Copyright © 2009 Davide Valenti. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

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.