vlasov-Moyal approximation of average acceleration flow has been built in phase space for a quantum system with electromagnetic field. The obtained approximation makes it possible to cut the Vlasov chain off at the second equation and also to analyze the Boltzmann H-function evolution.

quant-ph↗
arXiv2023

Investigation of the dynamics of transverse oscillations of a vertical rod under gravity, friction, and thermal expansion

The paper considers the mathematical formulation of the problem of transverse oscillations of a vertical rod under gravity, friction and external pulse effect, leading to thermal expansion of the rod. The dynamics of the system under consideration corresponds to the behavior of a fuel element (FE) in a pulsed reactor and is related to the dynamic stability of the processes occurring in it. The FE dynamics is described by inhomogeneous linear differential equation of the fourth order with non-constant coefficients. Initial boundary conditions are not smooth since they correspond to the instant heating of the part of the FE surface exposed to neutron pulse radiation. In the future the obtained exact solutions will be used as input data to simulate self-consistent system dynamics of more than 50 FE.

math-ph↗
arXiv2023

Is the Moyal equation for the Wigner function a quantum analogue of the Liouville equation?

The Moyal equation describes the evolution of the Wigner function of a quantum system in the phase space. The right-hand side of the equation contains an infinite series with coefficients proportional to powers of the Planck constant. There is an interpretation of the Moyal equation as a quantum analogue of the classical Liouville equation. Indeed, if one uses the notion of the classical passage to the limit as the Planck constant tends to zero, then formally the right-hand side of the Moyal equation tends to zero. As a result, the Moyal equation becomes the classical Liouville equation for the distribution function. In this paper, we show that the right side of the Moyal equation does not explicitly depend on the Planck constant, and all terms of the series can make a significant contribution. The transition between the classical and quantum descriptions is related not to the Planck constant, but to the spatial scale. For a model quantum system with a potential in the form of a «quadratic funnel», an exact 3D solution of the Schrödinger equation is found and the corresponding Wigner function is constructed in the paper. Using trajectory analysis in the phase space, based on the representation of the right-hand side of the Moyal equation, it is shown that on the spatial microscale there is an infinite number of «trajectories» of the particle motion (thereby the concept of a trajectory is indefinite), and when passing to the macroscale, all «trajectories» concentrate around the classical trajectory.

quant-ph↗
arXiv2023

The Wigner-Vlasov formalism for time-dependent quantum oscillator

This paper presents a comprehensive investigation of the problem of a harmonic oscillator with time-depending frequencies in the framework of the Vlasov theory and the Wigner function apparatus for quantum systems in the phase space. A new method is proposed to find an exact solution of this problem using a relation of the Vlasov equation chain with the Schrödinger equation and with the Moyal equation for the Wigner function. A method of averaging the energy function over the Wigner function in the phase space can be used to obtain time-dependent energy spectrum for a quantum system. The Vlasov equation solution can be represented in the form of characteristics satisfying the Hill equation. A particular case of the Hill equation, namely the Mathieu equation with unstable solutions, has been considered in details. An analysis of the dynamics of an unstable quantum system shows that the phase space square bounded with the Wigner function level line conserves in time, but the phase space square bounded with the energy function line increases. In this case the Vlasov equation characteristic is situated on the crosspoint of the Wigner function level line and the energy function line. This crosspoint moves in time with a trajectory that represents the unstable system dynamics. Each such trajectory has its own energy, and averaging these energies over the Wigner function results in time-dependent discreet energy spectrum for the whole system. An explicit expression has been obtained for the Wigner function of the 4th rank in the generalized phase space $\left\{ x,p,\dot{p},\ddot{p} \right\}.$

quant-ph↗
arXiv2022

PSI-Moyal equation

A full consideration of classical and quantum systems with radiation (electromagnetic/gravitational) requires the involvement of a mathematical description in the generalized phase space of high kinematical values. Based on the dispersion chain of equations of quantum mechanics, we construct a generalization of the von Neumann equation for the density matrix in the phase space of fourth-order kinematical values. The paper introduces a new extended definition of the fourth rank Wigner function, which is constructed from the wave functions of the second rank. A new extended Moyal equation (PSI-Moyal equation) for the Wigner function of the fourth rank is obtained. Theorems on the properties of the new PSI-Moyal equation and its solutions are proved. An example of a model quantum system is considered in detail.

