Heisenberg Uncertainty Relation in Quantum Liouville Equation

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 transformf(x,v,t) of a generic solutionψ(x;t) of the Schrödinger equation. We give a representation ofψ(x,t) by the Hermite functions. We show that the values of the variances ofxandvcalculated by using the Wigner functionf(x,v,t) coincide, respectively, with the variances of position operatorX^and conjugate momentum operatorP^obtained using the wave functionψ(x,t). Then we consider the Fourier transform of the density matrixρ(z,y,t) =ψ∗(z,t)ψ(y,t). We find again that the variances ofxandvobtained by usingρ(z,y,t) are respectively equal to the variances ofX^andP^calculated inψ(x,t). Finally we introduce the matrix∥Ann′(t)∥and we show that a generic square-integrable functiong(x,v,t) can be written as Fourier transform of a density matrix, provided that the matrix∥Ann′(t)∥is diagonalizable.