Heisenberg Uncertainty Relation in Quantum Liouville Equation
Author
Source
International Journal of Mathematics and Mathematical Sciences
Issue
Vol. 2009, Issue 2009 (31 Dec. 2009), pp.1-20, 20 p.
Publisher
Hindawi Publishing Corporation
Publication Date
2009-11-25
Country of Publication
Egypt
No. of Pages
20
Main Subjects
Abstract EN
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(x,v,t) of a generic solution ψ(x;t) of the Schrödinger equation.
We give a representation of ψ(x, t) by the Hermite functions.
We show that the values of the variances of x and v calculated by using the Wigner function f(x,v,t) coincide, respectively, with the variances of position operator X^ and conjugate momentum operator P^ obtained using the wave function ψ(x,t).
Then we consider the Fourier transform of the density matrix ρ(z,y,t) = ψ∗(z,t)ψ(y,t).
We find again that the variances of x and v obtained by using ρ(z, y,t) are respectively equal to the variances of X^ and P^ calculated in ψ(x,t).
Finally we introduce the matrix ∥Ann′(t)∥ and we show that a generic square-integrable function g(x,v,t) can be written as Fourier transform of a density matrix, provided that the matrix ∥Ann′(t)∥ is diagonalizable.
American Psychological Association (APA)
Valenti, Davide. 2009. Heisenberg Uncertainty Relation in Quantum Liouville Equation. International Journal of Mathematics and Mathematical Sciences،Vol. 2009, no. 2009, pp.1-20.
https://search.emarefa.net/detail/BIM-466565
Modern Language Association (MLA)
Valenti, Davide. Heisenberg Uncertainty Relation in Quantum Liouville Equation. International Journal of Mathematics and Mathematical Sciences No. 2009 (2009), pp.1-20.
https://search.emarefa.net/detail/BIM-466565
American Medical Association (AMA)
Valenti, Davide. Heisenberg Uncertainty Relation in Quantum Liouville Equation. International Journal of Mathematics and Mathematical Sciences. 2009. Vol. 2009, no. 2009, pp.1-20.
https://search.emarefa.net/detail/BIM-466565
Data Type
Journal Articles
Language
English
Notes
Includes bibliographical references
Record ID
BIM-466565