quant-ph↗
arXiv2022

Dispersion chain of quantum mechanics equations

Based on the dispersion chain of the Vlasov equations, the paper considers the construction of a new chain of equations of quantum mechanics of high kinematical values. The proposed approach can be applied to consideration of classical and quantum systems with radiation. A number of theorems are proved on the form of extensions of the Hamilton operators, Lagrange functions, Hamilton-Jacobi equations, and Maxwell equations to the case of a generalized phase space. In some special cases of lower dimensions, the dispersion chain of quantum mechanics is reduced to quantum mechanics in phase space (the Wigner function) and the de Broglie-Bohm «pilot wave» theory. An example of solving the Schrödinger equation of the second rank (for the phase space) is analyzed, which, in contrast to the Wigner function, gives a positive distribution density function.

math-ph↗
arXiv2022

PSI-Vlasov equation

A new equation for describing physical systems with radiation is obtained in this paper. Examples of such systems can be found in plasma physics, accelerator physics (synchrotron radiation) and astrophysics (gravitational waves). The new equation is written on the basis of the third Vlasov equation for the probability density distribution function of kinematic quantities: coordinates, velocities and accelerations. The constructed new Vlasov PSI - equation makes it possible to describe naturally dissipation systems instead of phenomenological modifications of the second Vlasov equation, and to construct conservative difference schemes in numerical simulation.

physics.plasm-ph↗
arXiv2021

Exact time-dependent solution of the Schrödinger equation, its generalization to the phase space and relation to the Gibbs distribution

Using the simplest but fundamental example, the problem of the infinite potential well, this paper makes an ideological attempt (supported by rigorous mathematical proofs) to approach the issue of «understanding» the mechanism of quantum mechanics processes, despite the well-known examples of the EPR paradox type. The new exact solution of the Schrödinger equation is analyzed from the perspective of quantum mechanics in the phase space. It is the phase space, which has been extensively used recently in quantum computing, quantum informatics and communications, that is the bridge towards classical physics, where understanding of physical reality is still possible. In this paper, an interpretation of time-dependent processes of energy redistribution in a quantum system, probability waves, the temperature and entropy of a quantum system, and the transition to a time-independent «frozen state» is obtained, which is understandable from the point of view of classical physics. The material of the paper clearly illustrates the solution of the problem from the standpoint of continuum mechanics, statistical physics and, of course, quantum mechanics in the phase space.

quant-ph↗
arXiv2021

Dispersion chain of Vlasov equations

On the basis of the Vlasov chain of equations, a new infinite dispersion chain of equations is obtained for the distribution functions of mixed higher order kinematical values. In contrast to the Vlasov chain, the dispersion chain contains distribution functions with an arbitrary set of kinematical values and has a tensor form of writing. For the dispersion chain, new equations for mixed Boltzmann functions and the corresponding chain of conservation laws for fluid dynamics are obtained. The probability is proved to be a constant value for a particle to belong the region where the quasi-probability density is negative (Wigner function).

math-ph↗
arXiv2020

Extended Wigner function for the harmonic oscillator in the phase space

New time dependent Wigner functions for the quantum harmonic oscillator have been obtained in this work. The Moyal equation for the harmonic oscillator has been presented as the wave equation of a 2D membrane in the phase plane. The values of the Wigner function are equal to the deviation values of the points on the surface of the membrane from the equilibrium state. The positive and negative values of the Wigner function correspond to the direction of the deviation from the equilibrium state. As an example, a time dependent Wigner function corresponding to the standing wave of quasi-probability density arising in the phase plane is considered.

quant-ph↗