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

doi:10.1016/j.aop.2017.05.014

Cited by 5 publications

(6 citation statements)

References 55 publications

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“…As an example, let us consider a quantum system described by a probability density function [13,14]:…”

Section: Numerical Resultsmentioning

confidence: 99%

“…Scalar potential Φ is not a smooth function, therefore curl r ⃗ v 1 = h 2m curl r ∇ r Φ = 0. The properties of the extended form of the vortex field (3.2) are considered in detail in [14].…”

Section: 7)mentioning

confidence: 99%

“…The solution (3.5) of the Schrödinger equation corresponds to ground state of the quantum system n = 0. Any function f 1 (⃗ r, t) = F (r) is solution f 1 of the first Vlasov equation (i.9) with vector field (3.2) [14]. Indeed,…”

Section: 7)mentioning

confidence: 99%

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

Perepelkin,

Sadovnikov,

Inozemtseva

et al. 2023

J. Stat. Mech.

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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.

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“…As an example, let us consider a quantum system described by a probability density function [13,14]:…”

Section: Numerical Resultsmentioning

confidence: 99%

Section: 7)mentioning

confidence: 99%

See 1 more Smart Citation

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

Perepelkin,

Sadovnikov,

Inozemtseva

et al. 2023

J. Stat. Mech.

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“…As a rule, the Schrödinger equation is used for description of quantum systems [1]. However, the fact that the Schrödinger equation can be reduced to the equation of continuity for the probability density function   , f r t indicates the possibility of applying it to the description of the classical systems of continuum mechanics [2][3][4].…”

Section: Introductionmentioning

confidence: 99%

“…Papers [3,4] consider the inverse problem, that is the construction of the Schrödinger equation solution out of the continuity equation solutions. Initially, the vector field of continuous t .…”

Section: Introductionmentioning

confidence: 99%

A new class of exact solutions of the Schrödinger equation

Perepelkin

Садовников

Иноземцева

et al. 2018

Continuum Mech. Thermodyn.

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The aim of this paper is to find the exact solutions of the Schrödinger equation. As is known, the Schrödinger equation can be reduced to the continuum equation. In this paper, using the non-linear Legendre transform the equation of continuity is linearized. Particular solutions of such a linear equation are found in the paper and an inverse Legendre transform is considered for them with subsequent construction of solutions of the Schrödinger equation. Examples of the classical and quantum systems are considered.

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