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 ( x , v , ) of a generic solution ( x ; ) of the Schrödinger equation. We give a representation of ( x , ) by the Hermite functions. We show that the values of the variances of x and v calculated by using the Wigner function ( x , v , ) coincide, respectively, with the variances of position operator and conjugate momentum operator obtained using the wave function ( x , ). Then we consider the Fourier transform of the density matrix ( z , y , ) = ( z , ) ( y , t ). We find again that the variances of x and v obtained by using ( z , y , ) are respectively equal to the variances of and calculated in ( x , ). Finally we introduce the matrix and we show that a generic square-integrable function ( x , v , ) can be written as Fourier transform of a density matrix, provided that the matrix is diagonalizable.