Heisenberg uncertainty relation in quantum Liouville equation
type: Item
instance of: scholarly article
title: Heisenberg uncertainty relation in quantum Liouville equation
zbMATH Open document ID: 1187.81172
DOI: 10.1155/2009/369482
author: Davide Valenti
published in: International Journal of Mathematics and Mathematical Sciences
publication date: 2010-04-20T00:00:00Z
full work available at URL: https://eudml.org/doc/231832
review text: Summary: We consider the quantum Liouville equation and give a characterization of the solutions which satisfy the Heisenberg uncertainty relation. We analyze three cases. Initially we consider a particular solution of the quantum Liouville equation: the Wigner transform \(f({\mathbf x},{\mathbf v},t)\) of a generic solution \(\psi({\mathbf x};t)\) of the Schrödinger equation. We give a representation of \(\psi({\mathbf x},t)\) by the Hermite functions. We show that the values of the variances of \textbf{x} and \textbf{v} calculated by using the Wigner function \(f({\mathbf x},{\mathbf v},t)\) coincide, respectively, with the variances of position operator \(\widehat{X}\) and conjugate momentum operator \(\widehat{P}\) obtained using the wave function \(\psi({\mathbf x},t)\). Then we consider the Fourier transform of the density matrix \(\rho({\mathbf z},{\mathbf y},t)= \psi^*({\mathbf z},t)\psi({\mathbf y},t)\). We find again that the variances of \textbf{x} and \textbf{v} obtained by using \(\rho({\mathbf z},{\mathbf y},t)\) are respectively equal to the variances of \(\widehat{X}\) and \(\widehat{P}\) calculated in \(\psi({\mathbf x},t)\). Finally we introduce the matrix \(\|A_{nn'}(t)\|\) and we show that a generic square-integrable function \(g({\mathbf x},{\mathbf v},t)\) can be written as Fourier transform of a density matrix, provided that the matrix \(\|A_{nn'}(t)\|\) is diagonalizable.
Mathematics Subject Classification ID: 81S05
Mathematics Subject Classification ID: 81S30
Mathematics Subject Classification ID: 81Q05
Mathematics Subject Classification ID: 33C45
zbMATH DE Number: 5695206
MaRDI profile type: Publication
OpenAlex ID: W2155055871
Wikidata QID: Q58648438
cites work: Q3953228
cites work: Q4023822
cites work: The Vlasov dynamics and its fluctuations in the \(1/N\) limit of interacting classical particles
cites work: Vlasov equations
cites work: The Motion of Slow Electrons in a Polar Crystal
cites work: Friedrichs model with virtual transitions. Exact solution and indirect spectroscopy.
cites work: On the Quantum Correction For Thermodynamic Equilibrium
cites work: Half-space problem of the Boltzmann equation for charged particles
cites work: On a one-dimensional Schrödinger-Poisson scattering model
cites work: Q4339594
cites work: Homogenization limits and Wigner transforms
cites work: An Analysis of the Quantum Liouville Equation
cites work: Q3243448
cites work: Q5517170
cites work: Q3293417
cites work: Q5558293
cites work: Mathematical methods for physicists.
DBLP publication ID: journals/ijmmsc/Valenti09
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description: scientific article; zbMATH DE number 5695206