Heisenberg Uncertainty Relation in Quantum Liouville Equation

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.