Heisenberg uncertainty relation in quantum Liouville equation
Summary: 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 \(f({\mathbf x},{\mathbf v},t)\) of a generic solution \(\psi({\mathbf x};t)\) of the Schrödinger equation. We give a representation of \(\psi({\mathbf x},t)\) by the Hermite functions. We show that the values of the variances of \textbf{x} and \textbf{v} calculated by using the Wigner function \(f({\mathbf x},{\mathbf v},t)\) coincide, respectively, with the variances of position operator \(\widehat{X}\) and conjugate momentum operator \(\widehat{P}\) obtained using the wave function \(\psi({\mathbf x},t)\). Then we consider the Fourier transform of the density matrix \(\rho({\mathbf z},{\mathbf y},t)= \psi^*({\mathbf z},t)\psi({\mathbf y},t)\). We find again that the variances of \textbf{x} and \textbf{v} obtained by using \(\rho({\mathbf z},{\mathbf y},t)\) are respectively equal to the variances of \(\widehat{X}\) and \(\widehat{P}\) calculated in \(\psi({\mathbf x},t)\). Finally we introduce the matrix \(\|A_{nn'}(t)\|\) and we show that a generic square-integrable function \(g({\mathbf x},{\mathbf v},t)\) can be written as Fourier transform of a density matrix, provided that the matrix \(\|A_{nn'}(t)\|\) is diagonalizable.
scientific article; zbMATH DE number 5695206
scientific article