Heisenberg Uncertainty Relation in Quantum Liouville Equation - Davide Valenti (International Journal of Mathematics and Mathematical Sciences)
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.