A new class of exact solutions of the Schrödinger equation

Perepelkin, E. E.; Садовников, Б. И.; Иноземцева, Н. Г.; Tarelkin, A.A.

doi:10.1007/s00161-018-0716-9

Cited by 8 publications

(4 citation statements)

References 16 publications

(39 reference statements)

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“…Each of these domains is a special case of the chain and is determined by what is meant by the distribution function and mean kinematical values. For instance, in continuum mechanics, the distribution function can be understood as the density of matter, in quantum mechanics-as the probability density, in field theory-as the charge density (the probability of detecting a charge) or even the magnetic permittivity function [18].…”

Section: Discussionmentioning

confidence: 99%

“…Such reasoning is explained by the hierarchical form of writing the Vlasov chain of equation (13). On the other hand, due to the independence of kinematical values r, v, ˙ v, ¨ v,• • •, it is possible to construct mixed distribution functions (17) and the corresponding average kinematical values (18). In this case, equation ( 12) may be interpreted as the dependence of the lowest kinematical value ˙ v on the highest kinematical value ¨ v. Similar reasoning may be carried out for equation (7).…”

Section: • • •mentioning

confidence: 99%

“…+ • • • and further. In this work, an infinite chain is constructed for the distribution functions of mixed type (17) and the kinematical values corresponding to them (18), which was called the dispersion chain of the Vlasov equations. By analogy with work [6], groups of equations for mixed H n -Boltzmann functions are constructed and their properties are investigated.…”

Section: • • •mentioning

confidence: 99%

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Dispersion chain of Vlasov equations

Perepelkin¹,

Садовников²,

Иноземцева³

et al. 2022

J. Stat. Mech.

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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).

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Section: Discussionmentioning

confidence: 99%

Section: • • •mentioning

confidence: 99%

Section: • • •mentioning

confidence: 99%

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Dispersion chain of Vlasov equations

Perepelkin¹,

Садовников²,

Иноземцева³

et al. 2022

J. Stat. Mech.

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show abstract

“…The use of expressions (i.5)-(i.14) makes it possible to obtain from solutions obtained for quantum systems solutions corresponding to classical mechanics, plasma physics, statistical physics, accelerator physics, field theory and continuum mechanics [22][23][24].…”

Section: Introductionmentioning

confidence: 99%

The Wigner-Vlasov formalism for time-dependent quantum oscillator

Perepelkin,

Sadovnikov,

Inozemtseva

et al. 2023

Phys. Scr.

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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 by 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 {r,p,p',p''}.

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A kind regularization method for solving Cauchy problem of the Schrödinger equation

Lv,

Feng

2025

Journal of Computational and Applied Mathematics

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A new class of exact solutions of the Schrödinger equation

Cited by 8 publications

References 16 publications

Dispersion chain of Vlasov equations

Dispersion chain of Vlasov equations

The Wigner-Vlasov formalism for time-dependent quantum oscillator

A kind regularization method for solving Cauchy problem of the Schrödinger equation

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